Homework for Math 501 will be of 2 kinds. The first kind will be the assigned problems that are listed below. These are the same for everyone in the class. Your solutions will be collected at lecture and some will be graded. The second kind will be assigned at lecture and won't be listed below. For the second kind, there will generally be just one problem per chapter, but each class member will get a different problem. After you solve yours, please give it to me, carefully written. I will collect all these and have copies made for each student of every solution.
Please always include your name on your solutions, and for the problems of the second kind, include the statement of your problem. Of course you may be asked by other students for further explanation of your solution (of the second kind of problem).
I encourage you to talk with anyone you'd like about the solutions of any of the problems, of either kind, but of course the solutions should be written in your own words.
CHAPTER 1: (Due in lecture on Monday, Sept. 17) 1a, 1d only for Thm 1.4(b) not for Thm 1.4(a), 1e, 1f, 1l, 1n only for Thm 1.14(a) not for Thm 1.14(b), 4bd, 6.
Here are 4 problems that are not in the book. They will be among the problems of type 2 that will be assigned to some lucky class members.
NEW 19. Prove the following special case of the Baire Category Theorem: The intersection of a countable number of open dense sets in R^1 is dense in R^1.
NEW 20. Show that the irrational numbers in R^1 are a set of type G_\delta, but that the rational numbers are not. [For the 2nd part, the 1st part of NEW 19 together with an argument by contradiction may help.]
NEW 21. Construct a single set in R^1 which is neither of type G_\delta nor F_\sigma. [How about the union of the negative rationals and the positive irrationals? Use NEW 20?]
NEW 22. Let \alpha be a real number, k=1,...,n, and H = {x = (x_1,...,x_n) in R^n: x_k = \alpha} (a hyperplane in R^n). Given \epsilon >0, show that there exist cubes {Q_j: j=1,2,...} in R^n with edges parallel to the coordinate axes such that H is a subset of the union of the Q_j and \sum v(Q_j) < \epsilon. (According to the terminology of Chapter 3, this means H has outermeasure 0 in R^n.)
CHAPTER 2 (problems for us all, due on Monday, Sept. 24): 1,4,5,6
Here are 5 new problems for chapter 2 that are not in the book. These are type 2 problems!
New 19: If [a',b'] is a subinterval of [a,b] and if \int_a^b f d\phi exists, show that \int_{a'}^{b'} fd\phi exists.
New 20: If lim_{|\Gamma| \to 0} S_\gamma[f;a,b] exists, show that it equals V[f;a,b].
New 21: If V[f;a,b] = +\infty, show that there is a point x_0 in [a,b] such that either V[f;I] = +\infty for every subinterval I of [a,b] having x_0 as left-hand endpoint or V[f,I] = +\infty for every subinterval I of [a,b] having x_0 as right-hand endpoint. (Note that for any x_0, a',b' such that a \le a'< x_0 < b' \le b, V[f;a',b'] is finite if both V[f;a',x_0] and V[f;x_0,b'] are finite.)
New 22: If V[f;a,b] = +\infty, show that there exist x_0 in [a,b] and a monotone sequence {x_k} in [a,b] such that x_k \to x_0 and \sum_{k} |f(x_{k+1}) - f(x_k)| = +\infty. (Use the result in New Exercise 21.)
New 23: If V[\phi;a,b] = +\infty, show that there is a continuous f on [a,b] such that \int_a^b f d\phi does not exist. (Use the result in New Exercise 22 together with the following fact: if {a_k} is a positive sequence with \sum_k a_k = +\infty, there is a positive sequence {\epsilon_k} converging to 0 with \sum_k \epsilon_k a_k = +\infty.)
New 24: Let f be continuous and \phi be of bounded variation on [a,b]. Show that lim_{\epsilon \to 0+} \int_a^{a+\epsilon} f d\phi = 0 if and only if either f(a) =0 or \phi is continuous at a. Deduce that the formula \int_a^b f d\phi = lim_{\epsilon \to 0+} \int_{a+\epsilon}^b f d\phi may not hold.
CHAPTER 3. Type 1 Problems for us all (Due at lecture on Thursday, Oct. 4): Read #1, and take it for granted; don't hand it in. But note that the number c that is mentioned in the last sentence about uniqueness should be restricted to c = 1,...,(b^k)-1. Also read and think about #4 but don't hand it in. Hand in these four: 2,3,5,6.
More Type 1 Problems for us all (Due at lecture on Thursday, Oct 11): 8,9,10,17
Here are some new problems that are Type 2 problems:
New 28: Prove the following assertion made in the proof of (3.33): If T: R^n \to R^n is a Lipschitz transformation, there is a constant c'>0 such that |TI| \le c'|I| for every interval I. (Consider first the case when I is a cube Q, noting that TQ is contained in a cube with edgelength c diam Q, where c is the Lipschitz constant of T. The case of general I can then be deduced by applying (1.11) to the interiors J^o of intervals J with I \subset J^o.)
New 29: Prove the following fact related to (3.35): |TE|_e = \delta |E|_e for any E \subset R^n, where T is a linear transformation of R^n and \delta = |det T|. (Use (3.36) and note that E = T^{-1}(TE) if \delta >0, where T^{-1} is the inverse of T. Note also that this leads to a shorter proof of (3.35) since TE is measurable is E is measurable, by (3.33).)
New 30: Let f: R^n \to R^1 be continuous. Show that the inverse image f^{-1}(B) of a Borel set B is a Borel set; see p. 52 for the definition of the inverse image of a set. (The collection of sets {E: f^{-1}(E) is a Borel set} is a \sigma-algebra and contains all open sets in R^1; cf. Exercise 10, Chapter 4, and (4.15).)
New 31: Construct a Lebesgue measurable set which is not a Borel set. (If f is the Cantor-Lebesgue function, than the function g(x) = x + f(x) is strictly increasing and continuous on [0,1]. Consider g^{-1}(E) for an appropriate E \subset g(C) where C is the Cantor set.)
New 32: Let \theta \in (0,1) and E be a set with 0<|E|_e<\infty. Show that there exists E' \subset E with |E'|_e = \theta |E|_e, and that E' can be chosen to be measurable if E is. (If Q_r denotes the cube with edgelength r centered at the origin, 0 < r < \infty, then |E \cap Q_r| is a continuous monotone function of r.)
Hint for #11: For the sufficiency, pick open sets G and G_1 with S \subset G, N_1 \subset G_1, |G-S| < \epsilon and |G_1| < |N_1|_e + \epsilon < 2\epsilon. Estimate |(G \cup G_1) - E|_e.
Addition to #15: As a consequence, using Exercise 13, show that if A \subset [0,1] and |A|_e + |[0,1] - A|_e =1, then A is measurable.
Restatement of #22: Replace #22 with this 2-part exercise: (a) Show that the outermeasure of a set is unchanged if in the definition of outermeasure we use coverings of the set by cubes with edges parallel to the coordinate axes, instead of coverings by intervals. (b) Show that outermeasure is also unchanged if coverings by parallelepipeds with a fixed orientation (i.e., with edges parallel to a fixed set of n linearly independent vectors) are used rather than coverings by intervals.
Chapter 4. (Type 1 problems for us all) 3,4,5 with these changes in numbers 3 and 5: In number 3, assume that both f and g are finite everywhere, and in number 5, add the following as a second part: Show that the same may be true even if f is continuous. (Hint: Let g(x) = x + F(x) where F is the Cantor-Lebesgue function, and consider f = g^{-1}.)
Also do the following problem:
New 22. Part a. Show that if f is measurable and B is a Borel set in R^1, then f^{-1}(B) is measurable.
Part b. If \phi is a Borel measurable function on R^1 and f is a finite measurable function on R^n, show that \phi(f(x)) is measurable on R^n.
These are due on Thursday, Oct. 25.
THE NEXT 4 PROBLEMS ARE DUE AT LECTURE ON Thursday, Nov. 1:
Chapter 4, numbers 6,11,19 plus the following one:
New 23. Let {f_k} be a sequence of measurable functions defined on a measurable set E. Show that the sets {x \in E: lim f_k(x) exists and is finite}, {x \in E: lim f_k(x) = + \infty} and {x \in E: lim f_k(x) = - \infty} are measurable.
ACCORDING TO THE SCHEDULES THAT YOU GAVE ME, WE ARE ALL FREE ON TUESDAY AND THURSDAY MORNINGS FROM 9:00-10:00 AM FOR EXTRA LECTURES IN ORDER TO MAKE-UP FOR THOSE WHICH WERE LOST DUE TO THE STORM. I HAVE SCHEDULE 4 OF THOSE TIMES FOR EXTRA LECTURES, EACH ABOUT 45 MINUTES LONG: NOV. 13, 15, 27, 29. THE ROOM WILL BE HILL 525.
HOMEWORK DUE THURSDAY, NOV. 15: Chapter 4: 13,14,15,20. In number 20, assume that f is finite a.e. in [a,b].
HOMEWORK DUE TUESDAY, NOV. 20. Chapter 4: 17,18 and the following problem:
New 21: Show that the necessity part of Lusin's theorem is not true for \epsilon =0, i.e., find a measurable set E and a finite measurable function f on E such that f is not continuous relative to E-Z for any set Z with |Z| =0. (Consider for example the characteristic function of the set E in Exercise 25, Chapter 3.)
Here are some new problems for Chapter 5 that are not in the book. They will be assigned in class, mostly as type 2 problems.
New 22. Show that the conclusion of the Lebesgue Dominated Convergence theorem can be strengthened to \int_E |f_k - f| dx \to 0.
New 23. Prove the following fact, sometimes referred to as the Sequential Version of the Lebesgue Dominated Convergence theorem. Let {f_k} and {\phi_k} be sequences of measurable functions on E satisfying f_k \to f a.e. in E, \phi_k \to \phi a.e. in E, and |f_k| \le \phi_k a.e. in E for all k. If f, \phi \in L(E) and \int_E \phi_k dx \to \int_E \phi dx, then \int_E |f_k - f| dx \to 0. (In case f=0 and all f_k \ge 0, apply Fatou's lemma to the sequence {\phi_k - f_k}.
New 24. A measurable function f on E is said to belong to weak L^p(E), 0 < p < \infty, if \omega_{|f|}(\alpha) \le A \alpha^{-p} for some positive constant A and all \alpha >0 (cf. (7.8) in case p=1).
(a) Show that if f \in L^p(E), then f \in weak L^p(E), but that the converse is false.
(b) Show that if 1< p < r< \infty, and f belongs to both weak L^1(E) and weak L^r(E), then f \in L^p(E).
(c) Show that f \in L^p(E) for 1 < p < \infty if f belongs to weak L^1(E) and f is bounded on E.
New 25. Give an example to show that the analogue of Theorem 5.8 with the roles of sup and inf interchanged is false.
New 26. Prove the following variant of Lebesgue's Dominated Convergence theorem: if {f_k} converges in measure to f on E and |f_k| \le \phi \in L(E), then f \in L(E) and \int_E f_k dx \to \int_E f dx.
New 27. The notion of equimeasurability can be extended to different sets E_1 and E_2, even in different dimensions, by saying that two measurable functions f_1, f_2 defined on sets E_1, E_2 respectively are equimeasurable if |{x\in E_1: f_1(x)>\alpha}| = |{y\in E_2: f_2(y)>\alpha}| for all \alpha. Show that if f is measurable and finite a. e. on E and \omega_f is strictly decreasing and continuous, then f and the inverse function of \omega_f are equimeasurable (on E and (0,|E|) respectively).
New 28. Let E be a measurable set in R^n with |E| < \infty. Suppose that f > 0 a.e. in E and f, log f \in L^1(E). Prove that
lim_{p \to 0+} (|E|^{-1} \int_E f^p dx)^{1/p} = exp(|E|^{-1} \int_E log f dx).
(Start by using theorem 5.36 to show that \int_E f^p dx \to |E| as p \to 0+. Note that (f^p - 1)/p \to log f.)
New 29. Let f be measurable, nonnegative, and finite a. e. on E. Prove that for any nonnegative constant c,
\int_E e^{cf} dx = |E| + c \int_0^\infty e^{c\alpha} \omega_f(\alpha) d\alpha.
Thus e^{cf} \in L(E) if |E| < \infty and there exist constants c_1 and C_1 with c_1 >c and \omega_f(\alpha) \le C_1 e^{-c_1\alpha} for all \alpha >0. (It may be helpful to note that \int_E e^{cf} dx = \infty if e^{cb} \omega_f(b) is unbounded as b \to \infty.)
HOMEWORK DUE THURSDAY, NOV. 29. Chapter 5: 1,2,3,21
All type 2 problems that are numbered less than or equal to 27 are due on Monday, Dec. 3. You may hand it in before Monday if you finish early.
HOMEWORK DUE MONDAY, DEC. 10. Chapter 5: 4,6,7,13,17
What's above will be the last graded assignment, but some exercises from Chapter 6 can help you see useful applications of Fubini-Tonelli, such as 1,2,5,10,11.
Our final exam will be on Monday, December 17, 9:30--12:30 in the morning. The room will be Hill 423, our regular room.