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%                   NEEDLE VARIATIONS AND ALMOST 
%            LOWER SEMICONTINUOUS DIFFERENTIAL INCLUSIONS
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%                       by Hector J. Sussmann 
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semicontinuous differential inclusions}}
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\footnotetext[1]{Most of this research was carried out while the
author was Weston Visiting Professor at the Weizmann Institute of
Science in Rehovot, Israel, during the Fall 2000 term. The author is
grateful to the Weizmann Institute for its generous hospitality and
exciting intellectual atmosphere.}  

\footnotetext[2]{Supported in part by NSF Grant DMS-9803411.}

\footnotetext[3]{\mbox{Department of Mathematics, Rutgers University,
Piscataway, NJ 08854, U.S.A.;} \mbox{{\it \mbox{E-mail}}:
sussmann@math.rutgers.edu; $\!${\it Web$\,$site}:
$\!$http:/$\!$/www.math.rutgers.edu/$\,\widetilde{\,\,\,}$sussmann}}

\begin{center}
{\Large \bf 
Needle Variations and Almost\\

\vspace{0.2cm}
Lower Semicontinuous Differential Inclusions\footnotemark[1]\\

\vspace{1cm}
{\it H\'ector J. Sussmann}}\footnotemark[2]$^,$\footnotemark[3]
\end{center}

\vspace{1cm}




\begin{minipage}{12cm}
\baselineskip=9pt
{\tiny\tableofcontents}
\end{minipage}

\baselineskip=13pt
\section{Introduction and overview}

The classical version of the ``Pontryagin Maximum Principle,''
presented in the book \cite{pontrybook}, relies on the construction of
``needle variations'' at \mbox{various} times $\tau$
in the domain $[a,b]$ of
the given reference trajectory $\xi_*$. The effects of these
variations are then \mbox{propagated} to the terminal time $b$
by means of the \mbox{differentials} $D\Phi_{b,\tau}(\xi_*(\tau))$
of the reference flow maps $\Phi_{b,\tau}$. This
produces a convex cone (the ``Pontryagin cone'') which in some sense
is a first-order approximation to the attainable set near the
point $\xi_*(b)$. The proof is
then concluded by means of a \mbox{topological} \mbox{argument}, based
on some variant of the Brower fixed point theorem, to
pass from the \mbox{separation} of two sets to the separation of their
approximating cones.

%\def\footloja{Lojasiewicz's ideas were kindly explained to us in an 
%unforgettable conversation that took place in Rochester, NY, in July 1993. 
%To our knowledge,  this path-breaking work regrettably
%remains unpublished to this day.}

The purpose of this note is to explain how the construction of 
\mbox{needle} \mbox{variations} can be carried out for {\em differential 
inclusions} rather than  for the case of vector fields of class
$C^1$ considered in \cite{pontrybook}.  This is part of a 
\mbox{broader} research program whose ultimate goal is to unify and generalize
the various existing versions of the finite-dimensional maximum principle
by \mbox{developing} a general ``primal'' technique based on mimicking the 
proof of \cite{pontrybook}, but \mbox{modifying} it by \mbox{using}
{\em flows} instead of vector fields, {\em generalized 
\mbox{differentials}}
instead of \mbox{ordinary} ones, $\,$and {\em $\,$abstract $\,$variations} 
$\,$instead $\,$of $\,$classical $\,$needle $\,$variations $\,$(cf. 
\mbox{Sussmann}  \cite{sussloja,sussinclu,sussmul,sussbro,susssanto,susshybrid,sussfrance,sussnewtheor}). 

A needle variation involves a {\em time-varying vector field}
(abbr. TVVF) \mbox{$\rr^n\times\rr\ni(x,t)\mapsto f(x,t)\in\rr^n$}.
The ``effect'' of the variation at a point $(\bar x,\bar t)$ is a
vector related to the differential $D\Phi^f(\bar q)$ of the flow map
$\Phi^f$ (cf. Definition \ref{defoftraj} in \S\ref{sectrajflow} for
the meaning of $\Phi^f$) at the point $\bar q=(\bar x,\bar
t,\bar t)$. More precisely, it is easy to
see that, if $f$ is regular enough (for example, of class $C^1$
as a function of $x$ and $t$) then $\Phi^f$ is differentiable at every
point $\bar q=(\bar x,\bar t,\bar{s})$ of its domain. If $\bar q$ is
of the special form $(\bar x,\bar t,\bar t)$, then the differential
$D\Phi^f(\bar q)$ is the linear map
$$
\rr^n\times\rr\times\rr\ni(v,h,k)\mapsto\Lambda^w(v,h,k)=
v+(h-k)w\in\rr^n\,,
$$ 
where $w=f(\bar x,\bar t)$, because 
$\Phi^f(\bar x+v,\bar t+h,\bar t+k)\sim \bar x+v+(h-k)f(\bar x,\bar t)$
and $\bar x=\Phi^f(\bar x,\bar t,\bar t)$, so that 
$\Phi^f(\bar x+v,\bar t+h,\bar t+k)-\Phi^f(\bar x,\bar t,\bar t)
\sim v+(h-k)f(\bar x,\bar t)$.  The vector $w$ 
is then the ``effect of the variation at $(\bar x,\bar t)$.''
(A detailed explanation of the notion of a needle variation,
and why it is important to differentiate the flow at points of the form
$(\bar x,\bar t,\bar t)$, is given in \S\ref{sec-expla} below.)

For the optimal control of a system $\dot x=f(x,u,t)$, $x\in\rr^n$,
$u\in U$, the TVVFs that are used to construct needle
variations are of the special form 
$\rr^n\times\rr\ni(x,t)\mapsto f(x,\eta(t),t)\equaldef f_\eta(x,t)$, 
where $t\mapsto\eta(t)$ is some open-loop control, which is usually
either the reference control or a constant control.  For such vector
fields, the argument given above to establish that $D\Phi^{f_\eta}(\bar
x,\bar t,\bar t)=\Lambda^{f_\eta(\bar x,\bar t)}$ is not
rigorous, even if the map $(x,u,t)\mapsto f(x,u,t)$ is very smooth,
because $\eta$ could be the reference control, which will typically
not be continuous.  It is possible, however, to render the argument
rigorous for almost all $\bar t$, by invoking the Scorza-Dragoni
theorem and using a notion of approximate continuity analogous to the
concept of a Lebesgue point.

The question discussed here is how to carry out the \mbox{construction} of
needle variations---and obtain flows that are differentiable at $(\bar
x,\bar t,\bar t)$ and have a \mbox{prescribed} effect $w$ for various
points $(\bar x,\bar t)$---when, instead of using \mbox{solutions} of a
differential equation $\dot x=f(x,u,t)$, we want to work with
\mbox{solutions} of {\em \mbox{differential} inclusions} $\dot x\in
F(x,t)$.  A natural idea would be to use \mbox{single-valued}
\mbox{selections} $f$ of the set-valued map $F$, and then study the
\mbox{differentiability} of the flows $\Phi^f$. Ideally, one might hope
that for every $(\bar x,\bar t)$ and every $w\in F(\bar x,\bar t)$ there
exists a good enough selection $f=f_{\bar x,\bar t,w}$ such that $f(\bar
x,\bar t)=w$.
However, this line of attack leads to dead end, because

\begin{itemize}
\uitem set-valued maps $(x,t)\mapsto F(x,t)$ with nonconvex values 
typically do not admit continuous selections, even if they are very 
regular (for example continuous, or Lipschitz), 
\end{itemize}

\vspace{-2mm}
\noindent while, on the other hand, 
\begin{itemize}
\oitem the application of the
Scorza-Dragoni theorem requires that one deal with vector fields
$(x,t)\mapsto f(x,t)$ that are continuous with respect to $x$ and
measurable with respect to $t$.  
\end{itemize}

It turns out, remarkably, that this difficulty can be overcome by
adopting a more sophisticated perspective and no longer requiring the
selections to be continuous with respect to $x$. The first evidence
that this approach might be feasible appeared in papers by Cambini and
\mbox{Querci} \cite{cambini}, and Pucci
\cite{pucci}, who studied some discontinuous ``\mbox{directionally}
\mbox{continuous}'' \mbox{vector} fields that are as good as continuous ones
as far as \mbox{existence} of \mbox{solutions} goes, and in addition have
other desirable related \mbox{properties} such as upper 
\mbox{semicontinuity} of the flow. Subsequently, A.~Bressan proved
that lower \mbox{semicontinuous} set-valued maps with nonempty closed values
admit selections in this class \mbox{(cf. Bressan
\cite{bressanondif,bressansolu,bressandir,bressanquali,bressan})}.

In this paper, we pursue these ideas further, by exhibiting a natural
class of discontinuous time-varying vector fields $(x,t)\mapsto
f(x,t)$---that we call ``admissible vector fields''---larger than that
studied by \mbox{Bressan}, that has all the properties needed to make
needle variations.  The \mbox{defining} property of these vector
fields is that the map that assigns to each curve \mbox{$[a,b]\ni
t\mapsto\xi(t)$} the indefinite integral \mbox{$[a,b]\ni
t\mapsto\int_a^tf(\xi(s),s)\,ds$} is \mbox{continuous} on a suitably
large \mbox{space} ${\cal A}$ of curves.  This condition is
\mbox{exactly} what is required to make it possible to apply the
Schauder fixed point theorem to establish existence of trajectories on
small time intervals.  (We prove existence in Theorem
\ref{theorexist}.)  These vector fields have other useful
properties, that we study in detail.

We prove that Bressan's directional continuity 
\mbox{implies} \mbox{admissibility}, but \mbox{admissibility} is 
a much more general
property. In \mbox{particular}, \mbox{admissible} vector fields have the
``measurable intertwining \mbox{property}'' (cf. \mbox{Theorem \ref{meassub}}
below). This makes it possible to construct plenty of \mbox{admissible} 
\mbox{selections} for set-valued maps $F$ that are
``\mbox{almost} lower semicontinuous'' (abbr. ALSC) by first
\mbox{constructing} \mbox{conically} \mbox{continuous} selections of lower
semicontinuous maps, and then intertwining them.

In order to be able to differentiate the flow $\Phi^f$ of an admissible
vector field $f$ at a point $(\bar{x},\bar{t},\bar{t})$, one needs 
$f$ to be ``approximately continuous'' at 
$(\bar{x},\bar{t})$. We prove that for ALSC maps there always exist many 
admissible selections with this extra property 

Finally, it is well-known that
differentiability of the flow in the most obvious sense (that is,
existence of a linear map that approximates the map to first order)
is not enough to make the topological arguments applicable.
For example, the function $f:\rr\mapsto\rr$ given by
$f(x)=\frac{1}{n}$ if $n\in\nn$, $n>0$, and 
$\frac{1}{n}\le x<\frac{1}{n-1}$, and $f(x)=x$ if $x\le 0$,
is differentiable at $0$, and its derivative at $0$ is equal to $1$.
However, it is not true that $f$ maps neighborhoods of $0$ to neighborhoods
of $0$. This illustrates the fact that in the open mapping 
theorem---that is, the statement that if 
\mbox{$f:\rr^n\mapsto\rr^m$} maps
$0$ to $0$ and $Df(0)$ exists and is surjective then $f$ maps neighborhoods
of $0$ to neighborhoods of $0$---it is essential to assume
that $f$ is \mbox{continuous} near $0$.  \mbox{Since}
this is already true for single-valued 
maps, something \mbox{similar} must be true for set-valued maps. That is, 
the set-valued analogue of the open \mbox{mapping} theorem, if it exists, 
cannot just involve the existence and \mbox{surjectivity} of the differential
at $0$, and must contain a supplementary 
\mbox{condition}, playing a similar role 
in the set-valued case as continuity does in the single-valued
case.  It turns out that the set-valued version of the open mapping theorem
does exist, and the supplementary condition that plays the role 
of continuity is ``regularity.''  (For a detailed account of this,
cf., for example, Sussmann \cite{sussmul}.)

Hence the differentiability property of the flow
needed for the \mbox{maximum} principle for differential
inclusions is {\em regular \mbox{differentiability}}, that is, the fact
that the flow is both differentiable at $(\bar{x},\bar{t},\bar{t})$ and regular
in a neighborhood of $(\bar{x},\bar{t},\bar{t})$. Our main result (\mbox{Theorem}
\ref{selectionalscprod-main} in \S \ref{sec-main}) says precisely
that {\em if a differential inclusion $\dot x\in F(x,t)$ in
$\rr^n$ has a right-hand side $F$ with nonempty closed values, then
for \mbox{almost}
\mbox{every} $\bar{t}$ it is possible to make a needle variation using
$F$ at any point $\bar{x}$ \mbox{generating}
the direction of any vector $\bar{y}\in
F(\bar{x},\bar{t})$, provided that $F$ is \mbox{almost} lower \mbox{semicontinuous} and satisfies in addition
a technical ``locally integrable \mbox{lower} \mbox{boundedness}''  
condition}.

The use of the results of this paper to prove a maximum
principle for differential inclusions is outlined in Sussmann
\cite{sussinclu}. The work carried out here is essentially that of
filling in all the details for the half of the results of
\cite{sussinclu} that has to do with the needle
variations.  The other part---not considered here---is the one dealing
with the differentiability of the reference flow along the reference
trajectory.  This requires a different set of tools, and will be the
subject of another paper.


\subssection{A brief outline of the logical structure of the paper}

\begin{itemize}
\item As indicated above, our main result is 
Theorem \ref{selectionalscprod-main} in \S \ref{sec-main},
in  which the set-valued
map $F$ is required to be ``almost lower semicontinuous'' and
``locally integrably lower bounded.''  

\uitem ``Almost lower semicontinuity'' is, essentially, the property that
would follow from the condition that the set-valued map $x\mapsto
F(x,t)$ is \mbox{lower} semicontinuous for each $t$, and the set-valued map
$t\mapsto F(x,t)$ is \mbox{measurable} for each $x$, if the Scorza-Dragoni
theorem was true for \mbox{lower} semicontinuous set-valued maps. Since 
the Scorza-Dragoni \mbox{theorem} is {\em not} true in  this situation,
its conclusion is turned into the definition of a new concept.
This is done in Definition \ref{defofalsc} in \S\ref{subsec-def-alsc}.

\uitem ``Local integrable lower boundedness'' means that, locally,
there exists an integrable function $t\mapsto\ffi(t)$ 
such that $\{y\in F(x,t):\|y\|\le\ffi(t)\}$ is nonempty, as explained
in Definition \ref{defoflilb} in \S\ref{sec-main}.

\uitem The conclusion of Theorem \ref{selectionalscprod-main} is
that for almost every $\bar{t}$ there exists, for each 
$\bar{x}$ and each $\bar{y}\in F(\bar{x},\bar{t})$, a single-valued selection
$f$ of $F$ which is such that the flow map
$(x,t,s)\mapsto\Phi^f(x,t,s)$ of the vector field
$f$ is ``regularly differentiable'' at $(\bar{x},\bar{t},\bar{t})$ 
and its differential is the linear map
$(v,h,k)\mapsto v+(h-k)\bar y\equaldef\Lambda^{\bar{y}}(v,h,k)$.

\uitem Flow maps are introduced in 
Definition \ref{defoftraj}  in \S\ref{sectrajflow}.
\uitem The meaning of ``regular differentiability'' is explained in 
Definition \ref{defregdif} in \S\ref{secregdif}. 
Basically, a set-valued map is regularly 
differentiable at a point $q$ if it is ``regular''  
$q$ and differentiable at $q$ in the usual sense of admitting 
a first-order linear approximation.  

\uitem The notion of a ``regular map'' is reviewed
in Definition \ref{defregu} in 
\S\ref{secregumaps}. As explained above, it is a natural
set-valued analogue of continuity of a single-valued map. 
(This is also discussed sketchily in \S\ref{secregumaps}---cf. 
\mbox{especially} Remark \ref{whygood} in \S\ref{secregumaps}---and in much
greater detail in, for \mbox{example}, Sussmann \cite{sussmul,
sussfrance,sussnewtheor}.)
A regular map is a set-valued map that can be approximated
in a particular way by ordinary continuous maps. 
The precise sense of the approximation is that of
``inward graph convergence,'' introduced in Definition
\ref{def-inward-graph} in \S\ref{secregdif}.  

\uitem The differentiability conclusion of 
Theorem \ref{selectionalscprod-main} is derived from the other conclusions
in conjunction with Theorem \ref{theosemidifreg} of \S\ref{secregdifflow}.

\uitem The other conclusions of Theorem \ref{selectionalscprod-main} assert 
the existence of an  ``admissible selection'' $f$ that has
$(\bar{x},\bar{t})$ as a ``point of approximate continuity'' 
(abbr. PAC) and is such that  $f(\bar{x},\bar{t})=\bar{y}$. 

\uitem Theorem \ref{theosemidifreg} says that, if a vector field 
$(x,t)\!\mapsto\!f(x,t)$ is \mbox{admissible} and
$(\bar x,\bar t)$ is a PAC of $f$, then the flow
map $(x,t,s)\!\mapsto\!\Phi^f(x,t,s)$ is regularly differentiable
at $(\bar x,\bar t,\bar t)$ and its differential is $\Lambda^{\bar{y}}$.

\uitem Theorem \ref{theosemidifreg} arises from  combining 
Theorem \ref{theorexist} of \S\ref{sec-adm}
with Theorem \ref{theodifreg} of \S\ref{sec-difflow}
and Theorem \ref{regpovar} of  \S\ref{sec-regul}.
Theorem \ref{theorexist} gives the local existence of solutions
and upper semicontinuity of the flow, 
Theorem \ref{theodifreg} gives the differentiability 
and Theorem \ref{regpovar} gives the regularity.

\uitem The concept of a ``point of approximate continuity'' is presented in 
\mbox{Definition} \ \ref{defapproxcont-true} \ in \S\ref{secpointapp}. 
It is, roughly, \ a generalization to maps 
\mbox{$(x,t)\mapsto f(x,t)$} of the usual 
notion of a Lebesgue point of a \mbox{function} $t\mapsto f(t)$.
For a Lebesgue point $\bar t$ of a function $t\mapsto f(t)$, the
bound $\int_{\bar t-h}^{\bar t+h}\|f(t)-f(\bar t)\|\,dt=
o(h)$ implies that the integral 
\mbox{$t\mapsto\int_a^tf(s)\,ds$}
is differentiable at $\bar t$ with derivative $f(\bar t)$. Theorem 
\ref{theodifreg} is proved by using a similar argument 
to establish differentiability of the flow at a point of approximate 
continuity and compute the differential of the flow.


\uitem  ``Admissibility'' is introduced in Definition \ref{def-adm} in 
\S\ref{sec-adm}. As explained above, it
is exactly what is required to be able to apply a fixed point 
argument to establish existence of trajectories on  small time intervals.
The existence result is Theorem \ref{theorexist} 
(cf. also Remark \ref{rem-exist} in \S\ref{sec-adm}).

\uitem Remarkably, the admissibility condition has several other important
\mbox{consequences} besides the existence of trajectories. One of them
is the result of Theorem \ref{regpovar}, which says that 
if $(x,t)\mapsto f(x,t)$ is an admissible vector field, then the 
flow map $(x,t,s)\mapsto\Phi^f(x,t,s)$ is regular on a sufficiently small
neighborhood of any point $(\bar x,\bar t,\bar t)$. This is proved by
showing that the 
regularized vector fields $f_\rho$ (obtained in the usual way, by convolution
with smooth functions $x\mapsto\ffi_\rho(x)$ that converge to a 
Delta function as $\rho\downarrow 0$) give rise to flows $\Phi^{f_\rho}$
(which are continuous single-valued maps) 
that converge to the flow $\Phi^f$ (which in general is set-valued, because
the equation $\dot x=f(x,t)$ need not have uniqueness of trajectories)
in the sense of inward graph convergence. 

\uitem Theorem \ref{selectionalscprod-main}, on the 
existence of an admissible selection that has a given point 
$(\bar{x},\bar{t})$ as a PAC and takes a prescribed
value $\bar{y}$ at $(\bar{x},\bar{t})$, is derived from 
Theorem \ref{selectionalscprod} of \S\ref{sec-selec-alsc}, which gives
the existence of selections that are ``integrally continuous''
on suitably large sets and satisfy the prescribed conditions
involving $\bar{x}$, $\bar{t}$, and $\bar{y}$.

\uitem Theorem \ref{selectionalscprod} is derived from 
Theorem \ref{selectionalsc} in \S\ref{sec-selec-alsc}, together with 
the ``\mbox{measurable} intertwining ''
Theorem \ref{meassub} of \S\ref{sectquasi} and 
\mbox{Theorem} \ref{theoweakly} of \S\ref{conicimplies}.

\uitem Theorem \ref{selectionalsc} gives the existence,
for a sequence $\{J_k\}_{k=1}^\infty$
of pairwise \mbox{disjoint} compact subsets of $\rr$ such that
$\msr\Big(\rr\backslash(\cup_{k=1}^\infty J_k)\Big)=0$, 
of a selection $f$ that has, when $t$ varies on each of the $J_k$, a 
``\mbox{conic} \mbox{continuity}'' property.  $\!$Using this, the argument
leading from \mbox{Theorem $\!$\ref{selectionalsc}} to \mbox{Theorem} 
\ref{selectionalscprod} proceeds
as follows: conic \mbox{continuity} implies integral \mbox{continuity} 
by  \mbox{Theorem} \ref{theoweakly}, and integral continuity 
on the individual sets $J_k$ implies full integral continuity
by Theorem \ref{meassub}.

\uitem Theorem \ref{selectionalsc} is derived from Theorem
\ref{selectionlsc} of \S\ref{sec-conic-exist}, on the existence of
\mbox{conically} continuous single-valued selections for lower
\mbox{semicontinuous} set-valued \mbox{maps}. This result is essentially due
to A.~Bressan. The main technical point where we depart
{}from Bressan's work is that we prove existence of conically
continuous selections that in addition are {\em continuous}
(rather than just directionally continuous) at a given point
$(\bar{x},\bar{t})$, and have a prescribed value there.
This technical improvement is the crucial step enabling us to 
get selections having a given point as a PAC,
and prove differentiability of the flow.
\end{itemize}

\subssection{Needle variations}\label{sec-expla}

One constructs {\em variations} for a triple $(f,\eta_*,\xi_*)$
consisting of a control system 
$\dot x=f(x,u,t)$, $x\in\rr^n$, $u\in U$, 
a ``reference control'' $[a,b]\ni t\mapsto\eta_*(t)\in U$,
and a corresponding ``reference trajectory'' 
$[a,b]\!\ni\! t\!\mapsto\!\xi_*(t)\!\in\!\rr^n$.  This is done as follows.
Starting with the flow 
\mbox{$\Phi^{*}\!=\!\{\Phi_{t,s}^*\}_{a\le s\le t\le b}$}
of the reference TVVF $(x,t)\mapsto f(x,\eta_*(t),t)$, one constructs 
a one-parameter family $\{\Phi^\eps\}_{0\le\eps\le\bar\eps}$
of flows \mbox{$\Phi^\eps=\{\Phi_{t,s}^\eps\}_{a\le s\le t\le b}$},
defined for small nonnegative $\eps$, such that 
$\Phi^0=\Phi^*$. (That is, one ``draws a curve $\eps\mapsto\Phi^\eps$
in the space of flows, starting at $\Phi^*$ for $\eps=0 $.'')
{\em Needle variations} are obtained by doing this in a particular way:
$\Phi^\eps$ is the flow of the control $\eta_\eps$ 
obtained from $\eta_*$ by replacing the value $\eta_*(t)$ by 
a constant control $\bar u$ on the interval $[\tau,\tau+\eps]$,
for some time $\tau\in[a,b[$.
In other words, we replace the reference flow by the flow
$\Psi=\{\Psi_{t,s}\}_{a\le s\le t\le b}$ of the time-varying vector
field $(x,t)\mapsto f(x,\bar u,t)$ 
on the interval $[\tau,\tau+\eps]$. Equivalently,
we define $\Phi_{t,s}^\eps$ by
$$
\Phi_{t,s}^\eps=\left\{\begin{array}{lcl}
\Phi_{t,s}^*&{\rm if}&a\le s\le t\le\tau\,\,{\rm or}\,\,
\tau+\eps\le s\le t\le b\,,\\
\Psi_{t,s}&{\rm if}&\tau\le s\le t\le\tau+\eps\,,\\
\Psi_{t,\tau}\compo\Phi_{\tau,s}^*&{\rm if}&a\le s\le\tau
\le t\le\tau+\eps\,,\\
\Phi_{t,\tau+\eps}^*\compo\Psi_{\tau+\eps,s}
&{\rm if}&\tau\le s\le\tau+\eps\le t\le b\,,\\
\Phi_{t,\tau+\eps}^*\compo\Psi_{\tau+\eps,\tau}
\compo\Phi_{\tau,t}^*
&{\rm if}&a\le s\le\tau\,\,{\rm and}\,\,\tau+\eps\le t\le b\,.
\end{array}\right.
$$

As explained before, the differential at a point $(\bar x,\bar
t)\in\rr^n\times\rr$ of the flow $\Phi^g$ of a vector field
$\rr^n\times\rr\ni(x,t)\mapsto g(x,t)\in\rr^n$ is the linear map
$\rr^n\times\rr\times\rr\ni(v,h,k)\mapsto v+(h-k)g(\bar x,\bar t)$, if $g$
is sufficiently regular. Applying this with $g(x,t)=f(x,\bar u,t)$, and
with $g(x,t)=f(x,\eta_*(t),t)$,  we find that
$\Psi_{\tau+\eps,\tau}(\bar x)=
\bar x+\eps f(\xi_*(\tau),\bar u,\tau)+o(\eps+\|\bar x-\xi_*(
\tau)\|)$ and also that
$\Phi_{\tau+\eps,\tau}^*(\bar x)=\bar x+\eps
f(\xi_*(\tau),\eta_*(\tau),\tau)
+o(\eps+\|\bar x-\xi_*(\tau)\|)$, so that
\begin{eqnarray*}
\Phi_{b,\tau}^\eps(\bar x)-\Phi_{b,\tau}^*(\bar x)
&=&\Phi_{b,\tau+\eps}^*\compo\Psi_{\tau+\eps,\tau}(\bar x)
-\Phi_{b,\tau+\eps}^*\compo\Phi_{\tau+\eps,\tau}^*(\bar x)\\
&\sim& D\Phi_{b,\tau+\eps}^*(\bar x)\cdot
\Big(\Psi_{\tau+\eps,\tau}(\bar x)
-\Phi_{\tau+\eps,\tau}^*(\bar x)\Big)\\
&\sim& D\Phi_{b,\tau}^*(\bar x)\cdot
\Big(\Psi_{\tau+\eps,\tau}(\bar x)
-\Phi_{\tau+\eps,\tau}^*(\bar x)\Big)\\
&\sim&\eps D\Phi_{b,\tau}^*(\bar x)\cdot w\,,
\end{eqnarray*}
where $w=f(\xi_*(\tau),\bar u,\tau)-f(\xi_*(\tau),\eta_*(\tau),\tau)$.
This gives the expansion
\begin{equation}\label{equ1}
\Phi_{b,\tau}^\eps(\bar x)\sim\Phi_{b,\tau}^*(\bar x)+
\eps D\Phi_{b,\tau}^*(\bar x)\cdot w\,,
\end{equation}
showing that the difference between ``following the modified flow starting
at $\bar x$ from time $\tau$ to time $b$ minus doing the same thing for the
unmodified flow'' is, to first order, equal to $\eps$ times the vector
$w$ propagated forward by the differential $D\Phi_{b,\tau}^*(\bar x)$.

In the proof of the maximum principle, $\bar x$ might be
$\xi_*(\tau)$, if we are applying our variation
directly to the reference trajectory. But $\bar x$ might also
be some other point close to $\xi_*(\tau)$ if, for example,
another variation---at an earlier time $\tau'<\tau$,
corresponding to a constant control $\bar u'$---has already been applied,
and we want to study the combined effect of both variations.

In that case, we really have to deal with a two-parameter variation,
and the terminal point map that we want to analyze is the
map
$$
x\mapsto\Phi_{b,\tau'}^{\eps_2,\eps_1}(x)=
\Phi_{b,\tau+\eps_2}^*\compo\Psi_{\tau+\eps_2,\tau}^{\bar u}
\compo\Phi_{\tau,\tau'+\eps_1}^*\compo\Psi_{\tau'+\eps_1,\tau'}^{\bar u'}(x)\,,
$$
where, naturally, $\Psi^{\bar u}$ and $\Psi^{\bar u'}$ are the flows of the
TVVFs $(x,t)\mapsto f(x,\bar u,t)$ and $(x,t)\mapsto f(x,\bar u',t)$,
respectively.

To get a linear approximation for 
$\Phi_{b,\tau'}^{\eps_2,\eps_1}(x)$, we use
(\ref{equ1}) taking $\eps=\eps_2$ and
$\bar x=\Phi_{\tau,\tau'}^{\eps_2,\eps_1}(x)
=\Phi_{\tau,\tau'+\eps_1}^*\compo\Psi_{\tau'+\eps_1,\tau'}
^{\bar u'}(x)$. This gives
\begin{equation}\label{equ2}
\Phi_{b,\tau'}^{\eps_2,\eps_1}(x)=
\Phi_{b,\tau}^{\eps_2}(\bar x)\sim\Phi_{b,\tau}^*(\bar x)+
\eps_2 D\Phi_{b,\tau}^*(\bar x)\cdot w\,,
\end{equation}
while on the other hand, using (\ref{equ1}) again, we find
$$
\bar x\sim \Phi_{\tau,\tau'}^*(x)+\eps_1
D\Phi_{\tau,\tau'}^*(x)(w')\,,
$$
where  $w'=f(\xi_*(\tau'),\bar u',\tau')-f(\xi_*(\tau'),\eta_*(\tau'),\tau')$.
Therefore 
\begin{eqnarray*}
\Phi_{b,\tau}^*(\bar x)&\sim& \Phi_{b,\tau}^*\Big(\Phi_{\tau,\tau'}^*(x)+\eps_1
D\Phi_{\tau,\tau'}^*(x)(w')\Big)\\
&\sim&\Phi_{b,\tau}^*\Big(\Phi_{\tau,\tau'}^*(x)\Big)+
\eps_1 D\Phi_{b,\tau}^*\Big(\Phi_{\tau,\tau'}^*(x)\Big)
\Big(D\Phi_{\tau,\tau'}^*(x)(w')\Big)\\
&\sim&\Phi_{b,\tau'}^*(x)+\eps_1 D\Phi_{b,\tau'}^*(x)(w')\,,
\end{eqnarray*}
using the flow identity $\Phi_{b,\tau}^*\compo=\Phi_{\tau,\tau'}^*
\Phi_{b,\tau'}^*$ as well as the chain rule 
$$
D\Phi_{b,\tau}^*\Big(\Phi_{\tau,\tau'}^*(x)\Big)\compo D\Phi_{\tau,\tau'}^*(x)
=D\Big(\Phi_{b,\tau}^*\compo\Phi_{\tau,\tau'}^*\Big)(x)
=D\Phi_{b,\tau'}^*(x)\,.
$$
If we substitute our approximate expression for 
$\Phi_{b,\tau}^*(\bar x)$ into (\ref{equ2}),  \mbox{observe} that 
$D\Phi_{b,\tau}^*(\bar x)\sim D\Phi_{b,\tau}^*(\xi_*(\tau))$ (because
$\bar x$ is close to $\xi_*(\tau)$), write 
\mbox{$\tau_2=\tau$}, $\tau_1=\tau'$, $\,x_2=\xi_*(\tau_2)$, 
$\,x_1=\xi_*(\tau_1)$, $\,u_2=\bar u$, $\,u_1=\bar u'$,
and then define \mbox{$w_i=f(x_i,u_i,\tau_i)-f(x_i,\eta_*(\tau_i),\tau_i)$}
for $i=1,2$, we find that 
\begin{equation}\label{equ3}
\Phi_{b,\tau_1}^{\eps_2,\eps_1}(x_1)
\sim\Phi_{b,\tau}^*(x_1)+
\eps_1 D\Phi_{b,\tau_1}^*(x_1)\cdot w_2+
\eps_2 D\Phi_{b,\tau_2}^*(x_2)\cdot w_2\,.
\end{equation}

It is clear that (\ref{equ3}) can be generalized to a formula
expressing the combined effect of several variations. For our purposes in
this paper, the crucial observation is that {\em for the flows 
$\Psi^{u_i}$ that are used to define the  \mbox{variations}, 
one needs approximate expressions not only for quantities such as
$\Psi_{\tau_i+\eps_i,\tau_i}^{u_i}(\xi_*(\tau_i))$, but also 
for $\Psi_{\tau_i+\eps_i,\tau_i}^{u_i}(x)$ where $x$ is close
to $\xi_*(\tau_i)$.}  The \mbox{reason} for this, as has been 
explained above, is that 
the point $x$ to which the \mbox{approximation} will be applied will not
in general lie on the reference \mbox{trajectory} $\xi_*$ itself,
due to the fact that $x$ will already contain the \mbox{cumulative}
effect of previously applied variations. In other words, 
{\em it is not enough to be able to differentiate flows
$(x,t,s)\mapsto\Phi_{t,s}(x)$ with respect to the time \mbox{variables}
$t$ and $s$, for a fixed $x$. One has to be able to differentiate 
the flow at a point with respect to $x$, $t$, and $s$.}

\begin{remark}
The preceding explanation may suggest that one only
needs to differentiate flow maps
$(x,t,s)\mapsto\Phi_{t,s}(x)$ with respect to $(x,t)$ for fixed
$s$, since in our discussion of needle variations $s$ (called 
$\tau$ or $\tau'$) was fixed. \mbox{However}, $\,$there $\,$are $\,$several
$\,$reasons $\,$for $\,$wanting $\,$to $\,$consider $\,$the $\,$joint 
\mbox{differentiability} 
of the flow map with respect to $x$, $t$, and $s$. The \mbox{simplest} one
is the fact that one may want to combine several
``needle variations \mbox{applied} at the same time $\tau$.'' For this purpose,
it turns out to be \mbox{convenient} to perturb the reference flow by inserting
different constant \mbox{controls} on \mbox{non-overlapping}
intervals close to a point $\tau$. For example, one may want to consider a
two-parameter variation such that 
$\Phi_{b,\tau}^{\eps_1,\eps_2}=
\Phi_{b,\tau+\eps_1+\eps_2}\compo
\Psi_{\tau+\eps_1+\eps_2,\tau+\eps_1}^{u_2}\compo
\Psi_{\tau+\eps_1,\tau}^{u_1}$. It is easy to see that, to compute the
effect of such a variation to first \mbox{order}, 
the flow map $(x,t,s)\mapsto\Psi_{t,s}^{u_2}(x)$ 
has to be differentiated jointly
with respect to $x$, $t$, and $s$, at the point 
$(\xi_*(\tau),\tau,\tau)$.\proofend
\end{remark}

\section{Notations and preliminary definitions}

\subssection{Numbers, intervals, metric spaces, neighborhoods}
\label{secnotations}
\noindent We use $\nn$ to denote the set $\{0,1,2,\,\ldots\,\}$ of all
nonnegative integers, and write $\overline{\nn}$ to denote the set
$\nn\cup\{+\infty\}$.  We use $\rr$ to denote the set of real
\mbox{numbers}. A {\em real interval\/} is an arbitrary connected subset of
$\rr$.  We use the notations $[a,b]$, $]a,b[$, $[a,b[$, $]a,b]$ for
the closed, open, left-closed right-open and right-closed left-open
intervals with endpoints $a,b$.  The \mbox{expression} $(a,b)$ will always
denote the ordered pair whose components are $a$ and $b$.  An \mbox{interval}
is {\em nontrivial} if it contains more than one point.
We write \mbox{$\rr_+=[0,+\infty[$}, $\rrinf=\rr\cup\{-\infty,+\infty\}$,
$\rrinf_+=\rr_+\cup\{+\infty\}=[0,+\infty]$.

If $X$ is a metric space, we use \idx{$d_X$}
$d_X$ to denote the distance function on $X$, and write 
$\bmath_X(x,r)$, $\overline{\bmath}_X(x,r)$,
to denote, respectively, the open and
closed balls with center $x$ and radius $r$.  We use
\idx{${\rm Comp}(X)$} ${\rm Comp}(X)$ to denote the set of all nonempty
compact subsets of $X$.  If $X=\rr^n$, we write $\bmath^n(x,r)$, 
$\overline{\bmath}^n(x,r)$, rather than 
$\bmath_{\rr^n}(x,r)$, $\overline{\bmath}_{\rr^n}(x,r)$.

The word \idx{neighborhood}``neighborhood'' will always be used in the
standard sense of point-set topology: a {\sl neighborhood} of a point $a$
(resp. of a set $S$) in a topological space $A$ is any set $U\subseteq A$
such that $a$ is an interior point of $U$ (resp. 
such that $S\subseteq {\rm Int}(U)$).

If $X$ and $Y$ are metric spaces, the product
$X\times Y$ is equipped with the {\em product metric} $d_{X\times Y}$, 
given by
$$
d_{X\times Y}\Big((x,y),(x',y')\Big)=d_X(x,x')+d_Y(y,y')\,\,\,\,\,{\rm if}
\,\,\,\,\,\,x,x'\in X\,,\,\, y,y'\in Y\,.
$$

\subssection{Set-valued maps}
\label{sectiononsvms}

A {\em set-valued map} is a triple $F=(X,Y,G)$ such that $X$ and $Y$ are
sets  and $G$ is a subset of $X\times Y$.  If $F=(X,Y,G)$ is a set-valued
map, then we refer to $X$, $Y$ and $G$ as the {\em source}, {\em target}
and {\em graph} of $F$, and write $X=\So(F)$, $Y=\Ta(F)$,
$G=\Gr(F)$.   We write $F:X\tto Y$
to indicate that $F$ is a set-valued map from $X$ to $Y$, and we use 
$SVM(X,Y)$ to denote the set of all such set-valued maps, so 
``$F\in SVM(X,Y)$'' is equivalent to ``$F:X\tto Y$.''

If $F=(X,Y,G)$ is a set-valued map, 
and $x$ is any object, we write $F(x)\equaldef\{y:(x,y)\in G\}$. 
$\,$More $\,$generally, $\,$if $\,$$S$ $\,$is $\,$any $\,$set, $\,$we 
$\,$define
\mbox{$F(S)=\bigcup_{x\in S}F(x)$}.
The {\em domain} $\Do(F)$ is the set $\{x:F(x)\neq\emptyset\}$,
and the {\em image} $\Im(F)$ of $F$ is the set $F(\So(F))$.
We call $F$ {\em everywhere defined} if \mbox{$\Do(F)=\So(F)$}, 
and say that $F$ is {\em surjective} if $\Im(F)=\Ta(F)$. 
Then $\Do(F)\subseteq\So(F)$ and $\Im(F)\subseteq\Ta(F)$,
and $F$ is everywhere defined iff $\Do(F)=\So(F)$. 

If $X$ is a set, we use $\id_X$ to denote the identity map
of $X$, i.e. the set-valued map $(X,X,{\rm Diag}(X))$, where 
${\rm Diag}(X)=\{(x,x):x\in X\}$.

If $F=(X,Y,G)$ is a set-valued map, and $S$ is a subset of $X$,
then the {\sl restriction} $F\restr S$ is the set-valued map
$(X\cap S,Y,G\cap(S\times Y))$.  Therefore $\So(F\restr S)=X\cap S$, 
$\Ta(F\restr S)=\Ta(F)$, and $(F\restr S)(x)=F(x)$ when $x\in S$,
while $(F\restr S)(x)=\emptyset$ if $x\notin S$.

The {\em composite} $F=F_2\compo F_1$ of two set-valued maps
$F_i=(X_i,Y_i,G_i)$, $i=1,2$, is \mbox{defined} if and only if $Y_1=X_2$. In that case, 
$F\equaldef (X_1,Y_2,G_2\compo G_1)$, where
$G_2\compo G_1\equaldef\{(x,z):(\exists y)((x,y)\in G_1\wedge
(y,z)\in G_2\}$. %It is clear that the composition operation
%is associative, in the sense that $(F_3\compo F_2)\compo F_1$
%is defined iff $F_3\compo(F_2\compo F_1)$ is, and in that case  
%$(F_3\compo F_2)\compo F_1=F_3\compo(F_2\compo F_1)$.

A set-valued map $F$ is {\em single-valued} if $F(x)$ consists
of a single point of $\Ta(F)$ for every $x\in\Do(F)$. 
An {\em ordinary map} is a set-valued map that is 
single-valued and everywhere defined. We write $F:X\mapsto Y$
to indicate that $F$ is an ordinary map with source $X$ (so that
$\Do(F)=X$ as well) and target $Y$.
Notice that according to these conventions the notation
$F:X\tto Y$ allows for the possibility that $F$ be multivalued, or
partially defined, or both, but the notation $F:X\mapsto Y$ automatically
implies that $F$ is both single-valued and defined on the whole set $X$. 
A {\em selection} of a set-valued map $F:X\tto Y$ is a single-valued
map \mbox{$f:X\mapsto Y$} such that $f(x)\in F(x)$ for all $x\in X$.
With our definitions, the Axiom of Choice implies that 
a set-valued map has selections iff it is
everywhere defined.

\subssection{Upper semicontinuous set-valued maps}\label{seconusc}

If $X$ and $Y$ are topological spaces, a set-valued map
\mbox{$F:X\tto Y$} is {\bf\em upper semicontinuous} (abbr. USC)
if the inverse image under $F$ of every closed subset of $Y$ is 
closed in $X$. The following observation is then easily verified.
\bfact\label{areq} 
$\,$Assume that $\,X\,$ and $\,Y\,$ are metric spaces, $\,X\,$ is
compact, and \mbox{$F:X\tto Y$}. Then 
$\Gr(F)$ is compact if and only if 
$F$ is USC and $F(x)$ is compact for each $x\in X$.\proofend
\efact 
\begin{remark}\label{remonuscscalar} 
For \ ordinary \ real-valued \ functions, \ the \ words \ ``upper
\mbox{semicontinuous}'' will always refer to upper semicontinuity 
in the classical sense of real function theory. (That is,
$f$ is upper semicontinuous if the set $\{x:f(x)<\alpha\}$ is open for every
$\alpha\in\rr$.) If we want to say that such an $f$ is USC 
in the set-valued sense, we will say
that $f$ is ``upper semicontinuous as a set-valued map.''\proofend
\end{remark}


\subssection{Regular Set-Valued Maps}\label{secregumaps}

As was explained in the introduction, the classical proof of the
\mbox{maximum} principle involves a topological argument, based on the
Brouwer fixed point theorem.  In order to carry out similar arguments
for set-valued maps, one needs a good set-valued analogue of the
notion of a continuous single-valued map.  This is provided by the
concept of a ``regular map,'' which we now proceed to define. (It will
be explained in Remark \ref{whygood} why this notion is the right
one.)

If $X$ is  a metric space and $L_1,L_2\in{\rm Comp}(X)$,
we define the ``Hausdorff semidistance'' $\Delta_X(L_1,L_2)$ from
$L_1$ to $L_2$ by letting  
\begin{equation}
\Delta_X(L_1,L_2)=\sup\Big\{\,{\rm{}dist}(x,L_2):x\in L_1\,\}\,.
\end{equation} 
This function is not in general symmetric. (For example, if 
$L_1\subseteq L_2$ and $L_1\neq L_2$, 
then $\Delta_X(L_1,L_2)=0$ but $\Delta_X(L_2,L_1)>0$.)
On the other hand, $\Delta_X$ satisfies the ``triangle  inequality'':
\begin{equation}\label{triangle-b} \Delta_X(L_1,L_3)\le
\Delta_X(L_1,L_2)+\Delta_X(L_2,L_3)\,\,\,
{\rm if}\,\,\,L_1,L_2,L_3\in{\rm Comp}(X)\,.
\end{equation} 


\begin{definition}\label{def-inward}
Let $X$ be a metric space, let $L\in{\rm Comp}(X)$, and
let $\{L_j\}_{j\in\nn}$  be a sequence in ${\rm Comp}(X)$.
We say that $L_j$ {\em inward converges} to $L$ if
$\lim_{j\to\infty}\Delta_X(L_j,L)=0$. \proofend
\end{definition}

If $K$, $Y$ are metric spaces, and $K$ is compact, we write
$SVM_{comp}(K,Y)$  to denote the set of all 
set-valued maps $F:K\tto Y$ whose graph is  \mbox{compact}
and nonempty. In view of our definitions, it is clear that
$SVM_{comp}(K,Y)$ is exactly the same
as the set $\{K\}\times\{Y\}\times{\rm{}Comp}(K\times Y)$, 
so $SVM_{comp}(K,Y)$ can be
\mbox{canonically} identified  with ${\rm{}Comp}(K\times Y)$.

\begin{definition}\label{def-inward-graph}
Let $K,Y$ be metric spaces such that $K$ is compact.  Let $F\in
SVM_{comp}(K,Y)$, and let $\{F_j\}_{j\in\nn}$ be a sequence in
$SVM_{comp}(K,Y)$.  We say that $\{F_j\}_{j\in\nn}$ {\em inward graph
converges to $F$ as $j\to\infty$} if the 
\mbox{graphs} $\Gr(F_j)$ $\,$inward
$\,$converge $\,$to $\,$$\Gr(F)$ $\,$in $\,$the $\,$space 
$\,$${\rm Comp}(K\times Y)$, $\,$i.e., $\,$if 
\mbox{$\lim_{j\to\infty}\Delta_{K\times Y}(\Gr(F_j),\Gr(F))=0$}. 
We write $F_j\togr F$ to indicate that the sequence 
$\{F_j\}_{j\in\nn}$ inward graph converges to $F$ as $j\to\infty$.\proofend
\end{definition}

\begin{definition}\label{defregu} 
Let $K,Y$ be metric spaces such that $K$ is compact.  
A {\em regular set-valued map from $K$ to $Y$} is a set-valued map 
$F\in{}SVM_{comp}(K,Y)$ which is a limit in
$SVM_{comp}(K,Y)$  of a sequence $\{F_j\}_{j\in\nn}$ 
of ordinary---i.e., single-valued and everywhere defined---continuous 
maps from $K$ to $Y$. 

We use ${\rm REG}(K,Y)$ to denote the set of all regular
set-valued  maps from $K$ to $Y$. \proofend
\end{definition}

\begin{remark}\label{whygood}
Regular maps have good fixed point properties analogous
to those of single-valued continuous maps. For example,
{\em a regular set-valued map 
$F:\bmath^n(0,1)\mapsto\bmath^n(0,1)$ must have a fixed point}. 
(Proof: let $\{F_j\}_{j\in\nn}$ be a sequence of continuous maps
{}from $\bmath^n(0,1)$ to $\bmath^n(0,1)$ such that 
$F_j\togr F$. For each $j$ pick $x_j$ such that $F_j(x_j)=x_j$.
Then $(x_j,x_j)\in\Gr(F_j)$. 
\mbox{Assume}, after passing to a subsequence if necessary, that
$x=\lim x_j$ exists. Pick $(x_j',y_j)\in Gr(F)$ 
closest to $(x_j,x_j)$. Then 
\mbox{$\|x_j'-x_j\|+\|y_j-x_j\|\to 0$}. So $(x_j',y_j)\to(x,x)$.
Therefore $(x,x)\in\Gr(F)$, because $Gr(F)$ is compact.)\proofend
\end{remark}

\subssection{Regular differentiability}\label{secregdif}

\begin{definition}\label{defregdif}
Assume that $n,m\in\nn$, $F:\rr^n\tto\rr^m$, $C$ is a closed convex cone
in $\rr^n$, and $\bar x\in\rr^n$. We say that $F$ is {\em regularly
differentiable at $\bar x$ in the direction of $C$} if 
\begin{itemize}\item[]\begin{itemize}
\item[(D\ref{defregdif}.1)] there exists a compact neighborhood
$N$ of $\bar x$ in $\rr^n$ such that the restriction
$F\restr(N\cap(\bar x+C))$ is a regular set-valued map from
$N\cap(\bar x+C)$ to $\rr^m$;
\item[(D\ref{defregdif}.2)] $F(\bar x)$ consists of a single point
$\bar y\in\rr^m$;
\item[(D\ref{defregdif}.3)] there exists a linear map
$L:\rr^n\mapsto\rr^m$ such that
\begin{equation}\label{eqdifmap}
\lim_{h\to 0\,,\,h\in C}
\frac{\sup\Big\{\|y-\bar y-L\cdot h\|:y\in F(\bar x+h)\Big\}}{\|h\|}=0\,.
\end{equation}
\end{itemize}\end{itemize}
If $L$ is a linear map such that (\ref{eqdifmap}), then $L$ is
said to be a {\em differential of $F$ at $\bar x$
in the direction of $C$}.\proofend
\end{definition}


\subssection{Essential measurability}
\noindent If $n\in\nn$, 
we use $\bor(\rr^n)$, $\leb(\rr^n)$ to denote, respectively, 
the $\sigma$-algebras of Borel and Lebesgue measurable subsets of $\rr^n$. 
We use $\blon$ to denote the $\sigma$-algebra whose members are the
Borel$\otimes$Lebesgue-measurable subsets of $\rr^n\times\rr$, that is, the 
$\sigma$-algebra of subsets of $\rr^n\times\rr$ generated by all the products
$A\times B$ such that $A\in\bor(\rr^n)$ and $B\in\leb(\rr)$.  
If $S\subseteq\rr^n\times\rr$, we use $\blons$ to denote the class of subsets
of $S$ that belong to $\blon$. Also, we use 
${\rm Bor}(\rr^n\times\rr)\cap S$,  ${\cal BL}(\rr^n,\rr)\cap S$,  
to denote, respectively, the $\sigma$-algebras 
$\{E\cap S:E\in{\rm Bor}(\rr^n\times\rr)\}$ and
$\{E\cap S:E\in{\cal BL}(\rr^n,\rr)\}$. 

We use $\msr(E)$ to denote the Lebesgue measure of a measurable
subset $E$ of $\rr$. A {\em null subset of $\rr$} is a  subset $E$ 
of $\rr$ such that $\msr(E)=0$. A {\em full subset of $\rr$} is a  subset $F$ 
of $\rr$ such that $\rr\backslash F$ is a null set.

If $S\subseteq X\times\rr$ for some set $X$, and $J\subseteq\rr$, we write
\begin{equation}
S_J\equaldef S\cap(X\times J)=\{(s,t)\in S:t\in J\}\,.
\end{equation}
\begin{definition}\label{defessen}
Let $n,m\in\nn$, and let $S$ be a subset of 
$\rr^n\times\rr$. We say that $S$ is {\em essentially measurable} if 
there exists a full subset $J$ of $\rr$ such that 
\mbox{$S_J\in\blon$}.
We use ${\cal EM}(\rr^n,\rr)$ to denote the set of essentially
measurable subsets of $\rr^n\times\rr$. 
If $S\subseteq\rr^n\times\rr$, we use 
${\cal EM}(\rr^n,\rr;S)$ to denote the set of all subsets of 
$S$ that belong to ${\cal EM}(\rr^n,\rr)$,
and write ${\cal EM}(\rr^n,\rr)\cap S$ $\,\,$to $\,\,$denote $\,\,$the 
$\,\,$set $\,\,$of $\,\,$all $\,$sets $\,\,$of $\,\,$the $\,\,$form 
$E\cap S$, \mbox{$E\in{\cal EM}(\rr^n,\rr)$}.\proofend
\end{definition}

It is clear that  ${\cal EM}(\rr^n,\rr)$ is a $\sigma$-algebra,
and that, if $S\subseteq\rr^n\times\rr$, then 
${\cal EM}(\rr^n,\rr;S)$ is a $\sigma$-algebra of subsets of
$S$ if and only if $S\in{\cal EM}(\rr^n,\rr)$, in which case
${\cal EM}(\rr^n,\rr;S)={\cal EM}(\rr^n,\rr)\cap S$. 
\mbox{Moreover}, the following  inclusions hold
$$
{\rm Bor}(\rr^n\times\rr)\subseteq{\cal BL}(\rr^n,\rr)
\subseteq{\cal EM}(\rr^n,\rr)
\subseteq{\rm Leb}(\rr^n\times\rr)\,,
$$
and they are all strict if $n>0$. 

\begin{remark}\label{remequi}
\forshort{It is easy to prove that {\em a}}\forlong{{\em A}} 
{\em subset $S$ of $\rr^n\times\rr$ is essentially 
\mbox{measurable} \forshort{iff} \forlong{if and
\mbox{only} if} there exists a full Borel subset $J$ of $\rr$ such that
\forlong{\mbox{$S_J\in{\rm Bor}(\rr^n\times\rr)$}}\forshort{$S_J$ 
belongs to ${\rm Bor}(\rr^n\times\rr)$}}.
(In \mbox{other} words, ``essential
Borel$\otimes$Lebesgue-measurability is equivalent to \mbox{essential}
Borel$\otimes$Borel-measurability.'') 
\forlong{Indeed, let $\Sigma$ be the set
of all \mbox{subsets} $S$ of $\rr^n\times\rr$ such that $S_J\in{\rm
Bor}(\rr^n\times\rr)$ for some full $J\in{\rm Bor}(\rr)$. Then
$\Sigma$ is obviously a $\sigma$-algebra, and $\Sigma\subseteq{\cal
EM}(\rr^n,\rr)$. Clearly, every set $S$ of the form $E\times F$, with
$E\in{\rm Bor}(\rr^n)$ and $F\in{\rm Leb}(\rr)$ belongs to
$\Sigma$. Therefore ${\cal BL}(\rr^n,\rr)\subseteq\Sigma$. If
$S\in{\cal EM}(\rr^n,\rr;S)$, then we can pick a
full subset $J$ of $\rr$ such that $S\cap(\rr^n\times J)\in{\cal
BL}(\rr^n,\rr)$. Then $S\cap(\rr^n\times J)\in\Sigma$, from which it
follows that $\Big(S\cap(\rr^n\times J)\Big)\cap(\rr^n\times L)
\in{\rm Bor}(\rr^n\times\rr)$ for some full $L\in{\rm Bor}(\rr)$.
If $M\in{\rm Bor}(\rr)$ is such that $M\subseteq J$ and
$\msr(J\backslash M)=0$, and we let $N=M\cap L$, then $N$
is a full Borel subset of $\rr$, and
$$
S_N=\Bigg(\Big(S\cap(\rr^n\times J)\Big)\cap(\rr^n\times L)\Bigg)
\cap(\rr^n\times M)\in{\rm Bor}(\rr^n\times\rr)\,,
$$
so $S\in\Sigma$. Therefore ${\cal EM}(\rr^n,\rr)\subseteq\Sigma$.
So $\Sigma={\cal EM}(\rr^n,\rr)$.}\proofend
\end{remark}

\begin{definition}
Let $n,m\in\nn$, and let $S$ be a subset of 
$\rr^n\times\rr$. A map \mbox{$f:S\mapsto\rr^m$} is said to be 
{\em essentially measurable on $S$} if 
$f^{-1}(B)$ belongs to ${\cal EM}(\rr^n,\rr;S)$ for every 
Borel subset of $\rr^m$.\proofend
\end{definition}
\forshort{It is easy to verify that}
\bfact
\ If $n,m\in\nn$, $S\subseteq\rr^n\times\rr$, and
$f:S\mapsto\rr^m$, then $f$ is essentially measurable if and only 
if there exists a full Borel subset $J$ of $\rr$ such that 
$S_J$ belongs to ${\rm Bor}(\rr^n\times\rr)$ and the restriction
\mbox{$f\restr S_J$} of $f$ to $S_J$ is Borel measurable.\forshort{\proofend} 
\efact
\forlong{
\begin{proof}
Assume that $f$ is essentially measurable. Let ${\cal A}$ be a countable
set of Borel subsets of $\rr^m$ that generates the $\sigma$-algebra
${\rm Bor}(\rr^m)$. For each $A\in{\cal A}$, the fact that
$f^{-1}(A)\in{\cal EM}(\rr^n,\rr)$, together with Remark \ref{remequi},
imply that we can  choose a full Borel subset
$J_A$ of $\rr$ such that 
$f^{-1}(A)\cap(\rr^n\times J_A)$ belongs to ${\rm Bor}(\rr^n\times\rr)$.
Let $J=\bigcap_{A\in{\cal A}}J_A$. Then 
$J$ is a full Borel subset of $\rr$, and 
$$
f^{-1}(A)\cap(\rr^n\times J)=
\Big(f^{-1}(A)\cap(\rr^n\times J_A)\Big)\cap(\rr^n\times J)
\in{\rm Bor}(\rr^n\times\rr)
$$
for every $A\in{\cal A}$. Therefore $f^{-1}(B)\cap(\rr^n\times J)
\in{\rm Bor}(\rr^n\times\rr)$ for every $B\in{\rm Bor}(\rr^m)$.
So $f\restr S_J$ is Borel measurable. This proves one of the two implications
of our statement. The other implication is trivial.\proofend
\end{proof}
}


\subssection{Curves, arcs, and spaces of arcs}\label{subsec-arcs}

\begin{definition}
{\rm Let $\,$$n\in\nn$. 
$\,$A {\em $\,$curve} $\,$in $\,$$\rr^n$ $\,$is $\,$a $\,$continuous 
$\,$map \mbox{$\xi:I\mapsto\rr^n$} 
whose domain $I=\Do(\xi)$ is a nonempty real interval.  
An {\em arc} in $\rr^n$ is a curve $\xi$ in 
$\rr^n$ whose domain is a compact subinterval
$[a_\xi,b_\xi]$ of $\rr$.
We use $ARC(\rr^n)$ to denote the set of all arcs in $\rr^n$,
regarded as a metric space with the distance 
$d_{ARC}:ARC(\rr^n)\times ARC(\rr^n)\mapsto\rr_+$ given by 
$$  
d_{ARC}(\xi,\eta)=|a_\xi-a_\eta|+|b_\xi-b_\eta|+
\sup\Big\{\|\tilde\xi(t)-\tilde\eta(t)\|:t\in\rr\Big\}\,,
$$
where, for $\zeta\in ARC(\rr^n)$, $\tilde\zeta$ denotes the extension
of the arc $\zeta$ to a map $\tilde\zeta:\rr\mapsto\rr^n$ given
by $\tilde\zeta(t)=\zeta\Big(\min(\max(t,a_\zeta),b_\zeta)\Big)$
for $t\in\rr\,$.

If $S$ is a subset of $\rr^n\times\rr$, an {\em arc in $S$} is a 
$\xi\in ARC(\rr^n)$ such that $(\xi(t),t)\in S$ for all 
$t\in\dom(\xi)$.  We use $ARC(\rr^n;S)$ to denote the set of all arcs in 
$S$, regarded as a subspace of the metric space $ARC(\rr^n)$.}\proofend
\end{definition}
It is clear that convergence in $ARC(\rr^n)$ is {\em uniform  convergence},
in the \mbox{following} precise sense:
\begin{itemize}\item[]\begin{itemize}
\item[(C$_{ARC}$)] {\sl If $\xi\in ARC(\rr^n)$, then a sequence 
$\{\xi_j\}_{j\in\nn}$
of arcs belonging to $ARC(\rr^n)$ converges to $\xi$ in 
$ARC(\rr^n)$ if and only if 
\begin{itemize}
\item[(i)] $a_{\xi_j}\to a_\xi$ and $b_{\xi_j}\to b_\xi$ as $j\to\infty$, 
\end{itemize}
and 
\begin{itemize}
\item[(ii)] whenever $\{t_j\}_{j\in\nn}$ is a sequence in $\rr$
such that $a_{\xi_j}\le t_j\le b_{\xi_j}$ for every 
$j\in\nn$, and $t\in\rr$ is such that $t_j\to t$ as $j\to\infty$ (so that
$a_\xi\le t\le b_\xi$), it follows
that $\xi_j(t_j)\to\xi(t)$ as $j\to\infty$.
\end{itemize}}\end{itemize}\end{itemize}

\medbreak\noindent
It is also clear that $ARC(\rr^n)$ is a complete metric space, and
$ARC(\rr^n;S)$ is a closed subset of $ARC(\rr^n)$ if and only if $S$
is a closed subset of $\rr^n\times\rr$.

\medbreak If $n,m\in\nn$, $S\subseteq\rr^n\times\rr$,
$f:S\mapsto\rr^m$,  and \mbox{$\xi\in ARC(\rr^n;S)$}, 
then $f\compo\xi$ denotes the 
map $[a_\xi,b_\xi]\ni t\to f(\xi(t),t)\in\rr^m$.   

\forshort{The following observation is easily verified.}
\bfact\label{remismeas}
If $n,m\in\nn$, $S\subseteq\rr^n\times\rr$, and
$f$ is ${\cal EM}(\rr^n,\rr)\cap S$-measurable, 
then $f\compo\xi$ is measurable for
every member $\xi$ of $ARC(\rr^n;S)$.\forshort{\proofend}
\efact
\forlong{
\begin{proof} Let $\xi:[a,b]\mapsto\rr^n$ be continuous
and such that $(\xi(t),t)\in S$ for each $t\in[a,b]$. Let
$K=\{(\xi(t),t):a\le t\le b\}$, so $K$ is a compact subset of $S$.

If $B$ is a 
Borel subset of $\rr^m$, then $f^{-1}(B)\in{\cal EM}(\rr^n,\rr)\cap S$, 
so $f^{-1}(B)=A\cap S$ for some $A\in{\cal EM}(\rr^n,\rr)$. 
Then 
\begin{eqnarray*}
(f\compo\xi)^{-1}(B)&=&\{t\in[a,b]:(\xi(t),t)\in f^{-1}(B)\}\\
&=&\{t\in[a,b]:(\xi(t),t)\in f^{-1}(B)\cap K\}\\
&=&\Xi^{-1}\Big(f^{-1}(B)\cap K\Big)\,,
\end{eqnarray*}
where $\Xi$ is the map
$[a,b]\ni t\mapsto(\xi(t),t)\in\rr^n\times\rr$. But
$$
\Xi^{-1}(f^{-1}(B)\cap K)=\Xi^{-1}(A\cap S\cap K)
=\Xi^{-1}(A\cap K)\,,
$$
since $K\subseteq S$. Clearly, $A\cap K\in{\cal EM}(\rr^n,\rr)$. 
So it suffices to
show that $\Xi^{-1}(E)$ is a Lebesgue measurable subset of $[a,b]$
whenever $E$ belongs to ${\cal EM}(\rr^n,\rr)$. 

Now, if $E\in{\cal EM}(\rr^n,\rr)$, then 
there exists a full Borel subset $J$ of $\rr$ such that 
$E_J$ is Borel measurable.
Then 
$$
\Xi^{-1}(E)\!=\!\Xi^{-1}(E_J)\!\cup\!\{t\!\in\! J\cap[a,b]\!:
\!\Xi(t)\!\in\! E\}\,,
$$
so $\Xi^{-1}(E)$ is Lebesgue measurable if $\Xi^{-1}(E_J)$ 
is Lebesgue measurable. So it suffices to show that the set
$\Xi^{-1}(E)$ is Lebesgue measurable whenever 
$E\in{\rm Bor}(\rr^n\times\rr)$. This conclusion will be true 
for all $E\in{\rm Bor}(\rr^n\times\rr)$ provided it is true for
all $E$ of the form $F\times G$, $F\in{\rm Bor}(\rr^n)$, 
$G\in{\rm Bor}(\rr)$. But, if $F\in{\rm Bor}(\rr^n)$ and 
$G\in{\rm Bor}(\rr)$, then $\Xi^{-1}(F\times G)=
\xi^{-1}(F)\cap G$, so $\Xi^{-1}(F\times G)$ is 
actually Borel measurable, completing our proof.\proofend
\end{proof}
}

\subssection{Points of density and Lebesgue points}

If $\ffi:\rr\mapsto\overline{\rr}$ is a measurable
function, a {\em Lebesgue point} of $\ffi$ is a point $t\in\rr$
such that $|\ffi(t)|<\infty$ and 
\begin{equation}
\lim_{h\to 0+}\frac{1}{h}\int_{[t-h,t+h]}
\Big|\ffi(s)-\ffi(t)\Big|\,ds =0\,.
\end{equation}

A {\em point of density} of a measurable subset 
$E\subseteq\rr$ is a point of $E$ that is a Lebesgue point of the indicator 
function\footnote[5]{The {\em indicator function} of $E$ is the function
whose value is $1$ on $E$ and $0$ outside $E$.}
$\chi_{_E}$ of $E$. Equivalently, a real number
$t$ is a  point of density of $E$ if and only if $t\in E$ and
\begin{equation}
\lim_{h\downarrow 0}\frac{1}{2h}\msr\Bigl(E\cap[t-h,t+h]\Bigr)=1\,.
\end{equation}

It is well known that if $\ffi:\rr\mapsto\overline{\rr}$ is 
locally integrable then almost every point of $\rr$ is a Lebesgue point
of $\ffi$.  In particular, almost every point of a measurable
subset $E$ of $\rr$ is a point of density of $E$. It follows 
that if $E\subseteq\rr$ is measurable and $\msr(E)>0$, then $E$ has a point 
of density. 

\subssection{Trajectories and flows}\label{sectrajflow}

\begin{definition}\label{defoftraj}  
Let $n\in\nn$, and let $F:\rr^n\times\rr\tto\rr^n$ be a set-valued map.
\begin{itemize}\item[]\begin{itemize}
\item[(D\ref{defoftraj}.i)] A {\em trajectory} of $F$ (or a ``trajectory
of the  differential inclusion \mbox{$\dot x\in F(x,t)$}'') 
is a locally absolutely continuous map \mbox{$\xi:I\mapsto\rr^n$}, 
whose domain $\Do(\xi)$ is a nonempty subinterval $I$ of $\rr$,
such that \mbox{$\dot\xi(t)\in F(\xi(t),t)$} for almost every $t\in I$. 

\oitem[(D\ref{defoftraj}.ii)] We  use
$\traj(F)$ to denote the set of all trajectories of $F$, and $\traj_c(F)$
to denote the set of all $\xi\in\traj(F)$ whose domain is a compact
interval.  

\oitem[(D\ref{defoftraj}.iii)] If $V$ is a subset of $\rr^n$, we write 
$\traj_c(F,V)$ to denote the set of all $\xi\in\traj_c(F)$ such that 
$(\xi(t),t)\in V$ for all $t\in\dom(\xi)$. 

\oitem[(D\ref{defoftraj}.iv)] A {\em maximal trajectory} of $F$ is a 
trajectory 
$\xi:I\mapsto\rr^n$ that \mbox{cannot} be extended to a trajectory of 
$F$ defined on a strictly larger \mbox{interval}. 

\oitem[(D\ref{defoftraj}.v)] For $(x,t,s)\in\rr^n\times\rr\times\rr$, we define
\begin{equation}
\Phi^F(x,t,s)=\{\xi(t):\xi\in\traj(F)\,,\,\xi(s)=x\}\,.
\end{equation}
The set-valued map 
$\Phi^F\!:\!\rr^n\!\times\!\rr\!\times\!\rr\tto\!\rr^n$ is called
the {\em flow} of $F$. 

\oitem[(D\ref{defoftraj}.vi)] For $(t,s)\in\rr\times\rr$, we define a 
set-valued map $\Phi_{t,s}^F:\rr^n\tto\rr^n$ by letting 
\begin{equation}
\Phi_{t,s}^F(x)\equaldef\Phi^F(x,t,s)\qquad{\rm for}\quad
x\in\rr^n\,.
\end{equation}
The set-valued maps $\Phi_{t,s}^F:\rr^n\tto\rr^n$ are the 
{\em flow maps} of $F$. 
\proofend
\end{itemize}\end{itemize}
\end{definition}

\noindent The sets $\traj_c(F)$ are subsets of the metric space
$ARC(\rr^n)$ defined in \S\ref{subsec-arcs}.

With our definitions, if $F:\rr^n\times\rr\tto\rr^n$,
$(x,t)\in\rr^n\times\rr$, $I=\{t\}$, and
$\xi:I\mapsto\rr^n$ is given by $\xi(t)=x$, then $\xi$ is a trajectory of 
$F$, even if $(x,t)\notin\Do(F)$. Therefore the identity
\begin{equation}\label{floid-a}
\Phi_{t,t}^F=\idrn
\end{equation}
holds.  In addition, the flow maps $\Phi_{t,s}^F$ also satisfy the {\em flow identity}
\begin{equation}\label{floid-b}
\Phi_{t,s}^F\compo\Phi_{s,r}^F=\Phi_{t,r}^F\qquad{\rm if}
\quad r\le s\le t\,.
\end{equation}
\begin{remark}
{\em The equality of (\ref{floid-b}) may fail to be  true
if it is not true that \mbox{$r\le s\le t$}}. For example, if
$F$ is a time-varying vector field on $\rr^n$ (i.e., a single-valued map
$F:\rr^n\times\rr\tto\rr^n$) that does not have unique \mbox{trajectories}, 
$a<b$, and $\xi_1,\xi_2:[a,b]\mapsto\rr^n$ are two trajectories
of $\dot x=F(x,t)$ that satisfy \mbox{$\xi_1(a)=\xi_2(a)$} and
$\xi_1(b)\neq\xi_2(b)$, then 
$\xi_2(b)$ belongs to $(\Phi_{b,a}^F\compo\Phi_{a,b}^F)(\xi_1(b))$,
so $\Phi_{b,a}^F\compo\Phi_{a,b}^F\neq\idrn=\Phi_{b,b}^F$.\proofend
\end{remark}

Zorn's Lemma easily implies that 
\bfact\label{exteexists}
If $n\in\nn$ and $F:\rr^n\times\rr\tto\rr^n$, then every trajectory of the
differential inclusion $\dot x\in F(x,t)$ can be \mbox{extended} 
to a \mbox{maximal} trajectory.\proofend
\efact
The following is an immediate consequence of Fact \ref{exteexists}.
\bfact\label{exteexists-a}
If $n\in\nn$ and $F:\rr^n\times\rr\tto\rr^n$, then for every 
$(x,t)\in\rr^n\times\rr^n$ there exists a maximal trajectory $\xi$ of 
$\dot x\in F(x,t)$ such that $\xi(t)=x$. \proofend
\efact
Naturally, it may turn out that ${\rm Dom}(\xi)$ is just $\{t\}$.

\section{Discontinuous vector fields and their flows}

\label{secdiscvecfields}

\subssection{Integrally continuous time-varying maps}\label{sectquasi}

\begin{definition}
Assume that $n,m\in\nn$, $S$ is a subset of $\rr^n\times\rr$, and
$f:S\mapsto\rr^m$. An {\em integral bound} for $f$ is
a  Lebesgue integrable function 
\mbox{$\rr\ni t\mapsto\ffi(t)\in[0,+\infty]$}  such that 
\begin{equation}\label{fboundo} 
\|f(x,t)\|\le\ffi(t)\,\,\,{\rm for\,\,\,all\,\,\,}(x,t)\in S\,. 
\end{equation}
We say that $f$ is {\em integrably bounded} 
if there exists an integral bound for $f$.
We say that $f$ is {\em locally integrably bounded} 
(abbr. LIB) if for every compact subset $K$ of $S$ the
restriction of $f$ to $K$ is integrably bounded.

A set ${\cal F}$ of maps from $S$ to $\rr^m$ is {\em uniformly LIB} 
if every $f\in{\cal F}$ is LIB and, in addition, the integral
bounds for the restriction of $f$ to $K$ can be chosen, for each 
compact subset $K$ of $S$, independently of $f$.  \proofend
\end{definition} 

\begin{definition}
Assume that $n,m\in\nn$, $S\subseteq\rr^n\times\rr$, 
$f:S\mapsto\rr^m$, and ${\cal A}\subseteq ARC(\rr^n;S)$. 
We say that $f$ is {\sl integrable along arcs on ${\cal A}$}, or
{\sl arc-integrable on ${\cal A}$}, if  
$f\compo\xi$ is Lebesgue integrable for every 
$\xi\in {\cal A}$.\proofend
\end{definition}

\bfact\label{iflib}
If $n,m\in\nn$, $S\subseteq\rr^n\times\rr$, 
$f:S\mapsto\rr^m$, and $f$ is ${\cal EM}(\rr^n,\rr)\cap S$-measurable 
and LIB, then $f$ is arc-integrable on $ARC(\rr^n;S)$.
\efact
\begin{proof}
If $\xi\in ARC(\rr^n;S)$, then Fact \ref{remismeas} implies that 
$f\compo\xi$ is measurable, and the integral bound (\ref{fboundo}) 
implies that $f\compo\xi$ is integrable.\proofend
\end{proof}

\noindent If $f$ is arc-integrable on ${\cal A}$, we can define a map 
${\cal T}_f^{{\cal A}}:{\cal A}\mapsto ARC(\rr^m)$---the 
{\em integral operator associated to $f$ on ${\cal A}$}---by 
letting ${\cal T}_f^{\cal A}(\xi)$ be, for each $\xi\in{\cal A}$, the 
arc $\eta:[a_\xi,b_\xi]\mapsto\rr^m$ given by
\begin{equation}\label{defoftf}
\eta(t)=\int_{a_\xi}^tf(\xi(s),s)\,ds\,.
\end{equation}

\begin{definition}
Assume that $n,m\in\nn$, $S\subseteq\rr^n\times\rr$, 
$f:S\mapsto\rr^m$, and ${\cal A}\subseteq ARC(\rr^n;S)$. 
We say that $f$ is {\em integrally continuous on ${\cal A}$} if  
\begin{itemize}\item[]\begin{itemize}
\item[(IC.1)] $f$ is ${\cal EM}(\rr^n,\rr)\cap S$-measurable, 

\item[(IC.2)] $f$ is arc-integrable on ${\cal A}$,

\item[(IC.3)] the map ${\cal T}_f^{{\cal A}}
:{\cal A}\mapsto ARC(\rr^m)$ is continuous.\proofend
\end{itemize}\end{itemize}
\end{definition}
We also define maps ${\cal T}_{f,E}^{\cal A}:{\cal A}\mapsto\rr^m$,
if $E$ is a measurable subset of $\rr$,  by
\begin{equation}
{\cal T}_{f,E}^{\cal A}(\xi)\equaldef\int_{E\cap\dom(\xi)}f(\xi(s),s)\,ds
\quad{\rm for}\quad\xi\in{\cal A}\,,
\end{equation}
and maps ${\cal T}_{f,a,b}^{\cal A}:{\cal A}\mapsto\rr^m$,
if $a,b\in\rr$ and $a\le b$, by
\begin{equation}
{\cal T}_{f,a,b}^{\cal A}(\xi)
\equaldef\int_{[a,b]\cap\dom(\xi)}f(\xi(s),s)\,ds\,
\quad{\rm for}\quad\xi\in{\cal A}\,.
\end{equation}
We observe that
\begin{equation}\label{follimm}
{\cal T}_{f,a,b}^{\cal A}(\xi)=
{\cal T}_{f}^{\cal A}(\xi)\Big(\lambda(\xi,b)\Big)
-{\cal T}_{f}^{\cal A}(\xi)\Big(\lambda(\xi,a)\Big)
\quad{\rm for}\quad\xi\in{\cal A}\,,
\end{equation}
where $\lambda(\xi,s)\equaldef\min\Big(\,\max(a_\xi,s)\,,\,b_\xi\,\Big)\,$.
\begin{proposition}\label{remquasi}
Assume that $n,m\in\nn$, $S\subseteq\rr^n\times\rr$, 
${\cal A}$ is a subset of $ARC(\rr^n;S)$, and $f$ is
${\cal EM}(\rr^n,\rr)\cap S$-measurable and LIB. Then Condition (IC.3) is
equivalent to the following ``strong integral continuity condition'':

\begin{itemize}\item[]\begin{itemize}
\oitem[(SIC)] for every $E\in{\rm Leb}(\rr)$,
the map ${\cal T}_{f,E}^{\cal A}:{\cal A}\mapsto\rr^m$ is continuous,
\end{itemize}\end{itemize}
\noindent as well as to the ``weak integral continuity condition'':
\begin{itemize}\item[]\begin{itemize}
\item[(WIC)]  the map  ${\cal T}_{f,a,b}^{\cal A}:{\cal A}\mapsto\rr^m$ is continuous
whenever $a,b\in\rr$ and $a\le b$.
\end{itemize}\end{itemize}
\end{proposition}

\begin{proof}
To prove the equivalence of (IC.3), (SIC) and (WIC),
we first observe that the implication (SIC)$\Rightarrow$(WIC) 
is trivial, and the implication 
(IC.3)$\Rightarrow$(WIC) follows immediately from (\ref{follimm}).

Next, we consider a sequence $\{\xi_j\}_{j\in\nn}$ 
of arcs $\xi_j$ that belong to ${\cal A}$ and converge 
to a $\xi_{\infty}\in{\cal A}$.  Then 
$a_{\xi_j}\to a_{\xi_\infty}$ and $b_{\xi_j}\to b_{\xi_\infty}$
as $j\to\infty$. In addition, it is easy to see that the set
$$
K=\Big\{\,(x,t):(\exists j\in\overline{\nn})\Big(\,t\in\dom(\xi_j)\wedge x=
\xi_j(t)\,\Big)\,\Big\}
$$
is a compact subset of $S$. Therefore there exists an integrable
function $\ffi_K:\rr\mapsto[0,\infty]$ such that the bound
$\|\dot\xi_j(t)\|\le\ffi_K(t)$ holds  for every pair $(j,t)$ such that
$j\in\overline{\nn}$ and $t\in\dom(\xi_j)$.  

Fix real numbers $\,\,\alpha\,$, $\,\,\beta\,$ such that 
$\,\,\alpha\le a_{\xi_j}\,\,$ and $\,\,\beta\ge b_{\xi_j}\,\,$ for 
\mbox{all} \mbox{$j\in\overline{\nn}$}. Define maps 
$\theta_j,\eta_j:[\alpha,\beta]\mapsto\rr^m$ by letting 
\begin{eqnarray*}
\theta_j(t)&=&\left\{\begin{array}{lcl}
f(\xi_j(t),t)&{\rm if}&t\in\,[a_{\xi_j},b_{\xi_j}]\,,\\
0&{\rm if}&t\in\,[\alpha,\beta]\,\backslash\,[a_{\xi_j},b_{\xi_j}]
\,,\end{array}\right.\\
\eta_j(t)&=&\int_{\alpha}^{t}\theta_j(s)\,ds\,.
\end{eqnarray*}

Then the $\theta_j\,$ satisfy \mbox{$\|\theta_j(t)\|\le\ffi_K(t)$} for
all $t,j$, and therefore the
$\eta_j$ are absolutely continuous and satisfy the
bound \mbox{$\|\dot\eta_j(t)\|\le\ffi_K(t)$} for all $t,j$. Therefore
the sequence $\{\eta_j\}_{j\in\nn}$ is uniformly bounded and
equicontinuous.  Moreover,
$\eta_j(b)-\eta_j(a)={\cal T}_{f,a,b}^{\cal A}(\xi_j)$
for all $j$ and all $a,b\in[\alpha,\beta]$ such that $a\le b$. If (WIC) holds,
it follows that the $\eta_j$ converge pointwise to
$\eta_{\infty}$, and the equicontinuity implies that the convergence is
uniform on $[\alpha,\beta]$.  In particular, if
$a_{\xi_j}\le t_j\le b_{\xi_j}$ and $t_j\to t$ as $j\to\infty$, then
$\eta_j(t_j)\to\eta_\infty(t)$, showing that 
${\cal T}_f^{{\cal A}}(\xi_j)\to{\cal T}_f^{{\cal A}}(\xi_\infty)$
in $ARC(\rr^m)$. This proves the implication
(WIC)$\Rightarrow$(IC.3). 

Furthermore, given any $\eps>0$ there is a $\delta =\delta(\eps)>0$ such
that $\int_G\ffi_K<\eps$ whenever $G$ is a measurable subset of
$[\alpha,\beta]$ of measure $<\delta$.  If $E$ is an arbitrary
measurable subset of $\rr$, and $\eps>0$, then we can let $\tilde
E=E\cap[\alpha,\beta]$, choose $\delta=\delta({\eps\over 3})$, and
find a subset $F$ of $[\alpha,\beta]$ which is a finite union of
pairwise disjoint intervals and is such that the symmetric difference
$\tilde E\Delta F$ has measure $<\delta$.  Then
$|\int_{\tilde E}\theta_j-\int_F\theta_j|\le{\eps\over 3}$ for all $j$.  
If (WIC) holds, then $\int_F\theta_j\to\int_F\theta_\infty$, so we can 
find a $j_0\in\nn$
such that $|\int_F\theta_j-\int_F\theta_\infty|\le{\eps\over 3}$ for $j\ge
j_0$, and then $|\int_{\tilde E}\theta_j-\int_{\tilde E}\theta_\infty|
\le\eps$ for $j\ge j_0$, from which it clearly follows that
${\cal T}_{f,E}^{{\cal A}}(\xi_j)-{\cal T}_{f,E}^{{\cal A}}(\xi_\infty)
\le\eps$ for $j\ge j_0$. 
This proves the implication (WIC)$\Rightarrow$(SIC).

We have thus established that
(WIC)$\Rightarrow$(SIC)$\Rightarrow$(WIC)$\Rightarrow$(IC.3)$\Rightarrow$(WIC),
which completes the proof of the equivalence of (IC.3), (SIC) and
(WIC).\proofend 
\end{proof}

The equivalence of (SIC)  and (IC.3) implies the following 
{\em measurable intertwining  property}:

\begin{theorem}\label{meassub}
Assume that $n,m\in\nn$ and $S\subseteq\rr^n\times\rr$.
Let $\,\,\{f_j\}_{j\in\nn}\,\,$ be a  uniformly $\,$LIB$\,$
sequence of ${\cal EM}(\rr^n,\rr)\cap S$-measurable maps 
$S\mapsto\rr^m$. Let ${\cal A}\subseteq ARC(\rr^n;S)$ 
be such that the $f_j$ are integrally continuous on ${\cal A}$.  
Let $\{E_j\}_{j\in\nn}$ be a sequence of pairwise disjoint
Lebesgue measurable subsets of $\rr$ such that 
$\rr=\bigcup_{j\in\nn}E_j\,$.
Define a map $f:S\mapsto\rr^m$ by letting
\begin{equation}
f(x,t)=f_j(x,t)\,\,\,\,{\rm for}\,\,\,\,j\in\nn\,,\,(x,t)\in S\,,\,t\in E_j\,.
\end{equation}
Then $f$ is integrally continuous on ${\cal A}$.
\end{theorem}

\begin{proof}
The ${\cal EM}(\rr^n,\rr)\cap S$-measurability of $f$ is obvious, 
and it is clear that $f$ is LIB, so $f$ is arc-integrable 
on $ARC(\rr^n;S)$ (cf. Fact \ref{iflib}).  
Let $\{\xi_\ell\}_{\ell\in\nn}$ be a sequence in ${\cal A}$ that converges 
to a $\xi_{\infty}\in{\cal A}$.  Let $E$ be a measurable 
subset of $\rr$. Let $K$ be a compact subset of $S$ such that 
$(\xi_\ell(t),t)\in K$ for all $\ell\in\overline{\nn}$ 
and all $t\in\dom(\xi_\ell)$. Let $\alpha,\beta$ be real numbers
such that $\alpha\le t\le\beta$ whenever $(x,t)\in K$. 
Let $\varphi:\rr\mapsto[0,+\infty]$ be integrable and such that
$\|f_j(x,t)\|\le\varphi(t)$ whenever $(x,t)\in K$	
and $j\in\nn$, and 
$\|f(x,t)\|\le\varphi(t)$ whenever $(x,t)\in K$.	
  
Let $\eps>0$. Let $\delta>0$ be such that $\int_H\varphi\le\eps$
whenever $H\subseteq\rr$ is measurable and ${\rm meas}(H)\le\delta$.
Let $j^*$ be such that 
$$
\sum_{j>j^*}{\rm meas}(E\cap E_j\cap[\alpha,\beta])\le\delta\,.
$$
Using (SIC), we find that for each $j$
\begin{equation}
\int_{E\cap E_j\cap\dom(\xi_\ell)}f\compo\xi_\ell\to
\int_{E\cap E_j\cap\dom(\xi_\infty)}f\compo\xi_\infty
\,\,\,{\rm as}\,\,\,\ell\to\infty\,.
\end{equation}
Therefore, if we let $\tilde E=E\cap(E_0\cup E_1\cup\ldots\cup E_{j^*})$, 
we see that 
\begin{equation}
\int_{\tilde E\cap\dom(\xi_\ell)}f\compo\xi_\ell\to 
\int_{\tilde E\cap\dom(\xi_{\infty})}
f\compo\xi_\infty \,\,\,{\rm as}\,\,\,\ell\to\infty\,,
\end{equation}
since $\tilde E=\cup_{j\le j^*}(E\cap E_j)$ and the
sets $E_j$ are pairwise disjoint.

On the other hand, ${\rm meas}((E\backslash\tilde E)\cap[\alpha,\beta])
\le\delta$, and $\dom(\xi_\ell)\subseteq[\alpha,\beta]$. Therefore
\begin{equation}
\Biggl\|\int_{(E\backslash\tilde E)\cap\dom(\xi_\ell)}f\compo\xi_\ell
\Biggr\| \le\eps
\,\,\,\,{\rm for}\,\,\,\,\ell\in\overline{\nn}\,.
\end{equation}
It follows that
\begin{equation}
\lim\sup_{\ell\to\infty}
\Biggl\|\int_{E\cap\dom(\xi_\ell)}f\compo\xi_\ell-
\int_{E\cap\dom(\xi_\infty)}f\compo\xi_\infty\Biggr\| \le2\eps\,.
\end{equation}
Since this is true for every $\eps>0$, we see that 
$$
\int_{E\cap\dom(\xi_\ell)}f\compo\xi_\ell\to
\int_{E\cap\dom(\xi_\infty)}f\compo\xi_\infty\,,
$$
and our proof is complete. \proofend
\end{proof}

\subssection{Conic continuity}

\noindent Let $n\!\in\!\nn$, $S\!\subseteq\!\rr^n\!\times\!\rr$.
If $\psi\!:\!S\!\mapsto\!\overline{\rr}_+$ is a function,  we use 
\mbox{$ARC_\psi(\rr^n;S)$ to} denote the set of all 
absolutely \mbox{continuous}
$\xi\in ARC(\rr^n;S)$ such that 
$\|\dot\xi(t)\|\!\le\!\psi(\xi(t),t)$ for almost all $t\!\in\!\Do(\xi)$.
Also, we write $\widetilde{ARC}_\psi(\rr^n;S)$ to denote the (much larger)
set of all $\xi\in ARC(\rr^n;S)$ such that 
\begin{equation}\label{limo}
\lim\sup_{t\downarrow\bar t}{\|\xi(t)-\xi(\bar t)\|\over
t-\bar t}\le\psi(\xi(\bar t),\bar t)\,\,\,{\rm for\,\,\,
a.e.\,\,\,}\bar t\in\Do(\xi)\,.
\end{equation}

\noindent If $C>0$, and $(\bar x,\bar t)\in\rr^n\times\rr$, we write
\begin{eqnarray*}
\Gamma_C(\bar x,\bar t)\!\!\!
&\equaldef&\!\!\Big\{(x,t)\in \rr^n\times \rr:t\ge \bar t\,,\,
\|x-\bar x\|\le C(t-\bar t)\Big\}\\
\Gamma_C^2(\bar x,\bar t)\!\!\!
&\equaldef&\!\!\Big\{(x,x',t)\!\in\!\rr^n\!\times\!\rr^n
\!\times\!\rr\colon(x,t)\!\in\!\Gamma_C(\bar x,\bar t),(x',t)\!\in\!
\Gamma_C(\bar x,\bar t)\Big\}\,.
\end{eqnarray*}


\begin{definition}\label{defconcon}
Let $n,m\in\nn$, $S\subseteq\rr^n\times\rr$, $f:S\mapsto\rr^m$. 
Let $C>0$, and let $(\bar x,\bar t)$ be a point of $S$.
\begin{itemize}
\item[1.] We say that $f$ is {\em forward
$\Gamma_C$-continuous} at $(\bar x,\bar t)$ if
\begin{equation}
\lim_{(x,t)\to (\bar x,\bar t)\,,\,(x,t)\in
\Gamma_{C}(\bar x,\bar t)\,\cap S}f(x,t)=f(\bar x,\bar t)\,.
\end{equation}
\item[2.] We call $f$ {\em weakly forward
$\Gamma_C$-continuous} at $(\bar x,\bar t)$ if 
\begin{equation}
\lim_{(x,x',t)\to (\bar x,\bar x,\bar t)\,,\,(x,x',t)\in
\Gamma_{C}^2(\bar x,\bar t)\cap\tilde S}
\Bigl(f(x,t)-f(x',t)\Bigr)=0\,,
\end{equation}
where $\tilde S=\{(x,x',t):(x,t)\in S\wedge(x',t)\in S\}$.
\end{itemize}
If $S\ni(x,t)\mapsto C(x,t)\in\overline{\rr}_+$ is a function, we 
say that the map $f$ is {\em forward \mbox{$\Gamma_C$-continuous}} (resp. 
{\em weakly forward $\Gamma_C$-continuous}) if it is 
forward (\mbox{resp.} weakly forward) $\Gamma_{C(\bar x,\bar t)}$-continuous 
at every point 
$(\bar x,\bar t)\in S$ such that $C(\bar x,\bar t)<\infty$.\proofend
\end{definition}

\forlong{
\begin{remark}
The forward $\Gamma_C$-continuity property of Definition \ref{defconcon}
is a ``conic continuity'' condition,
\mbox{because} it says that 
$f(x,t)$ approaches $f(\bar x,\bar t)$ when $(x,t)$ approaches
$(x,t)$ along the cone $\Gamma_C(\bar x,\bar t)$ in $\rr^n\times\rr$. \proofend
\end{remark}
}

\subssection{Conic continuity implies integral continuity}\label{conicimplies}

If $S\subseteq\rr^n\times\rr$, and $t\in\rr$, we write 
$S_t$ to denote the set 
$$
S_t\equaldef\{\,x\in\rr^n:(x,t)\in S\}\,.
$$
\begin{theorem}\label{theoweakly}
Let $n,m\in\nn$, $S\subseteq\rr^n\times\rr$,
$f:S\mapsto\rr^m$. Let $C$, $\psi$ be $\rr_+$-valued
functions on $S$ such that, 
%\xdef\implo{\equanosectno}
\begin{equation}\label{implo}
{\rm for\,\,\,a.e.\,\,\,}t\in\rr\,,\,\,\,
\psi(x,t)<C(x,t)<\infty\,\,\,\,{\rm for\,\,\,\,all\,\,\,}x\in S_t\,.
\end{equation}
Assume that $f$ is ${\cal EM}(\rr^n,\rr)\cap S$-measurable, 	
locally integrably bounded,  and 
weakly forward $\Gamma_C$-continuous. 
Then $f$ is integrally continuous on the set
$\widetilde{ARC}_\psi(\rr^n;S)$.
\end{theorem}


\def\implo{\ref{implo}}
\begin{proof}
It follows from Fact \ref{iflib} that $f$ is 
arc-integrable on $ARC(\rr^n;S)$, since $f$ is LIB and
${\cal EM}(\rr^n,\rr)\cap S$-measurable.

To prove integral continuity, we use Proposition \ref{remquasi}, 
and show that (SIC) holds. We will actually prove the stronger
fact that
\begin{itemize}
\item[(\#)] {\em if $\{\xi_j\}_{j\in\nn}$ is a sequence
in  $ARC(\rr^n;S)$ that converges in $ARC(\rr^n)$ to a \mbox{limit}
$\xi_\infty\in\widetilde{ARC}_\psi(\rr^n;S)$, then 
\begin{equation}\label{itholfore}
\int_{E\cap\Do(\xi_j)}f(\xi_j(s),s)\,ds\to
\int_{E\cap\Do(\xi_\infty)}f(\xi_\infty(s),s)\,ds
\end{equation}
for every measurable subset $E$ of $\rr$.}
\end{itemize}
For this purpose, we fix a sequence $\{\xi_j\}_{j\in\nn}$ in $ARC(\rr^n;S)$ 
that converges in $ARC(\rr^n)$ to a 
$\xi_\infty\in\widetilde{ARC}_\psi(\rr^n;S)$.
We write $a_j=a_{\xi_j}$, $b_j=b_{\xi_j}$, 
\mbox{$I_j=\Do(\xi_j)=[a_j,b_j]$}, for $j\in\overline{\nn}$. 

\medbreak Since $\xi_j\to\xi_\infty$ in $ARC(\rr^n;S)$, the sequences
$\{a_j\}_{j\in\nn}$ and $\{b_j\}_{j\in\nn}$ \mbox{converge} to the limits
$a_\infty$, $b_\infty$, respectively. So we can pick a compact interval
\mbox{$J=[a,b]$} such that $a\le a_j\le b_j\le b$ for all 
$j\in\overline{\nn}$. 
It is then clear that it suffices to prove that (\ref{itholfore})\
holds for all measurable subsets of $J$, since 
$\int_{E\cap I_j}f(\xi_j(s),s)\,ds=
\int_{(E\cap J)\cap I_j}f(\xi_j(s),s)\,ds$ 
for all $j\in\overline{\nn}$. 


For each $\eps$ such that $\eps>0$, let ${\cal E}(\eps)$ be the set of all 
measurable $E\subseteq J$ having  the property that there exists an
$N=N(\eps,E)\in\nn$ such that
$$
\Big\|\int_{E\cap I_j}f\compo\xi_j-\int_
{E\cap I_\infty}f\compo\xi_\infty\Big\|\le\eps\,\msr(E)
$$
for $j\ge N$.  It
is clear that ${\cal E}(\eps)$ is closed under finite disjoint unions.

Now fix $\eps$ and a measurable subset $E$ of $J$, and let 
${\cal H}(\eps,E)$ denote the collection of
all subsets ${\cal U}$ of ${\cal E}(\eps)$ such that 
\begin{itemize}
\item[(i)] every member $U$ of ${\cal U}$ is a measurable
subset of $J$ such that $U\subseteq E$ and $\msr(U)>0$, 
\item[(ii)] $U_1\cap U_2=\emptyset$ whenever 
$U_1,U_2\in{\cal U}$ and $U_1\neq U_2$.
\end{itemize}
It is clear that every ${\cal U}\in {\cal
H}(\eps,E)$ is finite or countable, since the members of ${\cal U}$ are
pairwise disjoint measurable subsets of $J$ of strictly positive measure.
Moreover, it is clear that every increasing union of members of ${\cal
H}(\eps,E)$ is in ${\cal H}(\eps,E)$. Finally,
${\cal H}(\eps,E)\neq\emptyset$,
since $\emptyset\in{\cal H}(\eps,E)$.  It then follows
{}from Zorn's Lemma that there is a ${\cal
U}\in{\cal H}(\eps,E)$ which is maximal with respect to inclusion.

We now show that 
%\xdef\sumu{\equanosectno}
\begin{equation}\label{sumu}
\sum_{U\in{\cal U}}\msr(U)=\msr(E)\,.
\end{equation}
\def\sumu{\ref{sumu}}
Suppose that (\ref{sumu}) does not hold. Let 
$$
\hat U=\bigcup\{U:U\in{\cal U}\}\,.
$$
Then $\hat U$ is measurable and satisfies
\begin{equation}
\hat U\subseteq E\qquad{\rm and}\qquad
\msr(\hat U)=\sum_{U\in{\cal U}}\msr(U)<\msr(E)\,,
\end{equation}
because the members of ${\cal U}$ are pairwise disjoint
measurable subsets of $E$. 

Let $V=E\backslash\hat U$. Then $\msr(V)>0$ and $V\subseteq E$.
Let $V'=V\backslash I_\infty$. We claim that 
\begin{equation}\label{sumu-gg}
\msr(V')=0\,.
\end{equation}
Indeed, assume that $\msr(V')>0$. Then there exist a compact interval 
$L$ such that $L\cap I_\infty=\emptyset$ and $\msr(V'\cap L)>0$. 
Since $a_j\to a_\infty$ and $b_j\to b_\infty$, there must 
exist a $j^*\in\nn$ such that
$L\cap I_j=\emptyset$ whenever $j\in\overline{\nn}$ and $j\ge j^*$. 
Then $\int_{V'\cap L\cap I_j}f\compo\xi_j=0$ if 
$j\in\overline{\nn}$ and $j\ge j^*$. Therefore 
$$
\lim_{j\to\infty}\int_{V'\cap L\cap I_j}f\compo\xi_j=
\int_{V'\cap L\cap I_\infty}f\compo\xi_\infty\,.
$$
Hence $V'\cap L\in{\cal E}(\eps)$. Since $V'\cap U=\emptyset$ 
whenever $U\in{\cal U}$, $V'\cap L\subseteq E$, and $\msr(V'\cap L)>0$, 
the set ${\cal U}\cup\{V'\cap L\}$ also belongs to ${\cal H}(\eps,E)$, 
contradicting the maximality of ${\cal U}$. This contradiction shows that
(\ref{sumu-gg}) holds. 

Now let $V_1=V\cap I_\infty$. Then $V=V'\cup V_1$, so (\ref{sumu-gg})
implies that $\msr(V_1)>0$. Let $V_2$ be the set of all $t\in\rr$
such that 
$$
a_\infty<t<b_\infty\,,
$$
$$
\psi(x,t)<C(x,t)<\infty\qquad{\rm for\,\,all}\quad x\in S_t
$$
and
$$
\lim\sup_{t'\to t+}\frac{\|\xi_\infty(t')-\xi_\infty(t)
\|}{t'-t}\le\psi(\xi_\infty(t),t)\,.
$$

It then follows from (\ref{implo}) and the fact that
$\xi_\infty\in\widetilde{ARC}_\psi(\rr^n;S)$ that
$\msr(I_\infty\backslash V_2)=0$, so $\msr(V_1\cap V_2)=\msr(V_1)>0$. 
We can therefore pick a point of density $\bar t$ of $V_1\cap V_2$. 

Let $\bar x=\xi_\infty(\bar t)$. 
Set $\bar C=C(\bar x,\bar t)$, $\bar B=\psi(\bar x,\bar t)$, and choose
$D$ such that $\bar B<D<\bar C$. 
Using the weak $\Gamma_{\bar C}$-continuity  of $f$ at 
$(\bar x,\bar t)$, find a $\delta>0$ such that 
\begin{eqnarray}
&\,\,&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
\Big(\,\bar t<t\le\bar t+\delta\wedge\|x-\bar x\|\le C(t-\bar t)\wedge
\|x'-\bar x\|\le C(t-\bar t)\,\Big)
\nonumber\\
&\Longrightarrow&\|f(x,t)-f(x',t)\|\le\eps\,.
\end{eqnarray}
By making $\delta$ smaller, if necessary, we may assume that 
%\xdef\densio{\equanosectno}
\begin{equation}\label{densio-gug}
a_\infty<\bar t-\delta<\bar t+\delta<b_\infty\,,
\end{equation}
\begin{equation}\label{densio}
\msr\Bigl(V_1\cap V_2\cap[\bar t-\delta,\bar t+\delta]\Bigr)
\ge{3\over 2}\delta\,,
\end{equation}
and 
\begin{equation}
\|\xi_\infty(t)-\xi_\infty(\bar t)\|\le 
D(t-\bar t)\,\,\,{\rm whenever}\,\,\,\bar t<t\le\bar t+\delta\,.
\end{equation}

\medbreak\noindent Let $\rho={\delta\over 3}$. Pick $N\in\nn$ such that 
$$
\|\xi_j(t)-\xi_\infty(t)\|\le (C-D)\rho\qquad{\rm for}\quad
\bar t\le t\le\bar t+\delta\,,\,j\in\nn\,,\,j\ge N\,.
$$
Then,  if $j\in\overline{\nn}$, $j\ge N$ and 
$\bar t+\rho\le t\le\bar t+\delta$,  
we have 
\begin{eqnarray*}
\|\xi_j(t)-\bar x\|&\le&\|\xi_j(t)-\xi_\infty(t)\|+\|\xi_\infty(t)-
\bar x\|\\
&\le&(C-D)\rho+D(t-\bar t)\\
&\le& C(t-\bar t)\,.
\end{eqnarray*}
Therefore
$$
\|f(\xi_j(t),t)-f(\xi_\infty(t),t)\|\le\eps\,\,\,\,{\rm whenever}
\,\,\bar t+\rho\le t\le \bar t+\delta\,,\,\,j\in\nn\,,\,\,j\ge N\,.
$$
So, if $j\ge N$, the inequality  
$$
\int_{V_1\cap V_2\cap[\bar t+\rho,\bar t+\delta]}\|f(\xi_j(t),t)-f(\xi_\infty
(t),t)\|\,dt\le\eps\,\msr(V_1\cap V_2\cap[\bar t+\rho,\bar t+\delta])\,.
$$
holds. On the other hand,   
\begin{equation}
\msr(V_1\cap V_2\cap[\bar t+\rho,\bar t+\delta])>0\,,
\end{equation}
because if $\msr(V_1\cap V_2\cap [\bar t+\rho,\bar t+\delta])=0$ 
then we would have
\begin{eqnarray*}
\msr(V_1\cap V_2\cap[\bar t-\delta,\bar t+\delta])&=&
\msr(V_1\cap V_2\cap[\bar t-\delta,\bar t+\rho])\\
&\le&\delta+\rho\\
&=&{4\delta\over 3}\,,
\end{eqnarray*}
contradicting (\ref{densio}). 
So, if we define $W=V_1\cap V_2\cap[\bar t+\rho,\bar t+\delta]$, we see that
$W\in{\cal E}(\eps)$ and $\msr(W)>0$.  Moreover, 
$W\cap U=\emptyset$ whenever $U\in{\cal U}$. Therefore
${\cal U}\,\cup\{W\}\in{\cal H}(\eps,E)$, contradicting the maximality of 
${\cal U}$.  So (\sumu) is proved.

Given any $\delta>0$, (\sumu) implies that we can find a finite subset 
${\cal V}$ of ${\cal U}$ such that 
\begin{equation}
\sum_{U\in{\cal V}}\msr(U)\ge \msr(E)-\delta\,.
\end{equation}
If we let 
$V=\bigcup\{U:U\in{\cal V}\}$, we see that $V\in{\cal E}(\eps)$.
So we have shown that 

\begin{itemize}
\item[(A)] {\sl If $E$ is a measurable subset of $J$ and $\eps>0$, 
$\delta>0$, then there is a measurable subset $V$ of $E$ such that 
$\msr(E\backslash V)\le\delta$ and
$$
\Big\|\int_Vf\compo\xi_j-\int_Vf\compo\xi_\infty\Big\|\le\eps\,\msr(V)
$$
for all sufficiently large $j$.}
\end{itemize}

\medbreak\noindent Now let 
$$
K=\bigcup_{j\in\overline{\nn}}\Big\{\,(\xi_j(t),t):
a_j\le t\le b_j\,\Big\}\,.
$$
Then $K$ is a compact subset of $S$. Using the fact that $f$ is LIB, 
choose an integrable function $\ffi:\rr\mapsto[0,+\infty]$ such that 
$\|f(x,t)\|\le\ffi(t)$ whenever $(x,t)\in K$.
Given $\eps$, choose $\delta$ such that $\int_L\ffi<\eps$
whenever $L$ is a measurable subset of $\rr$ such that 
$\msr(L)\le\delta$.   If we apply (A) with this choice of $\delta$, we see
that 
$$
\Big\|\int_Ef\compo\xi_j-\int_Ef\compo\xi_\infty\Big\|\le\eps(2+\msr(E))
$$ 
for all sufficiently large $j$. Since $\eps$ is arbitrary, we 
can conclude  that 
\begin{equation}
\Bigl\|\int_Ef\compo\xi_j-\int_Ef\compo\xi_\infty\Bigr\|\to 0\,\,\,{\rm
as}\,\,\,j\to\infty\,,
\end{equation} 
and the proof is complete.\proofend
\end{proof}

\subssection{Time-varying vector fields}\label{disctvvf}

\begin{definition} Let $n\in\nn$. 
A {\em time-varying vector field} (abbr. ``TVVF'') on $\rr^n$
is a---possibly partially defined---single-valued  map from  $\rr^n\times\rr$
to $\rr^n$. We use $\cvf(\rr^n)$ to denote the set of all TVVFs on 
$\rr^n$. If $S$ is a subset of $\rr^n\times\rr$, we use 
$\cvf(\rr^n;S)$ to denote the set of all TVVFs on $\rr^n$ whose domain
is $S$.\proofend
\end{definition}
A TVVF on $\rr^n$ is therefore a set-valued function 
$f:\rr^n\times\rr\tto\rr^n$ that happens to be single-valued.
It follows that all the definitions introduced in \S\ref{sectrajflow}
for differential inclusions $\dot x\in F(x,t)$,
and all the results stated there,  apply to TVVFs. 
So we have a well defined concept of ``trajectory of a TVVF $f$,''
or ``trajectory of the ordinary differential equation
$\dot x=f(x,t)$,'' and every TVVF $f$ has a {\em flow} $\Phi^f$.

We now prove some elementary estimates for LIB TVVFs.

If $\bar x\in\rr^n$, $\bar t\in\rr$, and $\delta>0$, we write
\begin{eqnarray*}
{\cal N}_{\delta}(\bar x,\bar t)&\equaldef&
\Big\{\,(x,t)\in\rr^n\times\rr:\|x-\bar x\|\le\delta\,\wedge
\,\bar t-\delta\le t\le\bar t+\delta\,\Big\}\,,\\
{\cal N}_{\delta}^{(2)}(\bar x,\bar t)
&\equaldef&\Big\{\,(x,t,s):(x,t)\in{\cal N}_{\delta}(\bar x,\bar t)\wedge
(x,s)\in{\cal N}_{\delta}(\bar x,\bar t)\,\Big\}\,.
\end{eqnarray*}

\begin{lemma}\label{lemismall}  
Let $\ffi:\rr\mapsto[0,+\infty]$ be an integrable function.
Let \mbox{$n\in\nn$}, and let $f\in TVVF(\rr^n)$. Let 
$\delta_1,\delta$ be  positive numbers such that
\begin{eqnarray}
\delta+\int_{\bar t-\delta}^{\bar t+\delta}\ffi&<&\delta_1\,,
\label{charadelta}  
\end{eqnarray}
and
\begin{equation}\label{charadeltaone}
\|f(x,t)\|\le\ffi(t)\qquad{\rm whenever}\quad(x,t)\in
{\cal N}_{\delta_1}(\bar x,\bar t)\cap\Do(f)\,.
\end{equation}
Then 
\begin{equation}\label{concludelta}
\|y\!-\!\bar x\|\le\delta_1\,\,\,{\rm whenever}\,\,\,
y\!\in\!\Phi^f(x,t,s)\,\,\,{\rm for\,\,\,some}
\,\,\,(x,t,s)\!\in\!{\cal N}_{\delta}^{(2)}(\bar x,\bar t)\,.
\end{equation}
\end{lemma}
\begin{proof}
Pick $x,t,s,y$ such that $(x,t,s)\in{\cal N}_{\delta}^{(2)}(\bar x,\bar t)$
and $y\in\Phi^f(x,t,s)$. Let $\zeta$ be a trajectory of
$f$ such that $\zeta(s)=x$ and $\zeta(t)=y$. Let 
$I$ be the compact interval $[\min(s,t),\max(s,t)]$. Let
$\xi$ be the restriction of $\zeta$ to $I$. Then
$I\subseteq[\bar t-\delta,\bar t+\delta]$, and
$\xi\in C^0(I,\rr^n)$. 

Let $L$ be the set of those $\tau\in I$ such that $\|\xi(\sigma)-\bar
x\|\le\delta_1$ for all $\sigma$ that lie between $s$ and $\tau$.
Then $L$ is a compact interval, and $s\in L\subseteq I$. So
$L=[\tau,\sigma]$, where $\tau\in I$, $\sigma\in I$, and
$s\in\{\tau,\sigma\}$.

Clearly, $(\xi(r),r)\in{\cal N}_{\delta_1}(\bar x,\bar t)$ whenever 
$r\in L$. Moreover, $(\xi(r),r)$ belongs to $\Do(f)$ and
$\dot\xi(r)=f(\xi(r),r)$ for almost all $r\in L$. 

It follows that, if $r\in L$, then
\begin{eqnarray*}
\|\xi(r)-\bar x\|&\le&\|\xi(r)-x\|+\|x-\bar x\|\\
&=&\|\xi(r)-\xi(s)\|+\|x-\bar x\|\\
&=&\Big\|\int_{s}^r f(\xi(u),u)\,du\Big\|+\|x-\bar x\|\\
&\le&\int_{\bar t-\delta}^{\bar t+\delta}\ffi(u)\,du+\delta\\
&<&\delta_1\,.
\end{eqnarray*}
We know that one of the endpoints of $L$ is $s$. Let 
$\rho$ be the other endpoint. Assume that $\rho\neq t$. Then
$\rho$ is an interior point of $I$. But \mbox{$\|\xi(\rho)-\bar x\|<\delta_1$}, 
because $\rho\in L$. Therefore there exists a number $\eps$
such that $\eps>0$, \mbox{$[\rho-\eps,\rho+\eps]\subseteq I$}, and 
$\|\xi(t)-\bar x\|<\delta_1$ whenever $t\in[\rho-\eps,\rho+\eps]$.
But then the interval $[\rho-\eps,\rho+\eps]$ must be entirely
contained in $L$, contradicting the fact that $\rho$ was a boundary
point of $L$. This contradiction shows that the assumption that
$\rho\neq t$ is false. So $\rho=t$, and then $L=I$.  Then
$t\in L$, so $\|y-\bar x\|=\|\xi(t)-\bar x\|\le\delta_1$.\proofend
\end{proof}

The following continuity property of the flow
is a trivial consequence of Lemma \ref{lemismall}:

\begin{corollary}\label{limfo}
Let $n\in\nn$, $f\in TVVF(\rr^n)$.
Assume that $f$ is LIB. \mbox{Assume} also that for some positive 
number $\delta$ the restriction of $f$ to 
\mbox{${\cal N}_{\delta}(\bar x,\bar t)\cap\dom(f)$} is integrably bounded.
Then 
\begin{equation}\label{ifnotoho}
\lim_{(x,t,s)\to(\bar x,\bar t,\bar t)\,,\,\Phi^f(x,t,s)\neq\emptyset}
\sup\Big\{\|y-\bar x\|:y\in\Phi^f(x,t,s)\Big\}=0\,.
\end{equation}
\end{corollary}
\begin{proof}
Let $\eps>0$. Pick $\delta_1$ such that $0<\delta_1\le\eps$ and 
the restriction of $f$ to 
\mbox{${\cal N}_{\delta_1}(\bar x,\bar t)\cap\dom(f)$} is integrably bounded.
Then choose an integrable $\ffi:\rr\mapsto[0,+\infty]$ such that
(\ref{charadeltaone}) is true, and pick $\delta$ for which
(\ref{charadelta}) is satisfied. Then 
Lemma \ref{lemismall} implies that 
$\|y-\bar x\|\le\eps$ whenever $\|x-\bar x\|\le\delta$, 
$\|t-\bar t\|\le\delta$, $\|s-\bar t\|\le\delta$, and
$y\in\Phi^f(x,t,s)$. So (\ref{ifnotoho}) is true.\proofend
\end{proof}

\subssection{Admissible time-varying vector fields}\label{sec-adm}

We will be interested in time-varying vector fields 
$f\in\cvf(\rr^n)$ that satisfy the following property:

\begin{itemize}\item[]\begin{itemize}
\item[(AD)] {\em The map $f$ is 
${\cal EM}(\rr^n,\rr)\cap\Do(f)$-measurable, and for \mbox{every} 
compact \mbox{subset}
$K$ of $\Do(f)$ there exists an \mbox{integrable} \mbox{function}
\mbox{$\ffi:\rr\mapsto[0,+\infty]$} such that
\begin{itemize}
\item[{\rm (AD.i)}] $\|f(x,t)\|\le\ffi(t)$ for all $(x,t)\in K$,
\item[{\rm (AD.ii)}] $f$ is integrally continuous on $ARC_\ffi(\rr^n;K)$.
\end{itemize}
}\end{itemize}\end{itemize}

\begin{definition}\label{def-adm}
A time-varying vector field on $\rr^n$ for which (AD) 
\mbox{holds} will be called
{\em admissible}.  We use $ADM(\rr^n)$ to denote the class
of \mbox{all} \mbox{admissible} time-varying vector fields on $\rr^n$. If
$S$ is a subset of $\rr^n\times\rr$, we use $ADM(\rr^n;S)$ to denote the class
of all $f\in ADM(\rr^n)$ such that $\Do(f)=S$. \proofend
\end{definition}

\begin{definition}
Let $n\in\nn$, $f\in\cvf(\rr^n)$. We call $f$ {\em trajectory-compact}
if the set $\traj_c(f,K)$ is compact for every compact subset $K$ of 
$\dom(f)$.\proofend
\end{definition}

In addition to the sets ${\cal N}_{\delta}(\bar x,\bar t)$,
${\cal N}_{\delta}^{(2)}(\bar x,\bar t)$   defined earlier, we
introduce their ``one-sided'' versions:
\begin{eqnarray*}
{\cal N}_{\delta,+}(\bar x,\bar t)&\equaldef&
\Big\{\,(x,t)\in\rr^n\times\rr:\|x-\bar x\|\le\delta\,\wedge
\,\bar t\le t\le\bar t+\delta\,\Big\}\,,\\
{\cal N}_{\delta,-}(\bar x,\bar t)&\equaldef&
\Big\{\,(x,t)\in\rr^n\times\rr:\|x-\bar x\|\le\delta\,\wedge
\,\bar t-\delta\le t\le\bar t\,\Big\}\,,\\
{\cal N}_{\delta,+}^{(2)}(\bar x,\bar t)
&\equaldef&\Big\{\,(x,t,s):(x,t)\in{\cal N}_{\delta,+}(\bar x,\bar t)\wedge
(x,s)\in{\cal N}_{\delta,+}(\bar x,\bar t)\,\Big\}\,,\\
{\cal N}_{\delta,-}^{(2)}(\bar x,\bar t)
&\equaldef&\Big\{\,(x,t,s):(x,t)\in{\cal N}_{\delta,-}(\bar x,\bar t)\wedge
(x,s)\in{\cal N}_{\delta,-}(\bar x,\bar t)\,\Big\}\,.
\end{eqnarray*}

\begin{definition}
Assume that $n\in\nn$, $f\in\cvf(\rr^n)$, $S=\dom(f)$, 
and \mbox{$(\bar x,\bar t)\in S$}. 
We say that $f$ is {\em flow-upper semicontinuous} (resp.
{\em \mbox{forward} flow-upper \mbox{semicontinuous}},  {\em backward 
flow-upper semicontinuous}) near $(\bar x,\bar t)$
if there \mbox{exists} a $\delta\in\rr$ such that 
$\delta>0$ and
the restriction to the set ${\cal N}_\delta^{(2)}(\bar x,\bar t)$
(resp. ${\cal N}_{\delta,+}^{(2)}(\bar x,\bar t)$, 
${\cal N}_{\delta,-}^{(2)}(\bar x,\bar t)$) of the
set-valued map $\Phi^f$ is upper \mbox{semicontinuous} and has nonempty 
compact values.\proofend
\end{definition}

\begin{theorem}\label{theorexist}   
Let $n\in\nn$, $f\in ADM(\rr^n)$, $S=\Do(f)$. Then $f$ is trajectory-compact.
Moreover, 

\begin{itemize}\item[]\begin{itemize}
\vspace{-1.5mm}
\item[(T\ref{theorexist}.i)] $f$ is flow-upper semicontinuous at every 
interior point $(\bar x,\bar t)$ of S, 

\vspace{-1.2mm}
\item[(T\ref{theorexist}.ii)] $f$ is forward flow-upper semicontinuous at 
every point $(\bar x,\bar t)$ such that 
${\cal N}_{\delta,+}(\bar x,\bar t)\subseteq S$ for some positive
$\delta$, 

\vspace{-1.2mm}
\item[(T\ref{theorexist}.iii)] $f$ is backward flow-upper semicontinuous 
at every point $(\bar x,\bar t)$ such that 
${\cal N}_{\delta,-}(\bar x,\bar t)\subseteq S$ for some positive
$\delta$.
\end{itemize}\end{itemize}
\end{theorem}

\begin{proof} We first prove trajectory-compactness.  Let $K\subseteq S$ be 
compact, and let $\{\xi_j\}_{j\in\nn}$ be a sequence in $\traj_c(f,K)$. 
Let $\ffi:\rr\to[0,+\infty]$ be integrable
and such that $\|f(x,t)\|\le\ffi(t)$ for all $(x,t)\in K$
and the integral map ${\cal T}_f^{ARC_\ffi(\rr^n;K)}:
ARC_\ffi(\rr^n;K)\mapsto ARC(\rr^n)$ is continuous. 

Then the arcs $\xi_j$ are entirely contained in $K$. Moreover,
the $\xi_j$ are \mbox{absolutely} continuous and satisfy 
$\|\dot\xi_j(t)\|\le\ffi(t)$ for almost all $t\in\Do(f)$.
Therefore the sequence $\{\xi_j\}_{j\in\nn}$ is uniformly bounded 
and equicontinuous, so
there is a subsequence $\{\xi_{j(\ell)}\}_{\ell\in\nn}$ that converges
in $ARC(\rr^n)$ to an \mbox{arc} $\xi_\infty$.   
Then $\xi_\infty\in ARC(\rr^n;K)$, since $K$ is compact.
Moreover, if we write $\dom(\xi_j)=[a_j,b_j]$ for $j\in\overline{\nn}$,
then $a_{j(\ell)}\to a_\infty$, $b_{j(\ell)}\to b_\infty$,
and \mbox{$\|\xi_{j(\ell})(t)-\xi_{j(\ell)}(s)\|\le\int_s^t\ffi$} whenever
$j\in\nn$ and $a_j\le s\le t\le b_j$. It \mbox{follows}
that $\|\xi_\infty(t)-\xi_\infty(s)\|\le\int_s^t\ffi$ whenever
$a_\infty\le s\le t\le b_\infty$. Therefore 
\mbox{$\xi_\infty\in ARC_\ffi(\rr^n;K)$}.

Let $\eta_j={\cal T}_f^{ARC_\ffi(\rr^n;K)}$ for $j\in\overline{\nn}$.
Then $\eta_j\in ARC(\rr^n)$, and the 
\mbox{continuity} of ${\cal T}_f^{ARC_\ffi(\rr^n;K)}$ on
$ARC_\ffi(\rr^n;K)$ implies that 
$\eta_{j(\ell)}\to\eta_\infty$ as $\ell\to\infty$.
Since $\xi_j\in\traj_c(f)$ for $j\in\nn$, the equalities
$$
\xi_j(t)=\xi_j(a_j)+\int_{a_j}^tf(\xi_j(s),s)\,ds=
\xi_j(a_j)+\eta_j(t)
$$
hold if $j\!\in\!\nn$, $a_j\!\le\!t\!\le\!b_j$. Taking $j\!=\!j(\ell)$ and letting
$\ell\!\to\!\infty$, we see that 
$$
\xi_\infty(t)=\xi_\infty(a_\infty)+\eta_\infty(t)
=\xi_\infty(a_\infty)+\int_{a_\infty}^tf(\xi_\infty(s),s)\,ds
$$
if $a_\infty\!\le\!t\!\le\!b_\infty$. So $\xi_\infty\!\in\!\traj_c(f,K)$,
proving that $f$ is trajectory-compact.

We now prove Conclusion (T\ref{theorexist}.i). Pick an interior
point $(\bar x,\bar t)$ of $S$.  For $\delta\in\rr$, $\delta>0$,
write ${\cal K}(\delta)\equaldef{\cal N}_{\delta}(\bar x,\bar t)$.
Let $\delta_1$ be  such that $\delta_1>0$ and 
${\cal K}(\delta_1)\subseteq S$.
Let $\ffi$ be an integrable function on $\rr$ such that 
$\|f(x,t)\|\le\ffi(t)$ whenever $(x,t)\in{\cal K}(\delta_1)$
and $f$ is integrally continuous on $ARC_\ffi(\rr^n;{\cal K}(\delta_1))$.
Let $\delta$ be such that $\delta>0$ and (\ref{charadelta}) holds.
Let $G$ be the graph of the restriction to 
${\cal N}_{\delta}^{(2)}(\bar x,\bar t)$ of the set-valued map $\Phi^f$, 
so that
$$
G=\Big\{\,(x,t,s,y)\!\in\!\rr^n\!\times\!\rr\!\times\!\rr\!\times\!\rr^n
\!:\!y\!\in\!\Phi^f(x,t,s),\,(x,t,s)\!\in\!{\cal N}_\delta^{(2)}
(\bar x,\bar t)\,\Big\}\,.
$$
We will prove that $G$ is compact. For this purpose, we pick a
sequence $\{(x_j,t_j,s_j,y_j)\}_{j\in\nn}$ of points of $G$, and
show that it has a subsequence that converges to a point 
$(x,t,s,y)\in G$.  It is clear that the sequences $\{x_j\}_{j\in\nn}$, 
$\{t_j\}_{j\in\nn}$, $\{s_j\}_{j\in\nn}$, are bounded. We may
therefore assume that the limits
$x_\infty=\lim_{j\to\infty}x_j$, $t_\infty=\lim_{j\to\infty}t_j$,
$s_\infty=\lim_{j\to\infty}s_j$, exist.

For each $j\in\nn$, let $\zeta_j$ be a trajectory of $f$ such that 
$\zeta_j(s_j)=x_j$ and $\zeta_j(t_j)=y_j$.  Let $I_j$ be the 
compact interval $[\min(s_j,t_j),\max(s_j,t_j)]$, and let
$\xi_j$ be the restriction of $\zeta_j$ to $I_j$. Then 
$\xi_j$ is a trajectory of $f$, $\xi_j(s_j)=x_j$, $\xi_j(t_j)=y_j$,
and $\dom(\xi_j)=I_j$. If $t\in I_j$, then 
$(x_j,t,s_j)\in{\cal N}_{\delta}^{(2)}(\bar x,\bar t)$, and 
Lemma \ref{lemismall} implies that $\|\xi_j(t)-\bar x\|\le\delta_1$,
so $(\xi_j(t),t)\in{\cal K}(\delta_1)$.

It follows that $\xi_j\in\traj_c(f,{\cal K}(\delta_1))$ 
for every $j\in\nn$.  Since $f$ is trajectory-compact, 
there is a subsequence $\{\xi_{j(\ell)}\}_{\ell\in\nn}$ of 
$\{\xi_{j}\}_{j\in\nn}$ that converges in 
$ARC(\rr^n)$ to a limit $\xi_\infty\in 
\traj_c(f,{\cal K}(\delta_1))$.  Let $y_\infty=\xi_\infty(t_\infty)$. 
Then 
$$
\lim_{\ell\to\infty}y_{j(\ell)}=
\lim_{\ell\to\infty}\xi_{j(\ell)}(t_{j(\ell)})
=\xi_\infty(t_\infty)=y_\infty\,,
$$
so $\lim_{\ell\to\infty}(x_{j(\ell)},t_{j(\ell)},s_{j(\ell)},y_{j(\ell)})
=(x_\infty,t_\infty,s_\infty,y_\infty)$. Moreover, 
$$
\xi_\infty(s_\infty)=\lim_{\ell\to\infty}\xi_{j(\ell)}(s_{j(\ell)})=
\lim_{\ell\to\infty}x_{j(\ell)}=x_\infty\,.
$$
Therefore $y_\infty\!\in\!\Phi^f(x_\infty,t_\infty,s_\infty)$.
Since $(x_j,t_j,s_j)\!\in\!{\cal N}_\delta^{(2)}(\bar x,\bar t)$ for every 
\mbox{$j\!\in\!\nn$}, it is clear that 
$(x_\infty,t_\infty,s_\infty)\in{\cal N}_\delta^{(2)}(\bar x,\bar t)$.
Therefore 
$(x_\infty,t_\infty,s_\infty,y_\infty)\in G$.
This completes the proof that $G$ is compact. 
Equivalently, we have shown that
the set-valued map $\Phi^f$ has compact values and is upper
semicontinuous on the set ${\cal N}_\delta^{(2)}(\bar x,\bar t)$. 

To conclude our proof, we have to show (cf. Remark \ref{rem-exist}) that 
\begin{equation}\label{isnonem}
\Phi^f(x,t,s)\neq\emptyset\qquad{\rm whenever}\quad
(x,t,s)\in{\cal N}_\delta^{(2)}(\bar x,\bar t)\,. 
\end{equation}
Fix a point $(x,t,s)$ of ${\cal N}_\delta^{(2)}(\bar x,\bar t)$. 
Let $J=[\bar t-\delta,\bar t+\delta]$. 
Let ${\cal X}$ be the Banach space of all continuous maps $\xi:J\mapsto\rr^n$, 
endowed with the sup norm. 
Let ${\cal C}$ be the subset of ${\cal X}$ consisting of those
$\xi\in{\cal X}$ that are absolutely continuous and satisfy 
$\|\xi(r)-\bar x\|\le\delta_1$ for all $r\in J$ and 
$\|\dot\xi(r)\|\le\ffi(r)$ for a.e. $r\in J$.
Then ${\cal C}$ is a compact convex subset of ${\cal X}$.   
Define a map $\mu:{\cal C}\mapsto{\cal X}$ by letting
\begin{equation}\label{defiofmu}
\mu(\xi)(r)=x+\int_{s}^rf(\xi(u),u)\,du\qquad
{\rm for}\quad\xi\in{\cal C}\,,\,r\in J\,.
\end{equation}
(The map $\mu$ is clearly well defined,
because (i)~if $\xi\in{\cal C}$ and $u\in J$
then $(\xi(u),u)\in{\cal K}(\delta_1)$, so $f(\xi(u),u)$ is defined;
(ii)~the essential measurability
of $f$ implies that the map $J\ni u\mapsto f(\xi(u),u)\in\rr^n$ is measurable, 
and (iii)~the integrable bound
\begin{equation}\label{intebo}
\|f(\xi(u),u)\|\le\ffi(u)
\end{equation}
implies that 
$J\ni u\mapsto f(\xi(u),u)\in\rr^n$ is integrable.)

If  $\xi\in{\cal C}$, and $\eta=\mu(\xi)$, then 
it follows from (\ref{defiofmu})
and (\ref{intebo}) that $\eta$ is absolutely continuous 
and $\|\dot\eta(r)\|\le\ffi(r)$ for all $r\in J$. Moreover,
since $\|x-\bar x\|\le\delta$, (\ref{charadelta}) implies
that $\|\eta(r)-\bar x\|\le\delta_1$ for all $r\in J$. Therefore
$\mu(\xi)\in{\cal C}$. So $\mu$ is a map from ${\cal C}$ to 
${\cal C}$. 

We now prove that the map $\mu$ is continuous. To see this, we write
\mbox{${\cal A}=ARC_\ffi(\rr^n;{\cal K}(\delta_1))$}, and observe
that ${\cal C}\subseteq{\cal A}$. The definition of $\mu$ 
implies the identity 
\begin{equation}\label{defiofmuid}
\mu(\xi)(r)=x\!+\!{\cal T}_f^{{\cal A}}(\xi)(r)\!-\!
{\cal T}_f^{{\cal A}}(\xi)(s)\quad
{\rm whenever}\,\,\,\xi\in{\cal C}\,,\,r\in J\,.
\end{equation}
Since ${\cal T}_f^{{\cal A}}$ is continuous on ${\cal A}$, the continuity
of $\mu$ follows.

By the Schauder Fixed Point Theorem, $\mu$ has a fixed point. 
Clearly, if $\xi$ is a fixed point of $\mu$, then 
\begin{equation}
\xi(r)=x+\int_{s}^tf(\xi(u),u)\,du\,\,\,{\rm for}\,\,\,r\in J\,.
\end{equation}
Then $\xi$ is a trajectory of $f$ defined on $J$, and 
$\xi(s)=x$.  If we let $y=\xi(t)$, then $y\in\Phi^f(x,t,s)$, so 
$\Phi^f(x,t,s)\neq\emptyset$.  This completes the proof of (T\ref{theorexist}.i).

The proofs of (T\ref{theorexist}.ii) and (T\ref{theorexist}.iii) 
are identical, with obvious trivial \mbox{modifications}. Alternatively, 
one can 
derive (T\ref{theorexist}.ii) and (T\ref{theorexist}.iii) directly from 
(T\ref{theorexist}.i) by applying (T\ref{theorexist}.i) to suitably chosen 
TVVFs. For example, to derive (T\ref{theorexist}.ii) we pick
$(\bar x,\bar t)$ and a positive $\delta$ such that 
${\cal N}_{\delta,+}(\bar x,\bar t)\subseteq S$, write
$$
\hat S=\Big(\rr^n\times\,]\!-\infty,\bar t\,[\,\Big)\bigcup\Big(
S\cap\Big(\rr^n\times[\bar t,+\infty\,[\,\Big)\,\Big)\,,
$$
and define a map $g:\hat S\mapsto\rr^n$ by letting
$$
g(x,t)=\left\{\begin{array}{lcl}
f(x,t)&{\rm if}&(x,t)\in S\,\,\,{\rm and}\,\,\,t\ge\bar t\,,\\  
f(\bar x,\bar t)&{\rm if}&x\in\rr^n\,\,\,{\rm and}\,\,\,t<
\bar t\,.\end{array}\right. 
$$
Then $g\in ADM(\rr^n)$, $\Do(g)=\hat S$,  and $(\bar x,\bar t)$ is an interior
point of $\hat S$. So we can apply (T\ref{theorexist}.i) and conclude that 
$\Phi^g$ is upper semicontinuous and has compact nonempty 
values on ${\cal N}_{\delta}^{(2)}(\bar x,\bar t)$ for some
positive $\delta$.  If we restrict $\Phi^g$ to 
${\cal N}_{\delta,+}^{(2)}(\bar x,\bar t)$, (T\ref{theorexist}.ii) follows.
The proof of (T\ref{theorexist}.iii) is similar. \proofend
\end{proof}

\begin{remark}\label{rem-exist}
The assertion that the values of the flow are nonempty (that is,
Statement \ref{isnonem}), is the {\em existence of solutions
theorem} for differential \mbox{equations} $\dot x=f(x,t)$ with
$f\in ADM(\rr^n)$. It is therefore not surprising that
the proof of this statement depends on a fixed point argument.\proofend
\end{remark}

A special case of Theorem \ref{theorexist} is the usual
Carath\'eodory \mbox{existence} theorem\forlong{:}\forshort{.
The proof is a trivial application of the Lebesgue dominated convergence
theorem.}

\begin{corollary}\label{corcar}  
Assume that $n\in\nn$, $\Om$ is open in $\rr^n$, and $I$ is a nonempty
subinterval of $\rr$. Let $S=\Om\times I$. Let $f:S\mapsto\rr^n$ be a
LIB time-varying vector field such that the map $\Om\ni x\mapsto
f(x,t)\in\rr^n$ is continuous for almost every $t\in I$, and the map
$I\ni t\mapsto f(x,t)\in\rr^n$ is measurable for every $x\in\Om$. Then
$f$ is integrally continuous on the space $ARC(\rr^n;S)$, so $f\in
ADM(\rr^n;S)$. In particular,
\begin{itemize}\item[]\begin{itemize}
\vspace{-1.2mm}
\item[(C\ref{corcar}.i)] $f$ is trajectory-compact,

\vspace{-1.2mm}
\item[(C\ref{corcar}.ii)] 
$f$ is flow-upper semicontinuous at every 
$(\bar x,\bar t)\in\Om\times{\rm Int}(I)$, 

\vspace{-1.2mm}
\item[(C\ref{corcar}.iii)] $f$ is forward flow-upper semicontinuous at every 
point $(\bar x,\bar t)$ such that $\bar x\in\Om$ and
$[\bar t,\bar t+\delta]\subseteq I$ for some positive
$\delta$, 

\vspace{-1.2mm}
\item[(C\ref{corcar}.iv)] $f$ is backward flow-upper semicontinuous at every
point $(\bar x,\bar t)$ such that $\bar x\in\Om$ and
$[\bar t-\delta,\bar t]\subseteq I$ for some positive $\delta$.\forshort{\proofend}
\end{itemize}\end{itemize}
\end{corollary}

\forlong{
\begin{proof}
It follows easily from the hypotheses that 
$f$ is \mbox{essentially} measurable. 
Let ${\cal A}=ARC(\rr^n;S)$.  Then Fact \ref{iflib} tells us
that $f$ is arc-integrable on ${\cal A}$. 

Let $\{\xi_j\}_{j\in\nn}$ be a sequence in ${\cal A}$ that converges 
in $ARC(\rr^n)$ to a $\xi_\infty\in{\cal A}$. For $j\in\overline{\nn}$,
let $L_j=\Do(\xi_j)=[a_j,b_j]$, and write 
$\eta_j={\cal T}_f^{{\cal A}}(\xi_j)$. Then 
$\lim_{j\to\infty}a_j=a_\infty$ and 
$\lim_{j\to\infty}b_j=b_\infty$.

Let $K$ be a compact subset of $S$ such that $(\xi_j(t),t)\in K$ 
whenever $j\in\overline{\nn}$ and $a_j\le t\le b_j$. Choose
an integrable function $\ffi$ on $\rr$ such that
$\|f(x,t)\|\le\ffi(t)$ whenever $(x,t)\in K$. Then, 
if $\{t_j\}_{j\in\nn}$ is a sequence such that
$a_j\le t_j\le b_j$ for every $j\in\nn$, and $t_j\to t$
as $j\to\infty$, we have 
\begin{equation}
\eta_j(t_j)=\int_{a_j}^{t_j}f(\xi_j(s),s)\,ds\to\int_{a}^{t}f
(\xi_\infty(s),s)\,ds={\cal T}_f^{\cal A}(a,t,\xi)
\end{equation}
by the Lebesgue Dominated Convergence Theorem, since 
$\xi_j(s)\to\xi_\infty(s)$ for each $s$, and 
$\|f(\xi_j(s),s)\|\le\ffi(s)$ for each $j$ and each $s\in L_j]$.
 
This proves the integral continuity of $f$ on ${\cal A}$. 
Therefore $f$ is admissible, and then all the other
conclusions are consequences of Theorem \ref{theorexist}.\proofend
\end{proof}
}

\subssection{Points of approximate continuity}\label{secpointapp}

\noindent 

\begin{definition}\label{defapproxcont-true}
Assume that $n,m\in\nn$, $S\subseteq\rr^n\times\rr$, 
$f:S\mapsto\rr^m$, and $(\bar x,\bar t)\in S$.   We say that  
$(\bar x,\bar t)$ is a {\em point of approximate \mbox{continuity}} 
(\mbox{abbreviated} PAC) of $f$ if  there exist positive numbers 
$\bar h,\bar\delta$ such that 
\begin{itemize}\item[]\begin{itemize}
\item[(D\ref{defapproxcont-true}.1)] 
$\bmath^n(\bar x,\bar\delta)\times[\bar t-\bar h,\bar t+\bar h]\subseteq S$,
\item[(D\ref{defapproxcont-true}.2)] there exist
measurable functions $\sigma_\delta:[\bar t-\bar h,\bar t+\bar h]\mapsto
[0,\infty]$, for $0<\delta<\bar\delta$, such  that 
\begin{itemize}
\item[(D\ref{defapproxcont-true}.2.1)] 
the bound $\,\|f(x,t)-f(\bar x,\bar t)\|\le\sigma_\delta(t)\,$ 
holds whenever $\,(x,t)\,$ belongs $\,$to 
\mbox{$\bmath^n(\bar x,\delta)\times[\bar t-\bar h,\bar t+\bar h]$}
\item[(D\ref{defapproxcont-true}.2.2)] 
$\lim_{\delta\downarrow 0\,,\,h\downarrow 0}\frac{1}{h}
\int_{\bar t-h}^{\bar t+h}\sigma_\delta(t)\,dt=0$.
\end{itemize}\end{itemize}\end{itemize}
The concepts of a {\em point of forward approximate continuity} 
and a {\em point of backward approximate continuity} are defined similarly, 
with the interval \mbox{$[\bar t-\bar h,\bar t+\bar h]$}
replaced by $[\bar t,\bar t+\bar h]$ and $[\bar t-\bar h,\bar t]$, 
respectively.\proofend
\end{definition}
\begin{proposition}\label{defapproxcont}
Assume that $n,m\in\nn$, $S\subseteq\rr^n\times\rr$, 
$f:S\mapsto\rr^m$, and $(\bar x,\bar t)\in S$.   Assume that   
there exist $K$, $E$, $\ffi$ such that 
\begin{itemize}\item[]\begin{itemize}
\item[(P\ref{defapproxcont}.a)] 
$E\in{\rm Leb}(\rr)$, $K$ is a compact subset of $S$, 
\mbox{$\ffi:\rr\mapsto[0,+\infty]$} is an integrable function, 
$(\bar x,\bar t)$  is an \mbox{interior} point of $K$, 
and $\bar t$ is a point of density of $E$ and
a Lebesgue point of $\ffi$, 
\item[(P\ref{defapproxcont}.b)] $\lim_{_{(x,t)\to(\bar x,\bar t)\,,\,
(x,t)\in K\,,\,t\in E}}f(x,t)=f(\bar x,\bar t)$;
\item[(P\ref{defapproxcont}.c)] $\|f(x,t)\|\le\ffi(t)$ for all
$(x,t)\in K$. 
\end{itemize}\end{itemize}
Then $(\bar x,\bar t)$ is a point of approximate continuity of $f$.\proofend 
\end{proposition}
\begin{proof}
Pick positive numbers $\bar\delta$, $\bar h$ such that
$\bmath^n(\bar x,\bar\delta)\times[\bar t-\bar h,\bar t+\bar h]\subseteq K$.
For $0<\delta\le\bar\delta$, $\bar t-\bar h\le t\le\bar t+\bar h$,
define $\sigma_\delta(t)$ by 
$$
\sigma_\delta(t)\!=\!\left\{\begin{array}{lcl}
\!\sup\Big\{\|f(x,s)\!-\!f(\bar x,\bar t)\|\!:\!\|x\!-\!\bar x\|\!\le\!\delta,
|s\!-\!\bar t|\!\le\!|t\!-\!\bar t|,s\!\in\!E\Big\}&\!{\rm if}\!&
\!t\!\in\! E,\\
\!\ffi(t)+\ffi(\bar t)&\!{\rm if}\!&\!t\!\notin\! E.\end{array}\right.
$$
Then $\sigma_\delta$ is measurable, because it is 
monotonically nondecreasing on the set $E\cap[\bar t,\bar t+\bar h]$
and monotonically nonincreasing on $E\cap[\bar t-\bar h,\bar t]$, 
and coincides with the integrable function 
$t\mapsto\ffi(t)+\ffi(\bar t)$ on $[\bar t-\bar h,\bar t+\bar h]\backslash E$.
It is clear that the bound of (D\ref{defapproxcont-true}.2.1) holds.
Moreover,
\begin{eqnarray*}
\int_{\bar t-h}^{\bar t+h}\sigma_\delta(t)dt&=&
\int_{[\bar t-h,\bar t+h]\cap E}\sigma_\delta(t)\,dt
+\int_{[\bar t-h,\bar t+h]\backslash E}\sigma_\delta(t)\,dt\\
&\le&\int_{[\bar t-h,\bar t+h]\cap E}\omega(\delta,h)\,dt
+\int_{[\bar t-h,\bar t+h]\backslash E}(\ffi(t)+\ffi(\bar t))\,dt\\
&\le&2\,h\,\omega(\delta,h)
+\int_{[\bar t-h,\bar t+h]\backslash E}(\ffi(t)-\ffi(\bar t)
+2\ffi(\bar t))\,dt\\
&\le&2\,h\,\omega(\delta,h)
+\int_{\bar t-h}^{\bar t+h}|\ffi(t)-\ffi(\bar t)|\,dt
+\int_{[\bar t-h,\bar t+h]\backslash E}2\,\ffi(\bar t)\,dt\\
&\le&2\,h\,\omega(\delta,h)
+\int_{\bar t-h}^{\bar t+h}|\ffi(t)-\ffi(\bar t)|\,dt
+2\,\ffi(\bar t)\,\mu(h)\,,
\end{eqnarray*}
where $\omega(\delta,h)\equaldef\sup\{\|f(x,t)-f(\bar x,\bar t)\|:
\|x-\bar x\|\le\delta,\|t-\bar t\|\le h,t\in E\}$
and $\mu(h)\equaldef\msr\Big([\bar t-h,\bar t+h]\backslash E\Big)$.


It follows from (P\ref{defapproxcont}.b) that 
$\lim_{\delta\downarrow 0,h\downarrow 0}\omega(\delta,h)=0$.
The fact that $\bar t$ is a Lebesgue point of $\ffi$ implies that
$\lim_{h\downarrow 0}\frac{1}{h}
\int_{\bar t-h}^{\bar t+h}|\ffi(t)-\ffi(\bar t)|\,dt=0$.
Finally, $\lim_{h\downarrow 0}\frac{1}{h}\mu(h)=0$,
because $\bar t$ is a point of density of $E$.
Therefore (D\ref{defapproxcont-true}.2.2) holds, and our proof is 
complete.\proofend
\end{proof}


\subssection{Differentiability of the flow}\label{sec-difflow}

The differentiation theorem will  say, roughly, that
if $(\bar x,\bar t)$ is a PAC of $f$, then 
\begin{equation}\label{requires}
\Phi^f(x,t,s)=x+(t-s)f(\bar x,\bar t)+o\Big(\,
\|x\!-\!\bar x\|\!+\!|t\!-\!\bar t|\!+\!|s\!-\!\bar t|\,\Big)  
\end{equation}
as $(x,t,s)\to(\bar x,\bar t,\bar t)$.

Since $\Phi^f$ is set-valued, Equation 
(\ref{requires}) requires interpretation.
The precise \mbox{meaning} of (\ref{requires}) is that,		
if we use $\Lambda^{w}$ to denote, for a given $w\in\rr^n$, the 
linear map $\Lambda^w:\rr^n\times\rr\times\rr\mapsto\rr^n$ given by
\begin{equation}\label{isdif}
\Lambda^{w}(v,h,k)=
v+(h-k)w\qquad{\rm for}\,\,\,v\in\rr^n\,,\,h,k\in\rr\,,
\end{equation}
then $\Lambda^{f(\bar x,\bar t)}(v,h,k)$ 
is a ``first-order approximation'' to the set-valued map
$$
(x,t,s)\mapsto\Phi^f(x,t,s)-\Phi^f(\bar x,\bar t,\bar t)=
\Phi^f(x,t,s)-\bar x
$$ 
near $(\bar x,\bar t,\bar t)$. Precisely, this means that
\begin{equation}\label{whatitmeans}
\lim\frac{\sup\Big\{\|y-\bar x-
\Lambda^{f(\bar x,\bar t)}(x-\bar x,t-\bar t,s-\bar t)\|
:y\in\Phi^f(x,s,t)\Big\}}
{\|x-\bar x\|+|t-\bar t|+|s-\bar t|}=0
\end{equation}
as $(x,s,t)\to(\bar x,\bar t,\bar t)$.  If we set $v=x-\bar x$,
$h=t-\bar t$, $k=s-\bar t$ in (\ref{isdif}), we see that 
(\ref{whatitmeans}) says that 
\begin{equation}\label{whatitmeans-a}
\lim_{(x,s,t)\to(\bar x,\bar t,\bar t)}  
\frac{\sup\Big\{\|y-x-(t-s)f(\bar x,\bar t)\|
:y\in\Phi^f(x,s,t)\Big\}}{\|x-\bar x\|+|t-\bar t|+|s-\bar t|}=0
\end{equation}
When this happens, we say that {\em $\Phi^f$ is differentiable
at $(\bar x,\bar t,\bar t)$ and the \mbox{differential} 
of $\Phi^f$ at $(\bar x,\bar t,\bar t)$ is the linear map}
$\Lambda^{f(\bar x,\bar t)}$.

The one-sided concepts of {\em forward and backward differentiability}
are \mbox{defined} \mbox{similarly}, replacing ``$(x,t,s)\to(\bar
x,\bar t,\bar t)$'' by ``$(x,t,s)\to(\bar x,\bar t,\bar t)$ via values
of $t,s$ such that $t\ge\bar t$ and $s\ge\bar t$'' and
``$(x,t,s)\to(\bar x,\bar t,\bar t)$ via values of $t,s$ such that
$t\le\bar t$ and $s\le\bar t$,'' respectively.

\begin{theorem}\label{theodifreg}
Assume that $n\in\nn$, $S\subseteq\rr^n\times\rr$, $f\in
TVVF(\rr^n;S)$, and $(\bar x,\bar t)\in S$.  Then, if $(\bar x,\bar
t)$ is a point of approximate continuity of $f$ it follows that the
flow $\Phi^f$ is differentiable at $(\bar x,\bar t,\bar t)$, and the
differential of $\Phi^f$ at $(\bar x,\bar t,\bar t)$ is the map
$\Lambda^{f(\bar x,\bar t)}$ defined by (\ref{isdif}).
\end{theorem}
\begin{proof}
Let $\bar\delta$, $\bar h$, and the functions 
$\sigma_\delta:[\bar t-\bar h,\bar t+\bar h]\mapsto[0,\infty]$
be as in Definition \ref{defapproxcont-true}.
For $0<\delta\le\bar\delta$, write
\begin{equation}
\omega(\delta)=\sup\Big\{\|y-\bar x\|:y\in\Phi^f(x,s,t):
(x,s,t)\in{\cal N}_\delta^{(2)}(\bar x,\bar t)\Big\}\,.
\end{equation}
Then $\omega(\delta)\to 0$ as $\delta\downarrow 0$ by Corollary \ref{limfo}.
Choose
$\delta^*$ such that $0<\delta^*\le\bar\delta$ and
$\omega(\delta^*)\le\bar\delta$.

Let $0<\delta\le\delta^*$. Suppose 
$y\in\Phi^f(x,s,t)$, $(x,s,t)\in{{\cal N}_\delta^{(2)}(\bar x,\bar t)}$. 
Let $\zeta$ be a trajectory of $f$ such that $\zeta(s)=x$ and $\zeta(t)=y$.
Let $I$ be the interval with endpoints $s,t$.   Let 
$\xi$ be the restriction of $\zeta$ to $I$. Then 
$$
y=x+\int_{s}^tf(\xi(r),r)\,dr=x+(t-s)f(\bar x,\bar t)+R\,,
$$
where the error term $R$ is given by 
$R=\int_{s}^t\Big(f(\xi(r),r)-f(\bar x,\bar t)\Big)\,dr$.

To estimate $R$, we first observe that for each $r\in I$ the point 
$\xi(r)$ belongs to  $\Phi^f(x,r,s)$, and 
$(x,r,s)\in{{\cal N}_\delta^{(2)}(\bar x,\bar t)}$, so 
$\|\xi(r)-\bar x\|\le\omega(\delta)$. Then
$$
r\in I\Longrightarrow\|f(\xi(r),r)-f(\bar x,\bar t)\|\le
\sigma_{\omega(\delta)}(r)\,.
$$
Therefore
\begin{equation}\label{thishas-a}
\|R\|\le\int_{\bar t-h}^{\bar t+h}\sigma_{\omega(\delta)}(r)\,dr\,,
\end{equation}
where $h=\max(|t-\bar t|,|s-\bar t|)$.

Inequality (\ref{thishas-a}) has been proved for arbitrary 
$(x,t,s)\in{{\cal N}_\delta^{(2)}(\bar x,\bar t)}$
and $y\in\Phi^f(x,t,s)$, provided that  $0<\delta\le\delta^*$.
For given $(x,t,s)$,  we can take $\delta=\|x-\bar x\|$,
and define $h=\max(|t-\bar t|,|s-\bar t|)$ as before. 
This yields 
\begin{eqnarray*}
\frac{\|y-x-(t-s)f(\bar x,\bar t)\|}
{\|x-\bar x\|+|t-\bar t|+|s-\bar t|}&=&
\frac{\|R\|}{\delta+|t-\bar t|+|s-\bar t|}\\
&\le&\frac{\|R\|}{h}\\
&\le&
\frac{1}{h}\int_{\bar t-h}^{\bar t+h}\sigma_{\omega(\|x-\bar x\|)}(r)\,dr\,,
\end{eqnarray*}
so (\ref{whatitmeans}) holds. \proofend
\end{proof}

\begin{corollary}\label{theodifregoneside}
Assume that $n\in\nn$, $S\subseteq\rr^n\times\rr$, $f\in
TVVF(\rr^n;S)$, and $(\bar x,\bar t)\in S$.  Then, if $(\bar x,\bar
t)$ is a point of forward (resp. backward) 
\mbox{approximate} continuity of $f$ it follows that the
flow $\Phi^f$ is forward (resp. backward) \mbox{differentiable} at 
$(\bar x,\bar t,\bar t)$, and the forward (resp. backward) 
differential of $\Phi^f$ at $(\bar x,\bar t,\bar t)$ is the map
$\Lambda^{f(\bar x,\bar t)}$ defined by (\ref{isdif}).
\end{corollary}
\begin{proof}
This can be proved using exactly the same arguments as in the proof
of Theorem \ref{theodifreg} with obvious one-sided modifications. 
Alternatively, the result can be derived from Theorem \ref{theodifreg}
by applying  Theorem \ref{theodifreg} to the TVVF $g$ considered in
the proof of Theorem \ref{theorexist}.\proofend
\end{proof}

\subssection{Regularization}\label{sec-regul}

Fix an $n\in\nn$, a nonempty open subset $\Om$ of $\rr^n$, and
a subinterval $I$ of $\rr$ of positive length. Let $S=\Om\times I$, 
and fix an admissible time-varying vector field  $f\in ADM(\rr^n;S)$. For 
$\rho>0$, let $\Om_\rho$ be the set of all $x\in\rr^n$
such that the closed ball 
\mbox{$\bmath^n(x,\rho)\equaldef\{x'\in\rr^n:\|x'-x\|\le\rho\}$}
is entirely contained in $\Om$. Then $\Om_{\rho}$ is open. 
Let $S_\rho=\Om_\rho\times I$.

Fix, once and for all, a function $\theta:\rr^n\to\rr$
of class $C^\infty$, such that $\theta(z)\ge 0$ for all $y$,
$\theta(z)=0$ whenever $\|z\|\ge 1$, and $\int_{\rr^n}\theta=1$.
Also fix, once and for all, locally integrable functions
$\ffi_K:I\mapsto[0,\infty]$, for each compact subset $K$ of $\Om$, 
such that $\|f(x,t)\|\le\ffi_K(t)$ whenever $x\in K$ and $t\in I$. 
If $K$ is a compact subset of $\Om$ and $\rho>0$, we write 
$K^\rho=\{x\in\rr^n:d(x,K)\le\rho\}$, and observe 
that $K^\rho\subseteq\Om$ if and only if $K\subseteq\Om_\rho$.

Define
\begin{equation}
f_{\rho}(x,t)=\int_{\rr^n}f(x-\rho z,t)\theta(z)\,dz\,\,\,
{\rm for}\,\,\,(x,t)\in S_\rho\,.
\end{equation}
Then for almost all $t\in I$ the integral is defined
for all $x\in\Om_\rho$, because for a.e. $t$ the map 
$x\mapsto f(x,t)$ is measurable and bounded on compact sets.
Moreover, if $K$ is a compact subset of $\Om_\rho$ then 
$f_\rho$ satisfies the bound
$$
\|f_\rho(x,t)\|\le\ffi_{K_\rho}(t)\quad{\rm whenever}
\quad x\in K\,,\,t\in I\,.
$$
In particular, $f_\rho$ is locally integrably bounded on
$S_\rho$.   Since 
\begin{equation}\label{flafornice}
f_{\rho}(x,t)=\rho^{-n}\int_{\rr^n}f(y,t)\theta\Big(\frac{x-y}{\rho}
\Big)dy\,\,\,{\rm for}\,\,\, x\in\Om_\rho\,,\,t\in I\,,
\end{equation}
it is easy to see that $f_\rho(x,t)$ is of class $C^\infty$ as a function
of $x$ for almost every $t$.  Moreover,  $f_\rho(x,t)$ is clearly measurable 
with respect to $t$ for each fixed $x$.  Furthermore,
using (\ref{flafornice}), we see that 
\begin{equation}\label{flaforlip}
f_{\rho}(x,t)-f_{\rho}(x',t)
=\rho^{-n}\int_{\rr^n}f(y,t)\Biggl(\theta\Big(\frac{x-y}{\rho}\Big)
-\theta\Big(\frac{x'-y}{\rho}\Big)\Biggr)dy
\end{equation}
for  $x,x'\in\Om_\rho$, $t\in I$. Therefore, 
if $K\subseteq\Om_\rho$ is compact, and
$x,x'\in K$, $t\in I$, we have
\begin{equation}\label{lipestprel}
\Big\|f_{\rho}(x,t)-f_{\rho}(x',t)\Big\|\le
\rho^{-n}\ffi_{K_\rho}(t)\int_{\rr^n}\Biggl|\theta\Big(\frac{x-y}{\rho}\Big)
-\theta\Big(\frac{x'-y}{\rho}\Big)\Biggr|dy\,.
\end{equation}
Then 
\begin{eqnarray}
&\,&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
\!\!\!
\Big(K\subseteq\Om_{\rho}\,\wedge\,K\,\,\,{\rm compact}\,\wedge\,
x,x'\in K\,\wedge\,t\in I\Big)\nonumber\\
&\,&\Longrightarrow\Big\|f_{\rho}(x,t)-f_{\rho}(x',t)\Big\|\le
\kappa\rho^{-n-1}\ffi_{K_\rho}(t)\|x-x'\|\,,\label{lipest}
\end{eqnarray}
where 
\begin{equation}
\kappa=\sup\Big\{\|\nabla\theta(z)\|:z\in\rr^n\Big\}\,.
\end{equation}
Clearly, (\ref{lipest}) implies that 
\bfact
\ If $\rho>0$ then the map $\Om_\rho\ni x\mapsto f_\rho(x,t)$ is
locally Lipschitz for almost all $t\in I$, and the 
Lipschitz constant $C_K(t)$ can be chosen, for each compact
subset $K$ of $\Om_\rho$, to be a locally integrable function
of $t$.\proofend
\efact
It then follows that $f_\rho$ has the usual uniqueness and 
continuous dependence properties. In particular: 
\bfact
\ The flow $\Phi^{f_{\rho}}$ is single-valued and continuous 
on its domain of \mbox{definition} $\dom(\Phi^{f_\rho})$, which is a 
relatively open subset
of $\Om_{\rho}\times I\times I$ containing the set 
$\{(x,t,t):x\in\Om_\rho\,,\,t\in I\}$. \proofend
\efact

If $(\bar x,\bar t)\in\rr^n\times I$ and $\delta>0$, we let 
\begin{eqnarray*}
\tilde{\cal N}_\delta(\bar x,\bar t)&=&\left\{\begin{array}{lcl}
{\cal N}_\delta(\bar x,\bar t)&{\rm if}&\bar t\in{\rm Int}(I)\,,\\
{\cal N}_{\delta,+}(\bar x,\bar t)&{\rm if}&\bar t=\min\,I\,,\\
{\cal N}_{\delta,-}(\bar x,\bar t)&{\rm if}&\bar t=\max\,I\,,
\end{array}\right.\\
\tilde{\cal N}_\delta^{(2)}(\bar x,\bar t)&=&\left\{\begin{array}{lcl}
{\cal N}_\delta^{(2)}(\bar x,\bar t)&{\rm if}&\bar t\in{\rm Int}(I)\,,\\
{\cal N}_{\delta,+}^{(2)}(\bar x,\bar t)&{\rm if}&\bar t=\min\,I\,,\\
{\cal N}_{\delta,-}^{(2)}(\bar x,\bar t)&{\rm if}&\bar t=\max\,I\,.
\end{array}\right.
\end{eqnarray*}

Fix $(\bar x,\bar t)\in\Om\times I$, and pick $\delta$  such that 
$\delta>0$ and
\begin{itemize}\item[]\begin{itemize}
\item[($\#.1$)] {\em 
${\tilde{\cal N}_{\delta}(\bar x,\bar t)}\subseteq S$
and the set-valued map $\Phi^f$ is upper semicontinuous
and has compact nonempty values on 
${\tilde{\cal N}_{\delta}^{(2)}(\bar x,\bar t)}$.}
\end{itemize}\end{itemize}    

\begin{theorem}\label{regpovar}
Let $\delta>0$ be such that ($\#.1$) holds. Then there exists 
a positive $\bar\rho$ such that 
\begin{itemize}\item[]\begin{itemize}
\item[($\#.2$)] {\em $\bmath^n(\bar x,\delta)\subseteq\Om_{\bar\rho}$},
\item[($\#.3$)] ${\tilde{\cal N}_\delta^{(2)}(\bar x,\bar t)}
\subseteq\dom(\Phi^{f_\rho})$ whenever $0<\rho\le\bar\rho$,
\end{itemize}\end{itemize}
and
\begin{itemize}\item[]\begin{itemize}
\item[($\#.4$)] the single-valued maps $\Phi^{f_\rho}\restr
{\tilde{\cal N}_\delta^{(2)}(\bar x,\bar t)}$ satisfy
\begin{equation}
\Phi^{f_\rho}\restr{\tilde{\cal N}_\delta^{(2)}(\bar x,\bar t)}
\,\,\,\,\togr\,\,\,\,\Phi^{f}\restr{\tilde{\cal N}}_\delta^{(2)}
(\bar x,\bar t)\,\,\,{\rm as\,\,\,}\rho\to 0\,.
\end{equation}
\end{itemize}\end{itemize}
In particular, the set-valued map $\Phi^{f}\restr{\tilde{\cal N}}_\delta^{(2)}:
\tilde{\cal N}_\delta^{(2)}\tto\rr^n$ is regular.
\end{theorem}

\begin{proof} 
The image $L=\Phi^f({\tilde{\cal N}_\delta^{(2)}(\bar x,\bar t)})$ is a
compact subset of $\Om$.
Let $K$ be a compact subset of $\Om$ such that $L\subseteq{\rm Int}(K)$.
We will show that
\begin{itemize}
\item[(*)] {\em if $\{(\rho_j,x_j,t_j,s_j)\}_{j\in\nn}$ is a 
sequence in $\rr\times\rr^n\times\rr\times\rr$ such that 
$\rho_j>0$, $(x_j,t_j,s_j)\in{\tilde{\cal N}_\delta^{(2)}
(\bar x,\bar t)}$, $\rho_j\to 0$, $(x_j,t_j,s_j)\to(x,t,s)$, then}
\begin{itemize}
\item[(*.1)] {\em $(x_j,t_j,s_j)\in\dom(\Phi^{f_{\rho_j}})$ for large
enough $j$}
\end{itemize}
{\it and}
\begin{itemize}
\item[(*.2)] {\em if $y_j=\Phi^{f_{\rho_j}}(x_j,t_j,s_j)$, then
$y_j\in K$ for large enough $j$, and every limit point 
$y$ of the sequence $\{y_j\}$ satisfies $y\in\Phi^f(x,t,s)$.}
\end{itemize}\end{itemize}

It is clear that (*) implies our desired conclusion.  To prove (*), we
first fix a $\bar\rho>0$ such that $K_{\bar\rho}\subseteq\Om$, and write
$\ffi=\ffi_{K_{\bar\rho}}$.

Assume that the hypothesis of (*) holds.  Define a compact interval
$\tilde I$ by letting
\mbox{$\tilde I=[\bar t-\delta,\bar t+\delta]$} if $t\in{\rm Int}(I)$, 
$\tilde I=[\bar t,\bar t+\delta]$ if $t=\min\,I$, 
$\tilde I=[\bar t-\delta,\bar t]$ if $t=\max\,I$. Let $\xi_j$ denote the
maximal \mbox{trajectory} of $f_{\rho_j}$ that goes through $x_j$ at time
$s_j$.  Let $J_j=\dom(\xi_j)$, so that $s_j$ lies in the interior
of $J_j$ relative to $I$. Let
$I_j=[\min(s_j,t_j),\max(s_j,t_j)]$, so that $I_j\subseteq\tilde I$.  
Let $H_j$ denote the set
of all $\tau\in I_j\cap J_j$ such that $\xi_j(u)\in K$ for all
$u\in[\min(s_j,\tau),\max(s_j\tau)]$.  Then $H_j$ is a compact
interval, $H_j\subseteq J_j\cap I_j$ and $s_j$ is one of the endpoints
of $H_j$. We let $\tau_j$ be the other endpoint, so
$H_j=[\min(s_j,\tau_j),\max(s_j,\tau_j)]$. Then, if we let $\partial
K=K\backslash{\rm Int}(K)$, it is easy to verify that
\bfact\label{dicho}
For each $j$, either $\tau_j=t_j$ or $\xi_j(\tau_j)\in\partial K$.
\efact
Let $z_j=\xi_j(\tau_j)$.   Then $(x_j,\tau_j,s_j,z_j)\in
{\tilde{\cal N}_\delta^{(2)}(\bar x,\bar t)}\times K$. 
Write \mbox{$\eta_j=\xi_j\restr H_j$}. Then $\eta_j\in\traj_c(f_{\rho_j})$,
and $\eta_j(t)\in K$ whenever $t\in H_J$, so $\eta_j$ belongs
to $ARC(\rr^n;K\times\tilde I)$, and
\begin{equation}
\|\dot\eta_j(r)\|\le\ffi(r)\,\,\,{\rm for}\,\,\,r\in H_j\,.
\end{equation}
It then follows that the sequence $\pmb{\eta}=\{\eta_j\}_{j\in\nn}$ is 
uniformly bounded and \mbox{equicontinuous}.
Therefore every subsequence of $\pmb{\eta}$
has a subsequence that converges to a limit in $ARC(\rr^n)$. Equivalently,
every infinite subset $V$ of $\nn$ contains an infinite subset
$W$ such that the sequence $\pmb{\eta}^W\equaldef\{\eta_j\}_{j\in W}$
converges in $ARC(\rr^n)$ to a limit $\eta^W$.

Let ${\cal W}$ be the set of all infinite subsets $W$ of $\nn$ be such that
$\pmb{\eta}^W$ converges. We will prove that
\begin{equation}\label{firstshow}
W\in{\cal W}\,\Longrightarrow\,\eta^W\in\traj(f)\,.
\end{equation}
%converges to a limit $(x_W,\tau_W,s_W,z_W,\eta_W)\in
%{{\cal N}_\delta^{(2)}(\bar x,\bar t)}\times K\times ARC(\rr^n)$.
%Clearly,   $\min(s,t)\le\tau\le\max(s,t)$. Let 
%$H=\dom(\eta)$, so $H=[s\wedge\tau,s\vee\tau]$. 

We first show that (\ref{firstshow}) implies our conclusion. Let
\begin{equation}
W'=\{j\in\nn:t_j\not\in J_j\}\,,\,\,\,W''=\{j\in\nn:t_j\neq\tau_j\}\,.
\end{equation}
Then
\begin{equation}\label{jincal}
j\in W'\Rightarrow j\in W''\Rightarrow z_j\in\partial K\,,
\end{equation}
where the first implication follows because $\tau_j\in J_j$, and the
second one is a consequence of Fact \ref{dicho}.  If the set $W''$ was
infinite, then we could pick $W$ to be a subset of $W''$ such that
$W\in{\cal W}$, and apply (\ref{firstshow}) to conclude that
$\eta^W\in\traj(f)$.  Since
$\dom(\eta_j)=H_j=[\min(s_j,\tau_j),\max(s_j,\tau_j)]$, 
and the sequence $\{\eta_j\}_{j\in W}$ \mbox{converges} to 
$\eta^W$ in $ARC(\rr^n)$,
it is clear that the limit $\tau^W=\lim_{j\to\infty,j\in W}$ exists,
and $\dom(\eta^W)=H^W=[\min(s,\tau^W),\max(s,\tau^W)]$.
Let $z^W=\eta^W(\tau^W)$. Then 
$z^W=\lim_{j\to\infty\,,\,j\in W}z_j$, so (\ref{jincal}) implies that
$z\in\partial K$.  Since $z^W=\eta^W(\tau^W)$, (\ref{firstshow}) implies
that $z^W\in\Phi^f(x,\tau^W,s)$. Since $(x,\tau^W,s)\in
{\tilde {\cal N}_\delta^{(2)}(\bar x,\bar t)}$ (because
$(x_j,\tau_j,s_j)\in{\tilde {\cal N}_\delta^{(2)}(\bar x,\bar t)}$
for every $j$), $z^W\in L\subseteq {\rm Int}\, K$.  So
we have reached a contradiction, proving that $W''$ is
finite. Then $W'$ is finite as well, and this shows that (*.1)
holds.  Moreover, the fact that $W''$ is finite shows that
$t_j=\tau_j$ for sufficiently large $j$, so $z_j=y_j$ for large $j$,
proving that $y_j\in K$ for large enough $j$. Finally, let $y$ be a
limit point of the sequence $\{y_j\}_{j\in\nn}$.  Pick an infinite
subset $W'$ of $\nn$ such that
$y_j\to y$ as $j\to\infty$ via values in $W'$. Then pick
$W\in{\cal W}$ such that \mbox{$W\subseteq W'$}.
Then $y=\eta^W(t)$.
This shows that $y\in\Phi^f(x,s,t)$, and the proof of (*) is complete.

To conclude our proof, we must establish (\ref{firstshow}).
Fix $W\in{\cal W}$, and let 
$\tau^W=\lim_{j\to\infty,j\in W}\tau_j$,
$H^W=\dom(\eta^W)=[\min(s,\tau^W),\max(s,\tau^W)]$,\
$z^W=\eta^W(\tau^W)$. Our conclusion is trivially true if 
$s=\tau^W$. Assume that $s\neq\tau^W$. Pick a number $u$ such that
$\min(s,\tau^W)<u<\max(s,\tau^W)$. 
Let $N=P\times Q$, where $P$ is a compact 
neighborhood of $\eta^W(u)$ in $\Om$, and $Q$ is a compact interval 
which is contained in the interior of $H^W$ and is such that
$u$ is an interior point of $Q$. Using the admissibility 
of $f$, choose an integrable function $\gamma:Q\mapsto[0,\infty]$ 
such that
$\|f(x,t)\|\le\gamma(t)$ for all $(x,t)\in N$, and
$f$ is integrally continuous on $ARC_\gamma(\rr^n;N)$. Write
${\cal A}=ARC_\gamma(\rr^n;N)$. 
Then there exist an $\eps>0$ and a $j^*\in\nn$
such that the following are true for $j\ge j^*$:
\begin{itemize}
\item[(1)] $\rho_j\le\eps$, 
\item[(2)] $Q\subseteq H_j$, 
\item[(3)] $(\eta_j(v)+z,v)\in N$ whenever $|v-u|\le\eps$, $\|z\|\le\eps$.
\end{itemize}
(Indeed, (2) holds if $j$ is large enough because the endpoints of 
$H_j$ converge to the endpoints of $H^W$  and $Q\subseteq{\rm Int}(H^W)$. 
Given $\eps$, (1) holds for large enough $j$ because $\rho_j\to 0$.
So all we need is to show that there exists an $\eps$ such that 
(3) holds for large enough $j$. If $\eps$ did not exist, 
there would exist, for each $k\in\nn$, a $j(k)$ such that 
$j(k)\ge k$ and a pair $(v_k,z_k)$ such that 
$\|z_k\|\le 2^{-k}$, $\|v_k-u\|\le 2^{-k}$, and 
$(\eta_{j(k)}(v_k)+z_k,v_k)\notin N$. But then 
$v_k\to u$, and $z_k\to 0$ as $k\to\infty$, so 
$(\eta_{j(k)}(v_k)+z_k,v_k)\to(\eta^W(u),u)$. Since $(\eta^W(u),u)$
is an interior point of $N$, this is a contradiction.) 

Then, for $j\ge j^*$, if $v_1,v_2\in[u-\eps,u+\eps]$ and $v_1<v_2$, we have
\begin{eqnarray}
\eta_j(v_2)-\eta_j(v_1)&=&
\int_{v_1}^{v_2}f_{\rho_j}(\eta_j(w),w)dw\nonumber\\
&=&\int_{v_1}^{v_2}\Big(\int_{\rr^n}f(\eta_j(w)-\rho_jz,w)
\theta(z)\,dz\Big)\,dw\nonumber\\
&=&\int_{\rr^n}\Big(\int_{v_1}^{v_2}f(\eta_j(w)-\rho_jz,w)\,dw
\Big)\theta(z)\,dz\nonumber\\
&=&\int_{\rr^n}\Big(\int_{v_1}^{v_2}f(\eta_j^z(w),w)\,dw\Big)
\theta(z)\,dz\nonumber\\
&=&\int_{\rr^n}\Big({\cal T}_f^{{\cal A}}(\eta_j^z)(v_2)-
{\cal T}_f^{{\cal A}}(\eta_j^z)(v_1)\Big)\theta(z)\,dz\,,\nonumber
\end{eqnarray}
where $\eta_j^z:[u-\eps,u+\eps]\to\Om$ is defined by
$$
\eta_j^z(w)=\eta_j(w)-\rho_jz\,\,\,{\rm for\,\,\,}|w-u|\le\eps\,,
$$
and the interchange of the integrals is justified by Fubini's Theorem,
using the bound $\|f(\xi_j(w)-\rho_jz,w)\|\le\gamma(w)$, which is valid
because the point $(\xi_j(w)-\rho_jz,w)$ belongs to $N$.   
It is clear that the curves 
$\eta_j^z$ belong to ${\cal A}$ whenever $\|z\|\le 1$ and $j\ge j^*$.
Moreover, the $\eta_j^z$ converge in $ARC(\rr^n)$ to 
the restriction $\eta^W\restr[u-\eps,u+\eps]$. Since $f$ is
integrally continuous on ${\cal A}$, it follows that
${\cal T}_f^{{\cal A}}(\eta_j^z)(v_2)-{\cal T}_f^{{\cal A}}(\eta_j^z)(v_1)
\to\int_{v_1}^{v_2}f(\eta^W(w),w)dw$
for each $z\in\rr^n$ such that $\|z\|\le 1$.  Since 
$\|{\cal T}_f^{{\cal A}}(\eta_j^z)(v_2)-{\cal T}_f^{{\cal A}}(\eta_j^z)(v_1)\|
\le\int_{u-\eps}^{u+\eps}\gamma(w)dw$, and 
$\theta$ is bounded and supported in $\{z:\|z\|\le 1\}$, we have
\begin{eqnarray}
\eta^W(v_2)-\eta^W(v_1)&=&\lim_{j\to\infty}\Big(\eta_j(v_2)-
\eta_j(v_1)\Big)\nonumber\\
&=&\lim_{j\to\infty}\int_{z\in\rr^n,\|z\|\le 1}
\Big({\cal T}_f^{{\cal A}}(\eta_j^z)(v_2)-
{\cal T}_f^{{\cal A}}(\eta_j^z)(v_1)\Big)\theta(z)\,dz\nonumber\\ 
&=&\int_{v_1}^{v_2}f(\eta^W(w),w)dw\,.\nonumber
\end{eqnarray}
So every interior point $u$ of $H^W$ has a neighborhood $[u-\eps,u+\eps]$
with the property that $[u-\eps,u+\eps]\subseteq H^W$ and
$\eta^W\restr[u-\eps,u+\eps]$ is a trajectory of 
$f$.  This clearly implies that $\eta^W\in\traj(f)$, completing the
proof.\proofend
\end{proof}

\subssection{Regular differentiability of the flow}\label{secregdifflow}

We now combine the results of the two previous subsections into a
\mbox{single} \mbox{statement}, asserting the regular
differentiability of the flow of an \mbox{admissible} vector field
$f\in ADM(\rr^n)$ at every point of approximate continuity of $f$, and
explicitly exhibiting a \mbox{differential}.

\begin{theorem}\label{theosemidifreg}
Assume that $n\in\nn$, $S\subseteq\rr^n\times\rr$, 
$f\in ADM(\rr^n;S)$. Let $(\bar x,\bar t)$ be a point of approximate
continuity of $f$. Then the flow $\Phi^f$ is regularly differentiable at 
$(\bar x,\bar t,\bar t)$, and 
the linear map $\Lambda^{f(\bar x,\bar t)}$ defined by (\ref{isdif})
is a differential of $\Phi^f$ at $(\bar x,\bar t,\bar t)$.
\end{theorem}
\begin{proof} 
Theorem \ref{regpovar} gives us a neighborhood of 
$(\bar x,\bar t,\bar t)$ where $\Phi^f$ is 
\mbox{regular}, and Theorem \ref{theodifreg} says 
that $\Lambda^{f(\bar x,\bar t)}$ is a first-order 
approximation to $\Phi^f$ near $(\bar x,\bar t,\bar t)$.\proofend
\end{proof}

\begin{corollary}\label{theodifregoneside-reg}
Assume that $n\in\nn$, $S\subseteq\rr^n\times\rr$, and $f\in
ADM(\rr^n;S)$. Let $(\bar x,\bar
t)$ be a point of forward (resp. backward) 
\mbox{approximate} continuity of $f$. Then the
flow $\Phi^f$ is forward (resp. backward) regularly \mbox{differentiable} at 
$(\bar x,\bar t,\bar t)$, and the forward (resp. backward) 
differential of $\Phi^f$ at $(\bar x,\bar t,\bar t)$ is the map
$\Lambda^{f(\bar x,\bar t)}$ defined by (\ref{isdif}).
\end{corollary}
\begin{proof}
This follows by applying Theorem \ref{theosemidifreg}
to the TVVF $g$ considered in
the proof of Theorem \ref{theorexist}.\proofend
\end{proof}




\section{Almost LSC differential inclusions}

\subssection{Lower semicontinuous set-valued maps}\label{seconlsc}

\begin{definition}\label{deflowersem}
Assume that $X$ and $Y$ are topological spaces.   A set-valued map
\mbox{$F:X\tto Y$} is {\bf\em lower semicontinuous} (abbr. LSC) 
if the inverse image under $F$ of every open subset of $Y$ is 
open in $X$. \proofend
\end{definition}
The following observation is a trivial consequence of the definition.
\bfact\label{sun1}
If $X$, $Y$ are topological spaces and 
$F:X\mapsto Y$ is an ordinary (i.e. single-valued, everywhere defined)
map, then $F$ is LSC as a set-valued map if and only 
if $F$ is a continuous map.\proofend
\efact
\forlong{In addition, the following fact is well known.}
\forshort{In addition, the following fact is well known and easy to prove.}
\begin{lemma}\label{isusc}
Let $S$ be a topological space and let $Q$  be a metric space.
Suppose that $F:S\tto Q$ is a set-valued map with nonempty values.
For each ordinary map $f:S\mapsto Q$, define a function 
$\rho_{f,F}:S\to\rr$ by letting
$$
\rho_{f,F}(s)\,\,\equaldef\,\,d_Q(f(s),F(s))
$$
Then the following three conditions are equivalent:
\begin{itemize}\item[]\begin{itemize}
\item[(L\ref{isusc}.a)] $F$ is lower semicontinuous;
\item[(L\ref{isusc}.b)] $\rho_{f,F}$ is upper semicontinuous for every
  continuous $f$;
\item[(L\ref{isusc}.c)] $\rho_{f,F}$ is upper semicontinuous for every
constant $f$.\forshort{\proofend}
\end{itemize}\end{itemize}
\end{lemma}
\forlong{
\begin{proof}
We first prove that (L\ref{isusc}.a) implies (L\ref{isusc}.b). 
Assume $F$ is LSC and $f$ is continuous.  Let
$s\in S$, and let $\eps>0$. Pick $q\in F(s)$ such that 
$d_Q(f(s),q)<\rho_{f,F}(s)+\frac{\eps}{3}$. Then choose a neighborhood $U$
of $s$ such that, whenever $s'\in U$, 
(a)~$F(s')\cap B_Q(q,\frac{\eps}{3})\neq\emptyset$, and 
(b)~$d_Q(f(s'),f(s))<\frac{\eps}{3}$. Then 
$d_Q(f(s'),F(s'))<\rho_{f,F}(s)+\eps$ whenever $s'\in U$, so 
$\rho_{f,F}(s')<\rho_{f,F}(s)+\eps$ for all $s'\in U$.  So
$\rho_{f,F}$ is upper semicontinuous.  

It is cleat that (L\ref{isusc}.b) implies (L\ref{isusc}.c).  We conclude
by proving that  (L\ref{isusc}.c) implies (L\ref{isusc}.a). Assume that 
(L\ref{isusc}.c) holds. 
Let $U\subseteq Q$ be open. Let \mbox{$s\in F^{-1}(U)$}, so 
$F(s)\cap U\neq\emptyset$.  Pick $q\in F(s)\cap U$. 
Let $\eps$ be such  that $\eps>0$ and $B_Q(q,\eps)\subseteq U$. 
Let $f:S\mapsto Q$ be
the constant function with value $q$. Then $\rho_{f,F}$ is continuous.
Since $\rho_{f,F}(s)=0$, because $q\in F(s)$, there is a neighborhood
$V$ of $s$ in $S$ such that $\rho_{f,F}(s')<\eps$ for all 
$s'\in  V$. It follows that $F(s')\cap B_Q(q,\eps)\neq\emptyset$ 
for all $s'\in V$. Since $B_Q(q,\eps)\subseteq U$, we see that 
$V\subseteq\{s'\in S:F(s')\cap U\neq\emptyset\}=F^{-1}(U)$.  
So $F^{-1}(U)$ is a neighborhood of $s$ whenever $s\in F^{-1}(U)$. 
Therefore $F^{-1}(U)$ is open. So $F$ is LSC.\proofend
\end{proof}
}

\subssection{Conically continuous selections of LSC set-valued 
maps}\label{sec-conic-exist}

\begin{theorem}\label{selectionlsc}
Assume that $n,m\in\nn$, $S\subseteq\rr^n\times\rr$, and 
\mbox{$F:S\tto\rr^m$} is a lower semicontinuous set-valued map with 
nonempty closed values.  Let \mbox{$g:S\mapsto\rr^m$},
$\beta:S\mapsto\rr$ be continuous maps such that 
\begin{equation}\label{ispos}
\beta(x,t)>\rho_{g,F}(x,t)\,\,\,{\rm for\,\,\,all\,\,\,}(x,t)\in S\,.
\end{equation}
Let
$(\bar{x},\bar{t})\in S$, $\bar{y}\in F(\bar{x},\bar{t})$, be such that 
\begin{equation}\label{ispos2}
\|g(\bar{x},\bar{t})-\bar{y}\|<\beta(\bar{x},\bar{t})\,.
\end{equation}   
Let $C:S\mapsto]\,0,\infty\,[$ be upper semicontinuous. Then there exists
a single-valued selection $f:S\mapsto\rr^m$ of $F$ such that 
\begin{itemize}\item[]\begin{itemize}
\item[(T\ref{selectionlsc}.1)] $f(\bar{x},\bar{t})=\bar{y}$,
\item[(T\ref{selectionlsc}.2)] $f$ is continuous at $(\bar{x},\bar{t})$,  
\item[(T\ref{selectionlsc}.3)] $f$ is forward $\Gamma_C$-continuous, 
\item[(T\ref{selectionlsc}.4)] $\|f(x,t)-g(x,t)\|<\beta(x,t)$ for all 
$(x,t)\in S$,
\item[(T\ref{selectionlsc}.5)] $f$ is
${\rm Bor}(\rr^n\times\rr)\cap S$-measurable.
\end{itemize}\end{itemize}
\end{theorem}

\begin{proof} Let $\tilde F$ be the set-valued function given by 
$$
\tilde F(x,t)=\left\{\begin{array}{lcl}
F(x,t)&{\rm if}&(x,t)\in S\backslash\{(\bar{x},\bar{t})\}\,,\\
\{\bar{y}\}&{\rm if}&(x,t)=(\bar{x},\bar{t})\,.\end{array}\right.
$$
Then $\tilde F$ is also LSC, and
\begin{eqnarray*}
\rho_{g,\tilde F}(x,t)&=&\rho_{g,F}(x,t)\quad{\rm when}\quad(x,t)
\neq(\bar{x},\bar{t})\,,\\
\rho_{g,\tilde F}(\bar{x},\bar{t})&=&\|g(\bar{x},\bar{t})-\bar{y}\|\,.
\end{eqnarray*}
Therefore $\rho_{g,\tilde F}(x,t)<\beta(x,t)$ for all $(x,t)\in S$. Then 
$\tilde F$ satisfies the same
hypotheses as $F$, and in addition $F(\bar{x},\bar{t})=\{\bar{y}\}$. So we may---and
will---assume, without loss of generality,  that $F(\bar{x},\bar{t})=\{\bar{y}\}$.

If $N\in\nn$ and $N>0$, we 
let $S_N=\{(x,t)\in S:C(x,t)<N\}$. Then  the upper semicontinuity
of $C$ implies that every $S_N$ is
relatively open in $S$, and the fact that $C$ has finite values
implies that and $S=\bigcup_{N=1}^{\infty}S_N$.  

We define, for each $(x,t)\in\rr^n\times\rr$, each $N\in\nn\backslash\{0\}$, 
and each $\al$ such that $\al>0$, a set $A(x,t,\al,N)$, by
\begin{equation}
A(x,t,\al,N)\!=\!\Big\{(x',t')\!\in\!\rr^n\!\times\!\rr\!:\!t\!\le\!t'
\!<\!t\!+\!\al,\|x'\!-\!x\|\!\le\!N(t'\!-\!t)\Big\}\,.
\end{equation}
We let ${\bf A}$ be the set of all sets $A(x,t,\al,N)$, for all 
$(x,t)\in\rr^n\times\rr$, $\al\in\,]0,1]$, $N\in\nn\backslash\{0\}$,
such that $A(x,t,\al,N)\cap S\subseteq S_{N}$.  

We then let ${\bf A}^*$ be the set of all those sets
$E\in{\bf A}$ such that \mbox{$(\bar{x},\bar{t})\notin\partial E$}.  (Here $\partial
E\equaldef{\rm Clos}(E)\backslash{\rm Int}(E)$, the closure and the
interior being taken relative to $\rr^n\times\rr$.)  We observe that 

\bfact\label{ogg}
If $(x,t)\in S$ is arbitrary, and $V$ is a neighborhood of $(x,t)$
in $\rr^n\times\rr$, then $V$ contains a set $E\in{\bf A}^*$ which is a 
neighborhood of $(x,t)$ in $\rr^n\times\rr$.  
\efact

Indeed, if suffices to pick $E=A(x,t-\frac{\al}{2},\al,N)$ where $N$
is such that $(x,t)\in S_N$, and $\al$ is sufficiently small.  If this
choice happens to result in a set $E$ such that $(\bar{x},\bar{t})\in\partial E$,
then we take instead the smaller set
$\tilde E=A(x,t-\frac{\al}{4},\frac{\al}{2},N)$, and observe that 
$\partial E\cap\partial\tilde E=\emptyset$, so $\tilde E$
has the desired property, and Fact \ref{ogg} is proved.

\medbreak
We then let ${\cal G}$ be the Boolean algebra of subsets of $S$ generated
by the sets $E\cap S$, $E\in{\bf A}^*$.

Let $\Phi_{{\cal G}}$ be the set of all maps $f:S\mapsto\rr^n$ 
such that there exists a partition ${\cal P}$ of $S$ into members of ${\cal
G}$ with the property that the restriction of $f$ to each $G\in{\cal P}$ is
continuous.  Let $\bar\Phi_{{\cal G}}$ be the closure of $\Phi_{{\cal G}}$
with respect to uniform convergence on compact subsets of $S$.  We claim that

\bfact\label{ugg}
If $f\!\!\in\!\bar\Phi_{{\cal G}}$, then $f$ is continuous 
at $(\bar{x},\bar{t})$ and forward \mbox{$\Gamma_C$-continuous}.
\efact

\noindent To prove Fact \ref{ugg}, we pick a sequence 
$\{(x^j,t^j)\}_{j\in\nn}$ in $S$ that converges to a limit
$(x,t)\in S$, and assume that either
\begin{equation}\label{casea}
x=\bar{x}\,\,\,{\rm and}\,\,\,t=\bar{t}\,,
\end{equation}
or
\begin{equation}\label{caseb}
\|x^j-x\|\le C(x,t)(t^j-t)\,\,\,{\rm for\,\,\,
all}\,\,\,j\,.
\end{equation}

We must show that $f(x^j,t^j)\to f(x,t)$.  Since $f$ is a uniform limit on
compact sets of functions in $\Phi_{{\cal G}}$, and the set
$\{(x,t)\}\cup\{(x^j,t^j):j\in\nn\}$ is compact, it suffices to assume that
$f\in\Phi_{{\cal G}}$.  For this purpose, we show

\bfact\label{igg}
Every member $H$ of
${\cal G}$ has the following property:
\begin{itemize}
\item[(P)] if $(x,t)\in H$ then $(x^j,t^j)\in H$ for sufficiently large $j$.
\end{itemize}
\efact

To prove Fact \ref{igg}, we let ${\cal Z}$ be the set of all subsets $H$ of
$S$ such that both $H$ and $S\backslash H$ have property
(P).  Then ${\cal Z}$ is
clearly closed \mbox{under} \mbox{complementation} (that is,
$H\in{\cal Z}\Rightarrow S\backslash H\in{\cal Z}\,$) and under the binary 
\mbox{operations} of union and intersection.  So to prove
that every $H\in{\cal G}$ has (P) it suffices to show that
$E\cap S\in{\cal Z}$ whenever $E\in{\bf A}^*$.  That is, we must pick
$E\in{\bf A}^*$ and show that (P) holds for $H=E\cap S$ and
also for  $H=S\backslash E$.  Let $E=A(x',t',\al,N)$.

Assume first that (\ref{casea}) holds, i.e that $(x,t)=(\bar{x},\bar{t})$.
Then $(x,t)$ does not belong to 
$\partial E$, so $(x,t)$ is an interior point of $E$ or of
$(\rr^n\times\rr)\backslash E$. So $(x^j,t^j)\in E$ 
for large $j$ if $(x,t)\in E$, and $(x^j,t^j)\in
(\rr^n\times\rr)\backslash E$ for large $j$ if $(x,t)\in
(\rr^n\times\rr)\backslash E$. Therefore $(x^j,t^j)\in E\cap S$ 
for large $j$ if $(x,t)\in E\cap S$, and $(x^j,t^j)\in S\backslash E$ for 
large $j$ if $(x,t)\in S\backslash E$. So (P) holds both for
$H=E\cap S$ and $H=S\backslash E$. 

Now consider the case when $(x,t)\neq (\bar{x},\bar{t})$, so that
(\ref{caseb}) holds. Then $t^j\ge t$ and $\|x^j-x\|\le C(x,t)(t^j-t)$
for all $j$.  If $(x,t)$ is an \mbox{interior} point of $E$ or of
$(\rr^n\times\rr)\backslash E$, 
then the desired conclusion follows as in the \mbox{previous}
case. So let us assume that $(x,t)\in\partial E$. Then either $t<t'+\al$ or
$t=t'+\al$. We will consider these two cases separately.

If $t<t'+\al$, then $\|x-x'\|=N(t-t')$, and
$(x,t)\in E$. But $(x,t)\in S$, and 
$E\cap S\subseteq S_N$, so $(x,t)$ belongs to $S_N$, and then
\mbox{$C(x,t)<N$}.
\mbox{Therefore} \mbox{$\|x^j-x\|\le N(t^j-t)$} for all $j$, from which it
follows that
$$
\|x^j-x'\|\le N(t^j-t)+N(t-t')=N(t^j-t')\quad{\rm for\,\,\,all}\,\,\,j\,.
$$
Since $t<t'+\al$ and $t^j\to t$, we have $t^j<t'+\al$ for large
$j$.  So $(x^j,t^j)\in E$ for large $j$, and then 
$(x^j,t^j)\in E\cap S$ for large $j$, so (P) holds if $H=E\cap S$. 
Furthermore, (P) also holds, vacuously, if $H=S\backslash E$, 
because the premiss ``$(x,t)\in H$'' is false.

If $t=t'+\al$, then $(x,t)\in S\backslash E$.  Moreover, 
$(x^j,t^j)\notin E$ for all $j$, since $t^j\ge t\ge
t'+\al$. So $(x^j,t^j)\in S\backslash E$ for all $j$, showing that 
(P) holds for $H=S\backslash E$. Since $(x,t)\notin E$, (P) holds vacuously
for $H=E\cap S$.

We have thus shown that property (P) holds in all possible cases
both for \mbox{$H=E\cap S$} and $H=S\backslash E$, whenever $E\in{\bf A}^*$.
Therefore (P) holds for all $H\in{\cal G}$.

Having shown that (P) holds for all $H\in{\cal G}$, Fact \ref{ugg}
follows easily: as explained above, it suffices to show for our given
$(x,t)$ and our given sequence $\{(x^j,t^j)\}$ that $f(x^j,t^j)\to
f(x,t)$ if $f\in\Phi_{{\cal G}}$.  But this is trivial, for $(x,t)$
belongs to a $G\in{\cal G}$ on which $f$ is continuous, and then
$(x^j,t^j)\in G$ for large $j$ in view of (P), so $f(x^j,t^j)\to
f(x,t)$.  Therefore Fact \ref{ugg} is proved.

\medbreak
We are now ready to construct the desired selection $f$ of $F$.

First, we observe that, since the function $\rho_{g,F}$ is upper
semicontinuous, (\ref{ispos}) implies that there exists a continuous
function $\gamma: S\mapsto\rr$ such that 
\begin{equation}\label{isposg}
\rho_{g,F}(x,t)<\gamma(x,t)<\beta(x,t)\,\,\,{\rm
for\,\,\,all}\,\,\,(x,t)\in S\,.
\end{equation}
\forlong{(Proof: for every $q=(x,t)\in S$, pick a relatively open
bounded subset
$U_{q}$ of $S$ such that $q\in U_q$ and a constant
$\kappa_q$ such that $\rho_{g,F}(q')<\kappa_{q}<
\beta(q')$ for all $q'\in U_q$. Then
${\cal U}=\{U_q:q\in S\}$ is an open covering of $S$. Since
$S$ is metric and hence paracompact, there exists a locally finite
set ${\cal V}$ of relatively open subsets of $S$ which is a
covering of $S$ and a refinement of ${\cal U}$. For each 
$V\in{\cal V}$, pick a point $q_V$ of $S$ such that 
$V\subseteq U_{q_V}$.  Define functions $\psi_V:S\mapsto[0,\infty\,[$ 
by $\psi_V(v)={\rm dist}(v,S\backslash V)$ for $v\in S$. Then the 
$\psi_V$ are continuous, nonnegative, and such that 
$\psi_V(v)>0$ if and only if $v\in V$. Let $\psi=\sum_{V\in{\cal V}}\psi_V$.
Define $\ffi_V=\frac{\psi_V}{\psi}$. 
Then the $\ffi_V$ are continuous, nonnegative, and such that 
$\ffi_V(v)>0$ if and only if $v\in V$, and 
$\sum_{V\in{\cal V}}\ffi_V\equiv 1$. Now define 
$\gamma(q)=\sum_{V\in{\cal V}}\ffi_V(q)\kappa_{q_V}$.)}


We then let $\delta=\frac{1}{2}(\beta-\gamma)$, and define
$\delta_k=(1-2^{-k})\delta$ for $k=0,1,\ldots$,
so $\delta$ and the $\delta_k$ are continuous functions 
{}from $S$ to $\rr$.

Pick, for each $(x,t)\in S$, a point $z_{x,t}\in F(x,t)$ such that
\begin{equation}\label{ispois}
\|g(x,t)-z_{x,t}\|<\gamma(x,t)\,.
\end{equation}
(This is possible because of (\ref{isposg})).
%Then $z_{\bar{x},\bar{t}}=\bar{y}$ since $F(\bar{x},\bar{t})=\{\bar{y}\}$. 
Then find neighborhoods $G_{x,t}$ of $(x,t)$ in $\rr^n\times\rr$ 
such that $G_{x,t}\in{\bf A}^*$, with the property that the inequalities
\begin{eqnarray}
d(z_{x,t},F(x',t'))&<&\frac{1}{2}\delta(x',t')\,,\label{almo}\\
\|g(x',t')-z_{x,t}\|&<&\gamma(x',t')\,,\label{almu}
\end{eqnarray}
hold for all $(x',t')\in G_{x,t}\cap S$.   (The fact that (\ref{almo}) 
can be achieved is a consequence of the lower semicontinuity  
of $F$, together with the continuity of $\delta$, since
$d(z_{x,t},F(x,t))=0$, and $\delta(x,t)>0$. The fact that
(\ref{almu}) can also be obtained follows from the continuity 
of $\gamma$ and $g$, together with (\ref{ispois})). 

Next, find a sequence $(x_1,t_1),(x_2,t_2),\ldots$ such that the
sets $G_{x_j,t_j}\cap S$, for $j=1,2,\ldots$ 
%---recall that $(\bar{x},\bar{t})$ is already defined---  
cover $S$. (This is possible because
$S$ is separable metric and hence second countable.)  Let 
$W_1^0=G_{x_1,t_1}\cap S$, and define inductively
\begin{equation}
W_j^0=(G_{x_j,t_j}\cap S)\backslash(W_1^0\cup W_2^0\cup\ldots
W_{j-1}^0)\,\,\,{\rm for}\,\,\,j=2,3,\ldots\,.
\end{equation}
Then 
${\cal P}^0=\{W_j^0:j=1,2,\ldots\}$ is a partition
of $S$ into members of ${\cal G}$.
Define $f_0(x,t)=z_{x_j,t_j}$ if $(x,t)\in W_j^0$. Then 
%$f_0(\bar{x},\bar{t})=\bar{y}$, 
$f_0$ is constant on each member of ${\cal P}^0$, and 
the inequalities  
\begin{equation}
d(f_0(x,t),F(x,t))<\frac{1}{2}\delta(x,t)\,,
\end{equation}
\begin{equation}
\|f_0(x,t)-g(x,t)\|<\gamma(x,t)\,,
\end{equation}
hold for all $(x,t)$. 

We now suppose that we have constructed maps $f_0,\ldots,f_k:S\mapsto
Y$, and finite or countable partitions ${\cal P}^0,\ldots,{\cal P}^k$
of $S$ into members of ${\cal G}$, such that
\begin{itemize}
\item[(i)] %$f_j(\bar{x},\bar{t})=\bar{y}$, and 
$f_j$ is constant on each member of ${\cal P}^j$ for
$j=0,\ldots,k$,  
\item[(ii)] the inequalities
$$
d(f_j(x,t),F(x,t)) < 2^{-j-1}\delta(x,t)\,
$$ 
$$
\|g(x,t)-f_j(x,t)\|<\gamma(x,t)+\delta_j(x,t)
$$
hold for all $(x,t)\in S$, $j\in\{0,\ldots,k\}$,
\item[(iii)] $\|f_j(x,t)-f_{j-1}(x,t)\| \le 2^{-j}\gamma(x,t)$ for all
$(x,t)\in S$, $j\in\{1,\ldots,k\}$, 
\item[(iv)] for each
$j\in\{1,\ldots,k\}$, ${\cal P}^j$ is a refinement of ${\cal P}^{j-1}$. 
\end{itemize}
(Notice that all these conditions hold for $k=0$, since (i) and (ii) 
have already been established, and (iii) and (iv) are vacuously true.)

We then construct $f_{k+1}$ as follows. Pick a $Q\in{\cal P}^k$.
%and let  $\bar Q={\rm Clos}(Q)\cap S$
Let $y^{k,Q}$ be the constant value of $f_k$ on $Q$.  Then 
\begin{equation}
d(y^{k,Q},F(x,t))=d(f_{k}(x,t),F(x,t))<2^{-k-1}\delta(x,t)
\end{equation}
and 
\begin{equation} 
\|g(x,t)-y^{k,Q}\|<\gamma(x,t)+\delta_k(x,t)
\end{equation}
for all $(x,t)\in Q$.
So we can pick, for each $(x,t)\in Q$,
a $z_{x,t}^{k,Q}\in F(x,t)$ such that 
\begin{equation}
\|y^{k,Q}-z_{x,t}^{k,Q}\|<2^{-k-1}\delta(x,t)\,.
\end{equation}
Then
\begin{eqnarray*}
\|g(x,t)-z_{x,t}^{k,Q}\|&<&\gamma(x,t)+\delta_k(x,t)+2^{-k-1}\delta(x,t)\\
&=&\gamma(x,t)+\delta_{k+1}(x,t)\,.
\end{eqnarray*}
Using the lower semicontinuity of $F$ and the continuity of $\delta$,
$\gamma$, $g$, and $\delta_{k+1}$, we can find for each $(x,t)\in Q$ a
relative neighborhood $W^{k,Q}(x,t)$ of $(x,t)$ in $S$ 
such that the inequalities 
\begin{eqnarray*}
d(z_{x,t}^{k,Q},F(x',t'))&<&2^{-k-2}\delta(x',t')\,,\\
\|y^{k,Q}-z_{x,t}^{k,S}\|&<&2^{-k-1}\delta(x',t')\,,\\
\|g(x',t')-z_{x,t}^{k,Q}\|&<&\gamma(x',t')+\delta_{k+1}(x',t')\,,
\end{eqnarray*}
hold for all $(x',t')\in W^{k,Q}(x,t)$. Using Fact \ref{ogg},
we can assume, after \mbox{shrinking} $W^{k,Q}(x,t)$ if necessary, that
$W^{k,Q}(x,t)\in{\cal G}$. 

We then find for each $Q\in{\cal P}^k$ a finite or countable subset
$M^{k,Q}$ of $Q$ such that 
$$
Q\subseteq\bigcup\,\Big\{\,W^{k,Q}(x,t):(x,t)\in M^{k,Q}\,\Big\}\,.
$$
We enumerate  each $M^{k,Q}$ in an arbitrary fashion
as a finite or countably infinite sequence 
$\Big\{(x_j^{k,Q},t_j^{k,Q})\Big\}_{1\le j<m(k,Q)}$, where 
$m(k,Q)\in\nn\cup\{\infty\}$.
%, subject only
%to the condition that when $Q=Q(k)$ then $(x_1^{k,Q},t_1^{k,Q})=(\bar{x},\bar{t})$.
We then define 
$$
\hat W_1^{k,Q}=Q\cap W^{k,Q}(x_1^{k,Q},t_1^{k,Q})
$$
and, for $1<\ell< m(k,Q)$, we let 
$$
\hat W_\ell^{k,Q}=\Big(Q\cap W^{k,Q}(x_\ell^{k,Q},t_\ell^{k,Q})\Big)\backslash 
(\hat W_1^{k,Q}\cup\ldots\cup\hat W_{\ell-1}^{k,Q})\,.
$$
We then let ${\cal P}^{k+1}$ be the collection of all sets $\hat
W_\ell^{k,Q}$, for all $Q\in{\cal P}^k$ and all $\ell\in\nn$ such that
$1\le\ell<m(k,Q)$.
Then ${\cal P}^{k+1}$ is a finite or countable partition of $S$ into
members of ${\cal G}$, and is a refinement of ${\cal P}^k$.  
%In particular, every member of ${\cal P}^{k+1}$ is bounded.  
If $(x,t)\in S$, we
define \begin{equation} f_{k+1}(x,t)=z_{x_{\ell}^{k,Q},t_\ell^{k,Q}}^{k,Q}\,, \end{equation}
where $Q$ is the unique member of ${\cal P}^k$ such that $(x,t)\in Q$, and
$\ell$ is the unique integer such that $1\le\ell< m(k,Q)$ and $(x,t)\in
\hat W_\ell^{k,Q}$.  Then $f_{k+1}$ is constant on each member of ${\cal
P}^{k+1}$. Moreover, if $(x,t)\in \hat W_\ell^{k,Q}$, then $(x,t)\in
W_\ell^{k,Q}(x_{\ell}^{k,Q},t_\ell^{k,Q})$, so that the inequalities
\begin{eqnarray*}
\|f_{k+1}(x,t)-f_{k}(x,t)\|&=&\|z_{x_{\ell}^{k,Q},t_\ell^{k,Q}}^{k,Q}
-y^{k,Q}\| \le 2^{-k-1}\delta(x,t)\,,\\
d(f_{k+1}(x,t),F(x,t))&=&d(z_{x_{\ell}^{k,Q},t_\ell^{k,Q}}^{k,Q},F(x,t))
<2^{-k-2}\delta(x,t)\,,\\
\|g(x,t)-f_{k+1}(x,t)\|&=&\|g(x,t)-z_{x_{\ell}^{k,Q},t_\ell^{k,Q}}^{k,Q}\|
<\gamma(x,t)+\delta_{k+1}(x,t)\,.
\end{eqnarray*}
are satisfied. It follows that the maps $f_0,\ldots,f_{k+1}$ and the
corresponding partitions ${\cal P}^0,\ldots,{\cal P}^{k+1}$ also satisfy
(i),$\ldots$,(iv).

The above inductive construction therefore leads to an infinite
sequence $\{f_j\}$ of members of $\Phi_{{\cal G}}$. In view of (iii),
this sequence converges uniformly on compact subsets of $S$ to a limit
$f$. It then follows from (ii), \mbox{together} with the fact that $F$
has closed values, that $f$ is a selection of $F$ and
$$
\|g(x,t)-f(x,t)\|\le\gamma(x,t)+\delta(x,t)<\beta(x,t)\quad
{\rm for\,\,\,all}\quad(x,t)\in S\,.
$$
In view of Fact \ref{ugg}, $f$ is continuous at $(\bar{x},\bar{t})$ and
forward $\Gamma_C$-continuous on $S$, since $f\in\bar\Phi_{{\cal G}}$ by
construction.

It is clear that $f(\bar{x},\bar{t})=\bar{y}$, because the map $f$
is a selection of $F$
and $F(\bar{x},\bar{t})=\{\bar{y}\}$. In addition, the maps $f_j$ are
obviously ${\rm Bor}(\rr^n\times\rr)\cap S$-measurable, because 
${\cal G}\subseteq{\rm Borel}(\rr^n\times\rr)\cap S$, since
${\bf A}^*\subseteq{\rm Borel}(\rr^n\times\rr)$.
Since $f_j\to f$ pointwise, we see that
$f$ is ${\rm Bor}(\rr^n\times\rr)\cap S$-measurable.
So $f$ satisfies all our
conditions, and our proof is complete.\proofend
\end{proof}



\subssection{Almost lower semicontinuous set-valued 
maps}\label{subsec-def-alsc}

\forlong{
The \ natural \ analogue \ of \ the \ well \ known \ Scorza-Dragoni
\ theorem \ for \mbox{continuous} set-valued maps is {\bf not} true in 
general for lower \mbox{semicontinuous} set-valued maps.
\begin{example} 
Let $h:[0,1]\mapsto[0,1]$ be a function which is not
Lebesgue-measurable. Define $F:[0,1]\times[0,1]\tto[0,1]$ by letting
$F(x,t)=[0,1]$ if $x\neq t$, $F(x,x)=\{h(x)\}$.   Then 
it is clear that $x\mapsto F(x,t)$ is LSC for each fixed $t$,
and $t\mapsto F(x,t)$ is measurable for each fixed $x$. If there existed
for each positive $n\in\nn$ a compact subset $K_n$ of $[0,1]$ such that
$\msr([0,1]\backslash K_n)\le\frac{1}{n}$ with the property 
that $F\restr([0,1]\times K_n)$ is LSC, then in particular 
$F_n=F\restr \tilde K_n$ would be LSC, where $\tilde K_n=\{(x,x):x\in K_n\}$.
But then $F_n$ would actually have to be continuous, because it is
single-valued. Since $F_n(x,x)=h(x)$, this implies that $h\restr K_n$ 
is continuous for each $n$. But then $h$ is Lebesgue-measurable, and we have
reached a contradiction.\proofend
\end{example}
In view of this fact, it is reasonable to turn the conclusion
of the Scorza-Dragoni theorem into a definition, and introduce the class
of ``almost lower semicontinuous'' set-valued maps.
}

\forshort{
It is well known that the natural analogue of the Scorza-Dragoni
theorem for \mbox{continuous} set-valued maps is {\bf not} true in 
general for lower \mbox{semicontinuous} set-valued maps.
It is therefore reasonable to turn the conclusion
of the Scorza-Dragoni theorem into a definition, and introduce the class
of ``almost lower semicontinuous'' set-valued maps.
}


\medbreak
Recall that, if $X$ is a set and $S$ is a subset of $X\times\rr$, then
$S_J$ denotes the set $S\cap(X\times J)$. If $X$, $Y$ are sets, 
$S\subseteq X\times\rr$, $F:S\tto Y$ is a set-valued map,
and $J\subseteq\rr$, we write 
$F_J$ to denote the restriction of $F$ to $S_J$, 
so $F_J$ is a set-valued map from $S_J$ to $Y$ 
$F_J(x,t)=F(x,t)$ whenever 
$(x,t)\in S_J$, and $\Do(F_J)=\Big(\Do(F)\Big)_J$.  
\begin{definition}\label{defofalsc}
Let $X$, $Y$ be topological spaces, let $S$ be a subset of 
the product \mbox{$X\times\rr$}, and let $F:S\tto Y$ be a set-valued map. 
We call $F$ {\em almost lower \mbox{semicontinuous}} (abbr. ALSC) if
\begin{itemize}\item[]\begin{itemize}
\item[(ALSC)] {\em for every $\eps>0$ there exists a closed subset $J$ of 
$\rr$ such that ${\rm meas}(\rr\backslash J)\le\eps$ and $F_J$ is
lower semicontinuous}.\proofend
\end{itemize}\end{itemize}
\end{definition}
The following characterization of the ALSC property 
is easily proved.
\bfact
\ If $X,Y,S,F$ are as in Definition \ref{defofalsc}, then 
$F$ is ALSC  iff

\begin{itemize}\item[]\begin{itemize}
\vspace{-1.5mm}
\item[(ALSC.a)] there exists a sequence  $\{J_k\}_{k=0}^\infty$ 
such that  

\begin{itemize}
\vspace{-1mm}
\item[(ALSC.a.i)\phantom{i}] the $J_k$ are pairwise disjoint subsets of 
$\rr$, $\rr=\bigcup_{k=0}^\infty J_k$, \mbox{$J_0$ has measure zero},  
and the $J_k$ for $k>0$ are compact, 

\vspace{-1mm}
\item[(ALSC.a.ii)] $F_{J_k}$ is LSC whenever 
$k>0$.\forshort{\proofend}
\end{itemize}
\end{itemize}\end{itemize}
\efact
\forlong{
\begin{proof}
It is clear that (ALSC) implies (ALSC.a).  To prove the
converse, assume that (ALSC.a) holds, and let 
$\{J_k\}_{k=0}^\infty$ be a family of sets that satisfies
(ALSC.a.i). Let $C$ be the set of all closed subsets
$L$ of $\rr$ such that $F_L$ is LSC. Then it is easy to see that $C$ is
closed under restrictions (i.e., if $L\in C$, $L'\subseteq S$, and 
$L'$ is closed, then $L'\in C$) and locally finite unions (i.e., if
$\{L^{\alpha}\}_{\alpha\in A}$ is a locally finite family of members
of $C$, then $L=\cup_{\alpha}L^{\alpha}$ also belongs to $C$).  
The sets $J_{k,n}=J_k\cap[n,n+1]$, for
$k\in\nn$, $k>0$, and $n$ an arbitrary integer, obviously belong to
$C$. Pick $\eps>0$, and then pick, for each $n$, a $k^*(n)$ such that
$$
{\rm  meas}\Biggl([n,n+1]\Big\backslash\Big(\bigcup_{k=1}^{k^*(n)}J_{k,n}
\Big)\Biggr)<\frac{\eps}{3\cdot 2^{|n|}}\,.
$$
Let $J=\bigcup_n\Big(\cup_{k=1}^{k^*(n)}J_{k,n}\Big)$. 
Then $J$ is a locally finite union
of members of $C$, so $J\in C$.  In addition, ${\rm meas}(\rr\backslash
J)\le\eps$.  So (ALSC) holds.\proofend
\end{proof}
}

In the special case when $F$ is single-valued, we know from 
Fact \ref{sun1} that the LSC
property is equivalent to ordinary continuity. Therefore
\bfact\label{forsin}
\ If $X,Y,S,F$ are as in Definition \ref{defofalsc}, and
in addition $F$ is single-valued and $\Do(F)\!=\!S$, then 
$F$ is ALSC as a set-valued map \mbox{if and only if} 
\begin{itemize}\item[]\begin{itemize}
\item[(ALSC.s)] there exists a sequence $\{J_k\}_{k=0}^\infty$ 
that satisfies (ALSC.a.i) and is such
that $F\restr S_{J_k}$ is continuous for every $k>0$.\proofend
\end{itemize}\end{itemize}
\efact

\begin{definition}\label{alcont}
If $X,Y,S,F$ are as in Fact \ref{forsin}, we call $F$ {\em almost 
continuous} if it is ALSC as a set-valued map, i.e., if 
it satisfies (ALSC.s).\proofend
\end{definition}

Under more special conditions (for example, if $X$ is a locally
compact \mbox{separable} metric space, $Y$ is a separable metric space, and
$S=\Omega\times I$ with $\Omega$ open in $X$ and $I$ an interval),
almost continuity is equivalent, by the Scorza-Dragoni Theorem, to
the much simpler statement that $g(x,t)$ is \mbox{continuous} in $x$
for almost every fixed $t$ and measurable in $t$ for all $x$.

By analogy with Definition \ref{alcont}, we can also introduce a concept
of ``\mbox{almost} upper semicontinuity'' for real-valued functions, related
in the obvious way to the notion of an upper semicontinuous function (which
is, of course, quite different from that of an 
``upper semicontinuous set-valued map that happens to be an ordinary
map with real values,'' cf. Remark \ref{remonuscscalar}).

\begin{definition}
Let $X$, $Y$ be topological spaces, let $S$ be a subset of 
\mbox{$X\times\rr$}, 
and let $f:S\mapsto \rr$ be a real-valued function. We call $f$ {\em almost 
upper \mbox{semicontinuous}} if there exists a sequence 
$\{J_k\}_{k=0}^\infty$ that satisfies (ALSC.a.i) and is such
that $f\restr S_{J_k}$ is upper semicontinuous for every $k>0$.\proofend
\end{definition}



\subssection{Selections of ALSC maps}\label{sec-selec-alsc}

\begin{definition}
Assume that $n,m\in\nn$, $S\subseteq\rr^n\times\rr$, 
\mbox{$f:S\mapsto\rr^m$}, and $C:S\mapsto\,]0,\infty[\,$. 
We call $f$ {\em almost forward $\Gamma_C$-continuous}
if there exists a \mbox{sequence} $\{J_k\}_{k=0}^\infty$ 
for which (ALSC.a.i) holds, such that
the restriction $f_{J_k}=f\restr S_{J_k}$
is forward \mbox{$\Gamma_C$-continuous} whenever $k>0$. \proofend
\end{definition}

\begin{theorem}\label{selectionalsc}
Assume that $n,m\in\nn$, $S\subseteq\rr^n\times\rr$, and 
$F:S\tto\rr^m$ is an ALSC set-valued map with nonempty closed values.  
Let $g:S\mapsto\rr^m$ be an almost continuous single-valued map
and let $\beta:S\mapsto\rr$ be almost continuous and such that 
\begin{equation}\label{isposnew}
{\rm for\,\,a.e.\,\,}t\in\rr: 
\beta(x,t)>\rho_{g,F}(x,t)\,\,{\rm whenever\,\,}x\in\rr^n\,,\,(x,t)\in S\,.
\end{equation}
Assume that both $g$ and $\beta$ are LIB. 
Let $C\!:\!S\!\mapsto\,]0,\infty[\,$ be almost upper
\mbox{semicontinuous}.  Then the \mbox{following} is true 
for almost all $\bar{t}\!\in\!\rr$: 
\begin{itemize}\item[]\begin{itemize}
\item[(T\ref{selectionalsc}.\#)] For every $\bar{x}\in\rr^n$ such that
$(\bar{x},\bar{t})$ is an interior point of $S$, 
and every $\bar{y}\in F(\bar{x},\bar{t})$ such that
\begin{equation}\label{isposnew2}
\|g(\bar{x},\bar{t})-\bar{y}\|<\beta(\bar{x},\bar{t})\,.
\end{equation}   
there exists a selection $f:S\mapsto\rr^m$ of $F$ such that 
\begin{itemize}
\item[(T\ref{selectionalsc}.1)] $f(\bar{x},\bar{t})=\bar{y}$;
\item[(T\ref{selectionalsc}.2)]  $(\bar{x},\bar{t})$ is a point of approximate 
continuity of $f$;
\item[(T\ref{selectionalsc}.3)] $f$ is almost forward $\Gamma_C$-continuous;
\item[(T\ref{selectionalsc}.4)] for a.e. $t\in\rr$, 
$\|f(x,t)-g(x,t)\|<\beta(x,t)$ for all $x\in\rr^n$ such that $(x,t)\in S$, 
\item[(T\ref{selectionalsc}.5)] $f$ is 
${\cal EM}(\rr^n,\rr)\cap S$-measurable.
\end{itemize}\end{itemize}\end{itemize}
\end{theorem}

\begin{proof}
Let $\Om={\rm Int}(S)$. Let ${\cal K}$ be a finite or countably infinite 
set of \mbox{compact} subsets of $\Om$ such that 
$\Om=\bigcup\{{\rm Int}(K):K\in{\cal K}\}$.  For each 
\mbox{$K\in{\cal K}$}, 
let \mbox{$\ffi_K:\rr\mapsto[0,\infty]$} be integrable and such that
\mbox{$\|g(x,t)\|+\beta(x,t)\le\ffi_K(t)$}
whenever $(x,t)\in K$,
and let $L_K$ be the set of Lebesgue points of $\ffi_K$. Let
$L=\bigcap\{L_K:K\in{\cal K}\}$.  Then $L$ is a full subset 
of $\rr$.

Let ${\cal J}=\{J_k\}_{k=0}^\infty$ be a sequence for which
(ALSC.a.i) holds, such that
that the maps $\beta_k=\beta\restr S_{J_k}$, 
$g_k=g\restr S_{J_k}$ are continuous, the functions
$C_k=C\restr S_{J_k}$ are upper semicontinuous, and the
set-valued maps $F_k=F\restr S_{J_k}$ are LSC. (The existence
of ${\cal J}$ is a trivial consequence of our hypotheses.)   

For $k>0$, let $\tilde J_k$ be the set of points of density $t$ of $J_k$
such that $t\in L$.
Let $\tilde J=\cup_{k=1}^\infty \tilde J_k$.  Then ${\rm
meas}(\rr\backslash\tilde J)=0$. We show that every $\bar{t}\in\tilde J$ has 
Property (T\ref{selectionalsc}.\#). 

Let $\bar{t}\in\tilde J$.  Let $k_0$ be the unique $k\in\nn$ such that
$\bar{t}\in J_k$, so that $\bar{t}\in\tilde J_{k_0}$, $k_0>0$, and $\bar{t}\in L$.
Let $\bar{x}\in\rr^n$ be such that $(\bar{x},\bar{t})\in{\rm Int}(S)$. 

Theorem \ref{selectionlsc} implies that for every $k\in\nn$ such that
$k>0$ there exists a selection \mbox{$f_k:S_{J_k}\mapsto\rr^m$} of
$F_k$ which is forward $\Gamma_{C_k}$-continuous and such that the
bound $\|g_k(x,t)-f_k(x,t)\|<\beta_k(x,t)$ holds for all $(x,t)\in
S_{J_k}$.  Moreover, when $k=k_0$ we can choose $f_k$ such that
$f_k(\bar{x},\bar{t})=\bar{y}$ and $f_k$ is continuous at
$(\bar{x},\bar{t})$.  Define $f_0$ to be an arbitrary selection of
$F\restr S_{J_0}$.  Then define $f$ by letting $f(x,t)=f_k(x,t)$ for
$(x,t)\in S_{J_k}$.

Clearly, $f$ satisfies 
(T\ref{selectionalsc}.1), (T\ref{selectionalsc}.3),
(T\ref{selectionalsc}.4), and
(T\ref{selectionalsc}.5). \mbox{Moreover}, if we let $E=J_{k_0}$, pick
a $K\in{\cal K}$ such that $(\bar{x},\bar{t})\in{\rm Int}(K)$, and define
$\ffi\equaldef\ffi_K$, then conditions (P\ref{defapproxcont}.a) and
(P\ref{defapproxcont}.b) of Proposition \ref{defapproxcont}
hold. Furthermore, since $\|f(x,t)-g(x,t)\|<\beta(x,t)$ for all
$(x,t)\in S$, and $\|g(x,t)\|+\beta(x,t)\le\ffi(t)$ for all $(x,t)\in
K$, condition (P\ref{defapproxcont}.c) holds as well.  Therefore
Proposition \ref{defapproxcont} implies that $(\bar{x},\bar{t})$ is a PAC of
$f$. So (T\ref{selectionalsc}.2) holds.\proofend
\end{proof}

\begin{theorem}\label{selectionalscprod}
Assume that $n,m\in\nn$ and $S\subseteq\rr^n\times\rr$. Let 
\mbox{$F:S\tto\rr^m$} be an ALSC set-valued map 
with $\,$nonempty $\,$closed $\,$values.  $\,\,$$\,$Let 
\mbox{$\psi:S\mapsto[0,\infty]$} be an almost continuous 
single-valued locally integrably bounded \mbox{function} such that 
$\min\Big\{\,\|y\|:y\in F(x,t)\,\Big\}<\psi(x,t)\quad{\rm for\,\,\,every}\quad
(x,t)\in S$.
Let \mbox{$\zeta:S\mapsto[0,\infty]$} be an almost upper semicontinuous 
single-valued function. 

Then the following is true for almost all $\bar{t}\in\rr$:
\begin{itemize}\item[]\begin{itemize}
\item[(T\ref{selectionalscprod}.\#)] For every pair $(\bar{x},\bar{y})$ 
such that $(\bar{x},\bar{t})\in{\rm Int}(S)$,
\mbox{$\bar{y}\in F(\bar{x},\bar{t})$}, and 
\mbox{$\|\bar{y}\|<\psi(\bar{x},\bar{t})$}, 
there exists a selection $f:S\mapsto\rr^m$ of $F$ such that 
\begin{itemize}
\item[(T\ref{selectionalscprod}.1)] $f(\bar{x},\bar{t})=\bar{y}$;
\item[(T\ref{selectionalscprod}.2)] $(\bar{x},\bar{t})$
is a point of approximate  continuity of $f$;
\item[(T\ref{selectionalscprod}.3)] $f$ is integrally continuous on 
$\widetilde{ARC}_{\zeta}(S)$,
\item[(T\ref{selectionalscprod}.4)] $\|f(x,t)\|<\psi(x,t)$ for all 
$(x,t)\in S$.
\end{itemize}\end{itemize}\end{itemize}
\end{theorem}

\begin{proof} We apply Theorem \ref{selectionalsc} with $g\equiv 0$, 
$\beta=\psi$, $C=1+\zeta$. This \mbox{yields} a selection $f$ that satisfies
(T\ref{selectionalscprod}.1), (T\ref{selectionalscprod}.2) and 
(T\ref{selectionalscprod}.4), and in addition is 
${\cal EM}(\rr^n,\rr)\cap S$-measurable and almost
forward $\Gamma_C$-continuous.  Then there exists a sequence
$\{J_k\}_{k=0}^\infty$ such that 
(ALSC.a.i) holds, having the property that the 
restrictions $\psi_k=\psi\restr S_{J_k}$,
$C_k=C\restr S_{J_k}$, $f_k\equaldef f\restr S_{J_k}$
are, \mbox{respectively}, continuous, upper semicontinuous,
and forward $\Gamma_{C_k}$-continuous.
Define maps $\tilde f_k(x,t):S\mapsto\rr^m$ by:
$$
\tilde f_k(x,t)=\left\{\begin{array}{lcl}
f_k(x,t)&{\rm if}&(x,t)\in S\,,\,t\in J_k\,,\\
0&{\rm if}&(x,t)\in S\,,\,t\notin J_k\,.\end{array}\right.
$$
(Naturally, $\tilde f_k$ may fail to be a selection
of $F$.)  We then have
\bfact\label{easyo}
$\tilde f_k$ is weakly forward $\Gamma_C$ continuous
on $S$ for every $k>0$.
\efact
To see this, we fix a $k$, and show that
\begin{itemize}
\item[(\&)] {\em if the following three conditions are true:
\begin{itemize}
\item[1.] $(x,t)\in S$, 
\item[2.] $(x_\ell,t_\ell)\in S$, \ $\,(x'_\ell,t_\ell)\in S$,
\ $\,\|x_\ell-x\|\le C(x,t)(t_\ell-t)\,$, \ and
\mbox{$\|x_\ell'-x\|\le C(x,t)(t_\ell-t)$} for $\ell=1,2,3,\ldots$,
\item[3.] \mbox{$x_\ell\to x$}, \mbox{$x_\ell'\to x$}, and $t_\ell\to t$, 
as $\ell\to\infty$, 
\end{itemize}
then $\tilde f_k(x_\ell,t_\ell)-\tilde f_k(x_\ell',t_\ell)\to 0$.}
\end{itemize}

Define $L_1=\{\ell:t_\ell\in J_k\}$, $L_2=\{\ell:t_\ell\notin J_k\}$.
If $L_1$ is infinite, then $\lim_{\ell\to\infty\,,\,\ell\in L_1}t_\ell=t$,
so $t\in J_k$, because $J_k$ is closed.  Then $C(x,t)=C_k(x,t)$. Clearly, 
\begin{equation}
\tilde f_k(x_\ell,t_\ell)-\tilde f_k(x_\ell',t_\ell)=
f_k(x_\ell,t_\ell)-f_k(x_\ell',t_\ell)
\end{equation}
if $\ell\in L_1$.  
Since $f_k$ is forward $\Gamma_{C_k}$-continuous,
$f_k(x_\ell,t_\ell)\to f_k(x,t)$ and $f_k(x_\ell',t_\ell)\to f_k(x,t)$
as $\ell\to\infty$ via values in $L_1$.  
So $\tilde f_k(x_\ell,t_\ell)-\tilde f_k(x_\ell',t_\ell)\to 0$ as
$\ell\to\infty$ via $L_1$.   If $L_2$ is infinite, then  
$\tilde f_k(x_\ell,t_\ell)-\tilde f_k(x_\ell',t_\ell)\to 0$ as 
$\ell\to\infty$ via $L_2$,  because 
$\tilde f_k(x_\ell,t_\ell)=\tilde f_k(x_\ell',t_\ell)=0$ when $\ell\in
L_2$.  This completes the proof of Fact \ref{easyo}.

Since each $\tilde f_k$ is LIB with integral bound $\psi$,
${\cal EM}(\rr^n,\rr)\cap S$-measurable, and weakly
$\Gamma_C$-continuous, 
Theorem \ref{theoweakly} implies that all the $\tilde f_k$ are
integrally continuous on $\widetilde{ARC}_\zeta(S)$.  
Moreover, the sequence $\{\tilde f_k\}_{k=1}^\infty$ is uniformly LIB.
It therefore follows from
the measurable intertwining property \mbox{(Theorem \ref{meassub})}  that $f$
is integrally continuous on $\widetilde{ARC}_\zeta(S)$. \proofend
\end{proof}

\subssection{The main theorem}\label{sec-main}

\begin{definition}\label{defoflilb}
Assume that $n,m\in\nn$, $S\subseteq\rr^n\times\rr$, and 
$F:S\tto\rr^m$ has nonempty values.
We say that $F$ is {\em locally integrably lower bounded}
(abbr. LILB) if for every compact subset $K$ of 
$S$ there exists an integrable function $\ffi\!:\!\rr\!\mapsto\![0,\infty]$
such that $\inf\Big\{\|y\|\!:\!y\!\in\!F(x,t)\Big\}\!\le\!\ffi(t)$
whenever $(x,t)\!\in\!K$.\proofend
\end{definition}  

We recall that a subset $S$ of a Euclidean space $\rr^k$ is 
locally compact if and only if it is the intersection
of an open set and a closed set. Moreover, if $S$ is locally compact
then there exists a sequence$\{K_i\}_{i=1}^\infty$ 
of compact subsets of $S$ such that $K_i\subseteq{\rm Int}(K_{i+1})$ 
for every $i$ and $S=\bigcup_{i=1}^\infty K_i$. Such a sequence then has 
the property that if $K$ is an arbitrary compact subset of $S$ then 
$K\subseteq{\rm Int}(K_i)$ for some $i$.

\begin{lemma}\label{lemloccomp}
Assume that $n,m\in\nn$, $S$ is a locally compact subset of 
$\rr^n\times\rr$, and
$F:S\tto\rr^m$ is a set-valued map with closed nonempty values.  Then $F$ is
LILB if and only if there exists an almost continuous single-valued
locally integrably bounded \mbox{function}
\mbox{$\psi:S\mapsto[0,\infty]$} such that
\begin{equation}\label{heresh}
\min\Big\{\,\|y\|:y\in F(x,t)\,\Big\}<\psi(x,t)\quad{\rm for\,\,\,every}\quad
(x,t)\in S\,.
\end{equation} 
\end{lemma}
\begin{proof}
If $\psi$ exists then it is clear that $F$ is LILB. Conversely, suppose
that $F$ is LILB. Let $\{K_i\}_{i=1}^\infty$ be a sequence
of compact subsets of $S$ such that $K_i\subseteq{\rm Int}(K_{i+1})$ 
for every $i$, and $S=\bigcup_{i=1}^\infty K_i$.  For each $i$,
let $\ffi_i:\rr\mapsto[0,\infty]$ be an integrable function
such that
$$
\rho_{0,F}(x,t)=
\min\Big\{\,\|y\|:y\in F(x,t)\,\Big\}<\ffi_i(t)\quad{\rm for\,\,\,every}\quad
(x,t)\in K_i\,.
$$ 
Let $L_1=K_1$, $L_i=K_i\backslash{\rm Int}(K_{i-1})$
if $i>1$. 
Also, let $\Om_1={\rm Int}(K_2)$, \mbox{$\Om_2={\rm Int}(K_3)$}, 
and  $\Om_i={\rm Int}(K_{i+1})\backslash K_{i-2}$ for $i>2$.
Then the $L_i$ are \mbox{compact}, the $\Om_i$ are relatively open 
subsets of $S$, and $L_i\subseteq\Om_i$ for \mbox{every $i$}.
\mbox{Moreover}, if $K$ is an arbitrary compact subset of $S$, then
there is an $i$ such that $K\subseteq{\rm Int}(K_i)$, and then
$K\cap\Om_j=\emptyset$ whenever $j\ge i+2$, so the \mbox{sequence}
$\{\Om_i\}_{i=1}^\infty$ is locally finite. Finally, it is clear that
$S=\bigcup_{i=1}^\infty L_i$. For each $i$, let $\eta_i:S\mapsto\rr$ be a 
nonnegative continuous function such that $\eta_i(x,t)=1$ whenever
$(x,t)\in L_i$ and ${\rm Clos}\Big(\{(x,t):\eta_i(x,t)\neq 0\}\Big)
\subseteq\Om_i$. Let $\eta(x,t)=\sum_{i=1}^\infty\eta_i(x,t)$, so
$\eta$ is a continuous strictly positive function on $S$. Let
$\theta_i(x,t)=\frac{\eta_i(x,t)}{\eta(x,t)}$, so the 
$\theta_i$ are nonnegative continuous functions on $S$
such that 
${\rm Clos}\Big(\{(x,t):\theta_i(x,t)>0\}\Big)\subseteq\Om_i$ for each
$i$, and $\sum_{i=1}^\infty\theta_i\equiv 1$.

Define $\psi(x,t)=\sum_{i=1}^\infty\ffi_{i+1}(t)\theta_i(x,t)$. 
Let $(x,t)\in S$. Let $I(x,t)$ be the set of all $i$ 
such that $\theta_i(x,t)>0$. If $i\in I(x,t)$, then $(x,t)\in\Om_i$,
so $(x,t)\in K_{i+1}$, and then $\rho_{0,F}(x,t)\le\ffi_{i+1}(t)$. 
Therefore $\psi(x,t)$ is a convex combination of real numbers $r$
such that  $\rho_{0,F}(x,t)\le r$. It follows that  
$\rho_{0,F}(x,t)\le\psi(x,t)$ for all $(x,t)\in S$, so 
(\ref{heresh}) holds.

If $K$ is a compact subset of $S$, and $i$ is such that
$K\subseteq{\rm Int}(K_i)$, then 
$\psi(x,t)=\sum_{j=1}^{i+1}\ffi_{j}(t)\theta_j(x,t)$, because 
$K\cap\Om_j=\emptyset$ whenever $j\ge i+2$.
So $\psi(x,t)$ is bounded above, for $(x,t)\in K$,
by the integrable function
\mbox{$\rr\ni t\mapsto\sum_{j=1}^{i+1}\ffi_{j}(t)$}. Therefore 
$\psi$ is locally integrably bounded. 

Finally, we show that $\psi$ is almost continuous. Given a positive
$\eps$, we can use Lusin's theorem to find, for each $i$, a
closed subset $C_i$ of $\rr$ such that 
$\msr(\rr\backslash C_i)<2^{-i}\eps$ and $\ffi_i\restr C_i$ is 
finite-valued and continuous.
Let $C=\bigcap_{i=1}^\infty C_i$. Then $S$ is closed,
$\msr(\rr\backslash C)<\eps$, and $\ffi_i\restr C$ is continuous.
It follows that the locally finite sum 
$\sum_{i=1}^\infty\ffi_{i+1}(t)\theta_i(x,t)$ is continuous  
as a function on $S\cap(\rr^n\times C)$. In other words, the
restriction of $\psi$ to $S_C$ is continuous, as desired.\proofend
\end{proof}

We are now ready to state and prove our main result.


\begin{theorem}\label{selectionalscprod-main}
Assume that $n\in\nn$ and $S$ is a locally compact subset of  
$\rr^n\times\rr$. Let 
\mbox{$F:S\tto\rr^n$} be an ALSC LILB set-valued map 
with $\,$nonempty $\,$closed $\,$values.  
Then the following is true for almost all $\bar{t}\in\rr$:
\begin{itemize}\item[]\begin{itemize}
\item[(T\ref{selectionalscprod-main}.\#)] For every $\bar{x}$ such that 
$(\bar{x},\bar{t})\in{\rm Int}(S)$ and every 
$\bar{y}\in F(\bar{x},\bar{t})$  
there exist a selection $f:S\mapsto\rr^n$ of $F$ and an almost
continuous LIB function $\psi:S\mapsto[0,\infty]$ such that 
\begin{itemize}
\item[(T\ref{selectionalscprod-main}.1)] $f(\bar{x},\bar{t})=\bar{y}$;
\item[(T\ref{selectionalscprod-main}.2)] $(\bar{x},\bar{t})$
is a point of approximate  continuity of $f$;
\item[(T\ref{selectionalscprod-main}.3)] $\|f(x,t)\|\le\psi(x,t)$ for all
$(x,t)\in S$;
\item[(T\ref{selectionalscprod-main}.4)] $f$ is integrally continuous
on $\widetilde{ARC}_\psi(\rr^n;S)$;
\item[(T\ref{selectionalscprod-main}.5)] $f$ $\,\,$is $\,\,$an 
$\,\,$admissible $\,\,$time-varying $\,\,$vector $\,\,$field $\,\,$on 
$\,\,$some \mbox{neighborhood} of $(\bar{x},\bar{t})$;
\item[(T\ref{selectionalscprod-main}.6)] the flow map $\Phi^f$ is regularly differentiable
at $(\bar{x},\bar{t},\bar{t})$ with \mbox{differential} equal to the linear map
$$
\rr^n\times\rr\times\rr\ni(v,h,k)\mapsto v+(h-k)\bar{y}\in\rr^n\,.
$$
\end{itemize}\end{itemize}\end{itemize}
\end{theorem}

\begin{proof}
It follows from Lemma \ref{lemloccomp} that there exists an
almost continuous LIB function $\psi:S\mapsto\rr$
such that (\ref{heresh}) holds.   Let $\Om={\rm Int}(S)$, 
and let $\{K_i\}_{i=1}^\infty$ be a sequence of compact subsets
of $\Om$ such that $K_i\subseteq{\rm Int}(K_{i+1})$ for every $i$
and $\Om=\bigcup_{i=1}^\infty K_i$.  For each $i$, pick an
integrable function $\ffi_i:\rr\mapsto[0,\infty]$ such that 
$\psi(x,t)\le\ffi_i(t)$ whenever $(x,t)\in K_{i+1}$. We define
$\zeta_i:S\mapsto[0,\infty]$ by letting
$$
\zeta_i(x,t)=\left\{\begin{array}{lcl}
\psi(x,t)&{\rm if}&(x,t)\in S\backslash K_i\,,\\
\ffi_i(t)&{\rm if}&(x,t)\in K_i\,,\end{array}\right.
$$
and remark that $\psi(x,t)\le\zeta_i(x,t)$ for all $(x,t)\in S$.

We claim that $\zeta_i$ is almost upper semicontinuous for every $i$. 
To see this, fix $i$ and a positive number $\eps$, and find a 
closed subset $C$ of $\rr$ such that $\msr(\rr\backslash C)<\eps$
and the functions 
$S\cap(\rr^n\times C)\ni(x,t)\mapsto\psi(x,t)\in[0,\infty]$
and $C\ni t\mapsto\ffi_i(t)\in[0,\infty]$ are finite-valued
and continuous.

Let us show that $\zeta_i$ is upper semicontinuous on 
$S_C=S\cap(\rr^n\times C)$. To prove this, 
we fix $\alpha\in\rr$, let $W=\{(x,t)\in S_C:\zeta_i(x,t)\ge\alpha\}$,
and prove that $W$ is relatively closed in $S_C$. Let 
$\{(x_j,t_j)\}_{j\in\nn}$ be a sequence of members of $W$ that converges
to a limit $(x,t)\in S_C$.  Then $\psi(x_j,t_j)\to\psi(x,t)$
and $\ffi_i(t_j)\to\ffi_i(t)$ as $j\to\infty$. If $(x,t)\notin K_i$,
then $(x_j,t_j)\notin K_i$ for large enough $j$. Therefore
$\zeta_i(x,t)=\psi(x,t)$ and $\zeta_i(x_j,t_j)=\psi(x_j,t_j)$ for large
$j$, so $\zeta_i(x_j,t_j)\to\zeta_i(x,t)$. Since 
$\zeta_i(x_j,t_j)\ge\alpha$, we find that $\zeta_i(x,t)\ge\alpha$,
so $(x,t)\in W$. Now suppose that $(x,t)\in K_i$. Then 
$(x_j,t_j)\in K_{i+1}$ for large enough $j$, because
$K_i\subseteq{\rm Int}(K_{i+1})$. Therefore 
$\psi(x_j,t_j)\le\ffi_i(t_j)$ for large $j$, and then the definition of
$\zeta_i$ shows that $\zeta_i(x_j,t_j)\le\ffi_j(t_j)$
for large $j$. It follows that
$\zeta_i(x,t)=\ffi_i(t)=\lim_{j\to\infty}\ffi_i(t_j)
\ge\lim\sup_{j\to\infty}\zeta_i(x_j,t_j)\ge\alpha$.
Therefore $(x,t)$ belongs to $W$ in this case as well, concluding the 
proof that $\zeta_i$ is almost upper semicontinuous.

Clearly, (\ref{heresh}) holds 
as well if $\psi$ is replaced by $\psi_N$, where 
$\psi_N\equaldef\psi+N$ and $N$ is an arbitrary nonnegative integer.
We can therefore apply Theorem \ref{selectionalscprod}, given any $N$ and 
$i$, with $\psi_N$ in the role of $\psi$, and 
$\zeta_i^N$ in the role of $\zeta$, where $\zeta_i^N(x,t)=
\zeta_i(t)+N$. We can then find full subsets $E_N^i$ of 
$\rr$ such that, whenever 
$\bar{t}\in E_N^i$, $(\bar{x},\bar{t})\in{\rm Int}(S)$, 
$\bar{y}\in F(\bar{x},\bar{t})$, and $\|\bar{y}\|<\psi_N(\bar{x},\bar{t})$,
there exists  a selection $f$ of $F$ that satisfies
(T\ref{selectionalscprod-main}.1) and 
(T\ref{selectionalscprod-main}.2), 
as well as the inequality $\|f(x,t)\|\le\psi_N(x,t)$ for all
$(x,t)\in S$, and is such that 
$f$ is integrally continuous on $\widetilde{ARC}_{\zeta_i^N}(\rr^n;S)$.

Let $E=\bigcap_{N=0}^\infty\bigcap_{i=1}^\infty E_N^i$. 
Then $E$ is a full subset of $\rr$. Let us show that
(T\ref{selectionalscprod-main}.\#) is true for every $\bar{t}\in E$.
To see this, pick a $\bar{t}\in E$, an $\bar{x}$ such that
$(\bar{x},\bar{t})\in{\rm Int}(S)$, and a
$\bar{y}\in F(\bar{x},\bar{t})$. Then choose an $N$
such that $\|\bar{y}\|<\psi_N(\bar{x},\bar{t})$ and an $i$ such that 
$(\bar{x},\bar{t})$ is an interior point of $K_i$. Then 
$t\in E_N^i$, so there
exists a selection $f$ of $F$ such that 
(T\ref{selectionalscprod-main}.1) and (T\ref{selectionalscprod-main}.2)
hold, (T\ref{selectionalscprod-main}.3) holds
with $\psi$ replaced by $\psi_N$, and 
$f$ is integrally continuous on $\widetilde{ARC}_{\zeta_i^N}(\rr^n;S)$.

Since $\psi(x,t)\le\zeta_i(x,t)$ for all $(x,t)\in S$, we also have 
$\psi_N(x,t)\le\zeta_i^N(x,t)$ for all $(x,t)\in S$, so 
$f$ is integrally continuous on $\widetilde{ARC}_{\psi^N}(\rr^n;S)$,
showing that (T\ref{selectionalscprod-main}.3) holds
with $\psi$ replaced by $\psi_N$.

Now, $(\bar{x},\bar{t})\in{\rm Int}(K_i)$, and 
$\|f(x,t)\|\le\psi_N(x,t)\le\ffi_i(t)+N=\zeta_i(x,t)$
when $(x,t)\in K_i$. So 
$f$ is bounded by the integrable function \mbox{$t\mapsto\ffi_i(t)+N$} on
$K_i$, and 
$f$ is integrally continuous on $\widetilde{ARC}_{\ffi_i+N}(\rr^n;K_i)$.
Therefore $f$ is {\it a fortiori} integrally continuous on 
$ARC_{\ffi_i+N}(\rr^n;K_i)$. So $f$ is admissible on
$K_i$. This completes the proof  that $f$ satisfies
(T\ref{selectionalscprod-main}.1), (T\ref{selectionalscprod-main}.2),
(T\ref{selectionalscprod-main}.3), (T\ref{selectionalscprod-main}.4), 
and (T\ref{selectionalscprod-main}.5), with 
$\psi$ replaced by $\psi_N$.

Finally, (T\ref{selectionalscprod-main}.6) follows from 
(T\ref{selectionalscprod-main}.5) and Theorem \ref{theosemidifreg}.
\proofend
\end{proof}

\vspace{-2mm}
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\end{document}

