TENTATIVE SYLLABUS, REAL ANALYSIS

Prerequisites and Overarching Goals

Prerequisites: An undergraduate analysis course.

You should be willing to learn (on your own), or have a working knowledge of, the following: Topology of Euclidean space (open/closed/compact ... sets), Heine-Borel theorem, Bolzano-Weierstrass theorem, notions of supremum, infimum, metric spaces ... etc ... (the standard stuff from a year-long course in undergrad real analysis).

Mathematical goals: Understand Measure Theory and Lesbesgue Integration.

Contact Information

Office: LN-2224, Extension: 7-3506, Email: paulXmathXbinghamtonXedu, where one must substitute in the variable X the appropriate symbol.

Office Hours: TBA.

Textbook

Main text: Probability and Measure by Patrick Billingsley.   Supplementary text: Measure Theory and Probability by Malcolm Adams and Victor Guillemin. I like both of these books a lot. We won't follow either book section by section but instead we'll do parts here and there, so it will be important to take careful lecture notes. It will be good for you to take detailed notes and rewrite them being careful to fill in missing steps and provide your own examples.

Extra readings

Here are some extra material that you might find fascinating --- at least I do. I've tried to do my best translating; if you find any mistakes, please let me know so I can make them accurate! If you want the original versions, please email me.

1. English translation of Lebesgue's 1901 paper "Sur une generalisation de l'integrale definie." Here, Lebesgue reveals his theory of integration for the first time!
2. English translation of Vitali's 1905 paper "Sul problema della misura dei gruppi di punti di una retta." Here, Vitali introduces the idea of a nonmeasurable set for the first time!

Lecture Notes

Please let me know of any mistakes you find (including grammatical one, as I'm terrible at grammar!). I hope to publish these notes one day as a book. When you read the notes, please keep in mind that the notes are geared to "average" first year grad students who don't have any previous knowledge in Lebesgue integration theory. If you find any portion of the notes that is not crystal clear or is too advanced or too slow for the intended audience, please let me know so that I can rewrite those portions of the notes.

1. Lecture Notes 1.
2. Lecture Notes 2.
3. Lecture Notes 3.
4. Lecture Notes 4.
5. Lecture Notes 5.
6. Lecture Notes 6.
7. Lecture Notes 7.
8. Lecture Notes 8.
9. Lecture Notes 9. Lecture Notes 9 contains revised versions of Lecture notes 6,7, and 8, plus new material.
10. Lecture Notes 10. Measurable functions.
11. Lecture Notes 11. Definition of integral.
12. Lecture Notes 12. Lebesgue's DCT and many applications of the DCT.
13. Lecture Notes 13. Lebesgue vs. Riemann integration.
14. Lecture Notes 14. Laws of Large Numbers. (In these lecture notes I mention the "Principle of Appropriate Functions." This principle is the same as "Lebesgue's Recipe," which I have now changed to the "Principle of Appropriate Functions".
15. The Pi-Lambda Theorem (Updated 12:13AM 5/3/08. Sorry, the previous version had some silly mistakes.) I have rewritten parts of the notes. The pdf file here is actually now part of a new version of notes and it's placed immediately preceding the proof of the Extension Theorem.

Homework

All the exercises are taken from the lecture notes (not from the books).

1. Homework 1. Due Monday Feb 18. In Exercises 1.3, do problem 2 and problem 8 (c). In Exercises 1.4, do problem 7. In Exercises 1.5, do problem 6. In problem 6, make sure to explicitly state the sample space, the probability measure, and the events whose probability you want to compute.
2. Homework 2. Due Friday Mar 7. In Lecture Notes 3: From Exercises 1.6, do problem 3 and from Exercises 1.7, do problem 3. In Lecture Notes 4: From Exercises 1.8, do problem 5 (explicitly state the probability set function too). In Lecture Notes 5: From Exercises 1.9, do part (iv) of problem 5 (Make sure to state the sample space and the prob. measure). From Exercises 1.9, do problem 3. (The point of problem 3 is to see that you understand the proof of Bernoulli's theorem.)
3. Homework 3. Due Friday Apr. 4. All the following problems are taken from Lecture Notes 9: From Exercises 2.1, do problem 6. From Exercises 2.2, do problem 3(c) and 6. From Exercises 2.3, do problem 2. From Exercises 2.4, do problem 11. From 2.6, do problem 12 (ii).
4. Homework 4. Due Monday, April 28. From Lec. Notes 10, Exercises 3.3, do 1(c). From Lec. Notes 11, Exercises 3.4, do 2(a) and 4. From Lec. Notes 12, Exercises 3.5, do 3(c). In Lec. Notes 12, Exercises 3.6, do 8(a) and 8(d). NOTE: In Exercises 3.3, problem 1 (c), let's suppose that 1 represents that the gambler wins a game and 0 he loses a game. If he wins a game he gets $1 and if he loses a game he gives the house $1. Please see Problem 2, Exercises 1.2 in Lec Notes 1. This problem might give you an idea on how to write the function B(x) as the infimum of a sequence of functions.
5. Homework 5 (Last but not least!). Due Wed, May 7-th. From Lec. Notes 12, Exercises 3.6, do Problem 10 (c). From Lec. Notes 13, Exercises 3.7, do Problem 1 or 7, but not both!. From Lec. Notes 14, Exercises 3.8, do Problem 8. (The reason you can choose either 1 or 7 in Exer. 3.7 is that you only have a few days to complete the HW, so please do 1 if you're pressed for time!)

Organization of Course and Schedule of Topics

We meet MWF 10:50AM-11:50AM in LN2205. 
I hope to cover the following topics:

1. 1/28: A motivational speech on why study Lebesgue integration theory.
2. 2/4: Semirings/sigma-algebras/Borel sets.
3. 2/11: Additive set functions.
4. 2/18: Integration on semirings and additive set functions on rings.
5. 2/25: Outer measure. The Weak Law of Large Numbers.
6. 3/3: The extension theorem.
7. 3/10: Geometric properties of Lebesgue measure.
8. 3/17: Measurable functions.
9. 3/31: Integration. Midterm.
10. 4/7: The Dominated Convergence Theorem. The Strong Law of Large Numbers.
11. 4/14: Fubini's Theorem or differentiation (Students choose)
12. 4/21: Fubini's Theorem or differentiation.
13. 4/28: Fourier series.
14. 5/5: Fourier series.

Final Exam Schedule

Grading

Add/Drop Deadline

General Comments

Class attendance is required. Questions are welcome at anytime. Please visit me in my office whenever you have a question or if you just want to talk.