Professor David Blecher

Welcome to Real Variables I (Math 6320)

Time and place : MW 1:00PM - 2:30PM in PGH 350

Office Hours MWF: 12-1pm (or by appointment)

Prerequisites: An undergraduate real analysis sequence (Math 4331, 4332) or equivalent, or content of instructor. A little topology and metric spaces would be useful.

TEXT: G.B. Folland, Real Analysis: Modern Techniques and Their Applications (Pure and Applied Mathematics: A Wiley-Interscience Series of Texts, Monographs and Tracts).

Recommended reading: Lebesgue Integration on Euclidean Spaces, Frank Jones, Jones & Bartlett. Real Analysis, H.L. Royden, (3rd Edition), Prentice Hall. Real and Complex Analysis, W. Rudin, McGraw Hill. Measure Theory, D. L. Cohn, Birkhauser.

This is the first semester of a 2 semester sequence. This semester we will be developing the basic principles of measure and integration. This body of knowledge is essential to most parts of mathematics (in particular to analysis and probability) and falls within the category of "What every graduate student has to know". The one test and the final exam will be based on the notes given in class, and on the homework. After each chapter we may try (it is often impossible because of students hours) schedule a problem solving workshop, based on the homework assigned for that chapter. The most important part of your task as a graduate student in this course is simply to reread the class notes making sure you understand everything. Please ask me about anything you don't follow. The second most important part of your task is to do as many as possible of the assigned homework. Final grade is approximately based on a total score of 400 points consisting of homework (100 points), a semester test (100 points), and a final exam (200 points), which may be in class close to the end of the semester. The instructor may change this at his discretion. The syllabus for the first semester will cover some but not all of the following topics: Measures. Measurable functions. Integration. Convergence of sequences of functions. The Lp spaces. Signed and complex measures. Product measures and Fubini's theorem. Differentiation and integration.

Homework (from Folland): Chapter 1: 1, 2, 4, 7--13, 17, 29, 30, 33. Other problems are given on the following homework sheet, of which you are asked to turn in questions 6,7,8,9,10,11, 3c, 13-15, 18b,d,e, together with the Folland questions above 2, 7, 9, 12a, 13, 17, 29, 30.

Homework sheet Chapter 1

Homework (from Folland): Chapter 2: 5, 9, 11, 13, 14, 15, 16, 20, 21, 25, 26, 34, 39, 40, 44, and p. 187 9, 10. Other problems are given on the following homework sheet.

Homework sheet Chapter 2

For Wednesday 10/14: turn in Folland: 5, 13, 14, 20, and sheet Q 2, 5, 6, 7, 8, 13, 14, 15.
Of the remaining CH 2 questions I will collect Folland 40, p. 187 9, 10, and sheet 16, 19, 21, 22, 24, 26, 27.
Homework session on above questions: Bill Folland 40, Manisha Folland p. 187 (9), Mike Folland p. 187 (10), HW Sheet : Martina 16, Letty 19, mauricio 21, Nicholas( and Manki?) 22, Sanat 24.
Tentative forthcoming homework list from Folland (this will change continuously and other examples added): Chapter 2: 32--35, 39, 41, 44, 45--51, Chapter 3: 2, 4, 7, 9--12, 17, 18--21, 24,25, 30, 37,38, 42, Chapter 6: 11, 13, 14, 20, Chapter 7: 7--9. 17, 20--22, 25.
This course is usually not taken as a first graduate course, and therefore I am assuming a certain level of mathematical sophistication. For example, if you are a new graduate student taking this class, I suggest reading some material about writing mathematics. You can find some books and web-advice on this here

Although I will be distributing most of the material in class (since this is a small graduate level class), students in this course should keep monitering this website, particularly just before tests (for example if there is a last minute change in a test date). Will be added to continuously...
Test 1 will be on Mon October 26. It will cover Chapters 1 and 2. It will be on up though the material we cover the previous Monday. I will give you a mock test. Basically you will have to answer 3 questions out of 8 or something like that. The questions will test your knowledge of the definitions, the statements of major results, a few questions from the homework, plus you will be asked to prove some results from the notes from the list below. Right now you should be reading through your class notes, and asking me questions about anything you do not understand there. You should also be reviewing old homework, noting your mistakes (I'll try to get it all back to you this week). I am quite aware that some of you will be having a panic attack just reading this; I assure you that everybody taking the test will have to admit that it was easy. If you keep current, and keep a positive attitude, you will not find the work burdensome I hope.
Proofs that I might ask on the test: 1.3.4, 1.4.1, the deductions in 1.5 that Lebesgue measure is complete, regular and $\sigma$-finite (these are just a few lines each), the proof that the completion of the Borel $\sigma$-algebra on $R^n$ is the Lebesgue $\sigma$-algebra, some of the parts of 2.1.3 (not anything very tricky), the proof of the existence of a set which is Lebesgue but not Borel mble (you do not need to construct the ternary function, just use it), and the last item in 2.1 (the increasing $s_n$, = Folland 2.10), 2.2.2, the Remarks in 2.2.3, 2.2.4 (Vanishing principle), Thm 2.2.9 (Beppo levi), 2.2.11 Thm and coroll., the complex version of 2.2.11, 2.2.14, Theorem A in 2.4. From 2.5, be able to prove in a fair detail that $L^1$ and $L^\infty$ are Banach spaces, density of simple functions in $L^p$.