Lebesgue Integration

Meeting times and Locations

  1. •Tuesday & Thursday, 10:00-11:30 am, 207 Cupples I


Instructor

  1. •Kabe Moen

  2. ‣Email:

  3. ‣Office: Cupples I 202

  4. ‣Office hours: Tuesday 2:00-4:00, Wednesday 3:00 -4:00.

  5. ‣Phone: (314) 935-6785


Introduction

  1. •Prerequisite: Math 4111.

  2. •Math 4121 is an introduction to measure theory and abstract integration.  Lebesgue measure will be the main example. Roughly speaking Lebsegue measure associates a non-negative number (size) to subsets of the Euclidean space.   Measure theory provides the theoretical basis for the tools in Probability and Statistics.  Some of the topics we will cover include:    

  3. -Riemann-Stieltjes integration and functions of bounded variation.

  4. -General measures, measurable functions, and the Lebesgue integral.

  5. -Construction of Lebesgue measure.

  6. -Lebesgue spaces of integrable functions.

  7. -Measures on product spaces and Fubini’s Theorem.

  8. -Decomposition of measures.

  9. -Assorted results in Fourier analysis.

  10. -Time permitting we may cover the following topics: differentiation theory, Hausdorff measure, elementary functional analysis.


Textbook

  1. •Measure and Integral: An Introduction to Real Analysis, by Richard Wheeden and Antoni Zygmund, CRC press.  


Supplemental texts


  1. •Real Analysis, by H. L. Royden, Macmillan Publishing.

  2. •Principles of Mathematical Analysis, by Walter Rudin.  Last semesters text is sort of a prerequisite for this course.  Chapter 11 also offers a nice hands-on construction of Lebesgue measure and integration.

  3. •Elements of Integration and Lebesgue Measure, by Robert Bartle, Wiley.

  4. •Real Analysis: Measure Theory, Integration, and Hilbert Spaces, by Elias Stein and Rami Shakarchi, Princeton University Press.


Exams and Homework

  1. •There will be one in-class exam and a take-home final.     

  2. •Homework will be collected weekly.  Usually the homework will be assigned on Thursday and collected the following Thursday.  Each homework assignment will consist of six to eight problems of varying difficulty.  Most of the problems will be theoretical, to teach you how to perform rigorous proofs. 

  3. •One of the worst forms of cheating is to copy solutions verbatim off of a website or chat room.  This is flagrant academic dishonesty, tantamount to plagiarism; when there is clear evidence that it occurred, the evidence will be forwarded to the Arts and Sciences Integrity Committee with a recommendation for a stern penalty.  Penalties can be avoided by acknowledging that the solution came from a specific website, but the grader will be asked to give only a small amount of credit for such solutions in whose devising the student played no role.

  4. •A lesser form of cheating on homework is for two or more students to work together and submit essentially the same solutions without acknowledging the help of others. We encourage students to discuss homework and course materials together, but every individual student must write up his or her own homework.  Therefore, no solutions from two students should look too much alike.  A good way to avoid copying, even inadvertently, from another student is to talk about problems together without taking any notes away from the conversation.  This lets you share understanding and ideas, but forces you to reconstruct your own understanding on paper.


Grading

  1. •The homework will be worth 30%, the midterm will be worth 30%, and the final will be worth 40% of your grade.


Academic Integrity

  1. •This link gives the general policies of the University on academic integrity.  Please also see the comments above about homework collaboration.