Theory of Functions of a Real Variable I, Math 501
This course provides an introduction to the fundamental topics in the analysis of functions of a real variable, including applications. The main text is Real Analysis by Folland. I will also make frequent reference to the analysis text of Lieb and Loss. I will put a copy on reserve, but this book is a real bargain, especially if purchased directly from the AMS, which I recommend. I will also post lecture notes on a number of topics. In particular, the first 4 lectures are based on lecture notes for which a link is provided below.
Here is the syllabus for the the course, The syllabus refers to various class notes and homework assignments that will be posted on the assignments page.
(1.)Here are notes for the first four lectures.
(2.)Here are detialed solutions for all of the topology exercises.
(3.)Here are detialed solutions for selected exercises in Chapter 1 and 2 of Folland.
(4.)Here are notes on unifirm integrability and Vitali' s Theorem.
(5.)Here are notes on sigma algebras, measurability and measures.
(6.)Here are solutions to the problems on Test1.
(7.)Here are notes on integration.
(8.)Here are notes on the construction of measures, with additional exercises for the Wednesday meetings.
(9.)Here are notes on product measures, Lebesgue measure on R^n and related topics.
(10.)Here are notes on Hilbert Sapce methods.
(11.)Here are notes on the Lebesge Decomposition Theorem, the Lebesgue Differentiation Thoerem and the Radon-Nikodym Theorem and related topics.