Math 5052

Measure Theory and Functional Analysis II, Spring 2024

Course Description

Continuation of Math 5051. Prerequisites: Math 5051 or permission of the instructor.

Contact Information

Instructor: Ari Stern
Email: stern@wustl.edu
Office: Cupples I 211B
Office Hours: TuTh 11:00am-12:30pm

Lectures

MWF 10:00-10:50am, Eads 215. The first class will be on Friday, January 19 (due to a scheduling conflict on January 17), and the last will be on Friday, April 26. Class will be canceled for Spring Break (week of March 10-16).

Homework Assignments

Problem sets will be posted to Canvas approximately weekly, and students will submit their solutions electronically by uploading them to Gradescope before the specified due date and time. You are encouraged to discuss the homework with your fellow students, but your final write-up must be your own. Please make sure that your solutions are written clearly and legibly. (Typing up solutions in LaTeX is encouraged and is valuable practice for mathematical writing later in your career.)

Exams

There will be one in-class midterm exam on Wednesday, March 6. The final exam will be held on Monday, May 6, from 10:30am-12:30pm.

Grading

Grades will be based on a weighted average of homework (40%, lowest score dropped), midterm exam (20%), and final exam (40%).

Required and Supplemental Texts

The required textbook for this course is Real Analysis: Modern Techniques and Their Applications, by Gerald B. Folland (second edition, Wiley, 1999). This book has more than a few typographical errors, so it's a good idea to check the list of errata on Folland's homepage.

As a supplemental text, I also recommend Measure, Integration, & Real Analysis, by Sheldon Axler, which is freely available online (although it is also published by Springer in hardcopy). This is a very recent text, but it appears to do a nice job of expanding some of the material that Folland treats more tersely, and giving concrete motivation where Folland is more abstract.

I have also asked the library to place the following supplemental texts on reserve:

• H. L. Royden, Real Analysis (3rd edition).
• E. M. Stein and R. Shakarchi, Real Analysis: Measure Theory, Integration, and Hilbert Spaces.
• E. M. Stein and R. Shakarchi, Functional Analysis: Introduction to Further Topics in Analysis.
• T. Tao, An Introduction to Measure Theory (based on his freely-available course notes).
• R. L. Wheeden and A. Zygmund, Measure and Integral: An Introduction to Real Analysis.

Do not feel obligated to purchase any of these non-required books (although each one is excellent in its own way). I am making them available simply because it can be helpful to see alternative treatments of the same material.

Course Outline

I plan to cover the topics discussed in Folland chapters 6-10 and (time permitting) parts of 7 and 11:

• Chapter 6: L^p Spaces
• Chapter 8: Elements of Fourier Analysis
• Chapter 9: Elements of Distribution Theory
• Chapter 10: Topics in Probability Theory
• Sections 7.1 and 11.1: Radon Measures and Haar Measure (time permitting)

Academic Integrity

All students are expected to adhere to high standards of academic integrity, as specified in the undergraduate student academic integrity policy and graduate school academic and professional integrity policy. Any violations of this policy will be referred to the relevant Academic Integrity Officer. Violations of this policy include, but are not limited to:

• homework collaboration exceeding that permitted in the Homework Assignments section above (e.g., preparing your final write-up side-by-side with another student rather than independently, copying another student's solutions);
• submitting the work of others as your own (e.g., copying solutions found online);
• exam misconduct (e.g., copying another student's exam or knowingly allowing your exam to be copied, giving or receiving unauthorized assistance, use of unauthorized books/notes or other materials, unauthorized use of electronic devices, etc.).

If you have any questions, or are unsure about what is permitted/prohibited by this policy, please ask me.

In many cases, academic integrity violations are the result of getting behind in coursework and making bad decisions under pressure. Keep up with your assignments, ask questions when you are unsure what is expected of you, and do not give in to the temptation to cut corners.

COVID-19 Health and Safety Protocols

Students experiencing symptoms consistent with COVID-19 or concerned about a possible exposure should contact Habif Health and Wellness Center (314-935-6666) to arrange for testing as indicated. If instructed by Habif to quarantine or isolate, students should notify their instructor as soon as possible by forwarding the email they received from Habif. Any accommodation needs for COVID-related absence not covered in an instructor’s standard course policies should be discussed between the student and instructor.

While on campus, it is imperative that students follow all public health guidelines established to reduce the risk of COVID-19 transmission within our community. The full set of University protocols can be found at https://covid19.wustl.edu/health-safety/.

Access to Classroom Content During Excused Absences (Including COVID-19)

To encourage in-person attendance and participation, WashU no longer requires that lectures be recorded or otherwise made available outside the classroom, aside from certain exceptional cases: "In the case of excused student absences due to COVID and other factors (e.g., illness, religious holidays, family emergencies, etc.), instructors should develop strategies for providing students access to the fundamental content of a given class session so that students are able to make progress in the course while complying with public health and university guidelines around quarantine and isolation and managing other challenges that disrupt their ability to attend class."

If you cannot attend class due to COVID or other factors similar to those listed above, and if you contact me at least 2 hours prior to the start of class, I will try to arrange for video recording of the lecture. If I am unable to record the lecture (e.g., due to technical difficulties) or do not receive advance notice to do so, I will provide written lecture notes instead.