Math 720: Measure Theory and Integration

Homework and Solutions

Assignment 15.
Previous assignments.
Solutions

Scanned Student Solutions

Perfect / nearly perfect student solutions will be scanned and posted for benefit of others. All identifying information will be removed before posting. If for some reason you don't your solutions scanned, please make a note of it on the homework.

You must log in with your Andrew ID to get access to solutions. If you get a "Permission Denied" message, then send me an email with your Andrew ID to get access for the semester.
LaTeX sources

If you're interested in typing your homework using LaTeX, you can find the LaTeX sources of all the homework here. Be sure you also download the files gimac.sty and giexam.sty, and put them in the same directory.

WARNING: These files are provided as is, with no warranty whatsoever. The files all LaTeX cleanly, with no errors on my system. If they don't compile cleanly on your system, you're on your own.

Handouts

Lecture schedule (will continually be updated).
Lecture notes (by students):
Few handwritten notes
The ergodic theorem, and an application to continued fractions
A more general change of variables
Midterm (from 2013) and solutions.
Your midterm with solutions and selfies.
Final (from 2012) and solutions.
Final (from 2013) and solutions.
Your final and solutions.
720 (Spring 2013) website

Logistical Info

Instructor	Gautam Iyer. Office Wean Hall #6121 412 268 8419 eMail For MINOR clarifications and logistical queries, my address is: `gi1242+720@NoSPAM.edu` (replace "`NoSPAM`" with "`cmu`"). For hints or longer questions, come by my office hours and speak to me in person. Anonymous Feedback Feedback at any time (either anonymous or signed) is always appreciated. Use this form to send me anonymous (or signed) feedback. [PS: In a desparate attempt to thwart evil spammers, I've restricted access to this form. Your public IP address should resolve to a hostname ending in `cmu.edu`.]
Lectures	MWF 2:30--3:20 in WEH 8220.
Office Hours	Mondays 10:30-12:20.
Mailing list	math-720 This list will be used for all announcements related to this class. All students (and anyone auditing) should subscribe to this list using the above link. I will NOT use BlackBoard. NOTE: Any student who is registered for this course on the first day of class will be automatically subscribed this mailing list. Everyone else should subscribe themselves using the above link. Please subscribe to this list!

Homework Schedule, Exams Dates and Grading Rubric

Homework due

Wednesdays, beginning of class.

Late homework will NEVER be accepted. Really.

Midterm

Fri, Oct 10 (closed book).

Either	2:00-3:20 in WEH 8201
or	2:30-3:50 in WEH 8220

Final

Fri, Dec 12, 5:30pm--8:30pm in WEH 7201 (and NOT PH A18C). (Closed book.)

Grading

Homework: 20%, midterm: 30%, final: 50%.

Syllabus

This is a first graduate course on Measure Theory, and will at least include the following.

Outer measures, measures, $\sigma$-algebras, Carathéodory's extension theorem.
Borel measures, Lebesgue measures.
Measurable functions, Lebesgue integral (Monotone Convergence Theorem, Fatou's Lemma, Dominated Convergence Theorem).
Modes of Convergence (Egoroff's Theorem, Lusin's Theorem)
Product Measures (Fubini-Tonelli Theorems), $n$-dimensional Lebesgue integral
Signed Measures (Hahn Decomposition, Jordan Decomposition, Radon-Nikodym Theorem, change of variables)
Differentiation (Lebesgue Differentiation Theorem)
$L^p$ Spaces, Hölder's inequality, Minkowskii's inequality, completeness, uniform integrability, Vitali's convergence theorem.

This will be followed by some special topics (e.g. Fourier Analysis).

Course Outline.

The course will start by constructing the Lebesgue measure on $\mathbb{R}^n$, roughly following Bartle, chapters 11--16.
After this, we will develop integration on abstract measure spaces roughly roughly following Cohn, chapters 1--6 or Folland.
If time permits, I will continue with some Fourier Analysis roughly following Folland chapter 8.

References

Since measure theory is fundamental to modern analysis, there is no dearth of references (translation: I'm not writing lecture notes). Here are a few other nice references I recommend.

I recommend buying either Cohn or Folland. Folland has a few nice additional topics (topology, functional analysis, Fourier analysis and probability). Cohn on the other hand is a slightly easier read and more focussed. If you're going to do your Ph.D. in something analysis related, buy Folland. If not, Cohn should suffice for this course.
An excellent (but harder) alternative to Folland is Rudin, which I strongly recommend.
Alternately, two slightly easier books are Jones or Royden. (If you find this course a little brisk / hard, I recommend reading Royden.)
If you prefer learning from lecture notes, here are some by Lenya Ryzhik and Terry Tao. (The last one is available as a PDF, and also as a regular published book.) Alternately, contact Giovanni Leoni for measure theory lecture notes from 2011.
An excellent treatment of Fourier Series can be found in Chapter 1 of Wilhelm Schlag's notes. (This has many advanced Harmonic Analysis topics, which I recommend reading later.)

Class policies

Homework

Late homework will never be accepted.
Homework will be assigned every Wednesday, and due the following Wednesday at the beginning of class.
Working in groups is encouraged, but solutions must be written up on your own.

Exams

All exams are closed book, in class exams.
Exams from previous years are/will be hosted here, in the Handouts section.