Math 711 Notes

Supplementary Notes

I will post additional notes or information on this page as the semester progresses. Class exercise handouts will be kept at the top of the page. Web links could also be posted here, so let me know if you find something that you think would be useful or interesting to your classmates. If I agree, I will post it.

Errata to the primary textbook

(8/25) Last year, Thomas Ventimiglia pointed me to a PDF file of errata to Folland’s Real Analysis. (Thanks, Thomas!) The linked file is for the 2nd edition, 6th and later printings, which should be appropriate for your copy. There is also a file for the 5th and earlier printings on Folland’s home page, but this earlier list had already been corrected in last year’s copies of the book.

Class Exercise Handouts

(12/08) Functions of Bounded Variation

(11/12) The Hardy-Littlewood Maximal Function

(11/10) Singularity and Absolute Continuity

(10/27) Lipschitz Functions and Change of Variables

(10/22) Product Spaces and Integrals

(10/06) Evaluating (Lebesgue) Integrals

(10/06) More on Measurable Functions

(09/29) Measurable Functions

(09/24) Measures and Monotone Functions

(09/15) Dirac and Counting Measures

(09/10) σ-algebras, Gδ Sets

(09/08) Vitali Sets

(09/03) Limits and Integrals

Additional Notes

11/17
  1. Here are some notes on the Hahn-Banach Theorem, and an interesting example of a dual space (the dual of Lp when 0 < p < 1).
10/27
  1. Here is a proof that every Radon measure on Rn is regular.
10/08
  1. There is a nice discussion on Mathematics Stack Exchange of constructions of a Borel set with positive but not full measure on each interval, including Walter Rudin’s own solution to the problem! (Thanks to Loren Wagner for this link.)
09/17
  1. Here is the Wikipedia page on the Cantor function referred to in today’s homework assignment.
09/10
  1. Here is the Wikipedia page on Gδ Sets referred to in today’s class exercise sheet.
09/03
  1. Here is a nice Wikipedia page on Vitali sets.
  2. The question of the existence of non-Lebesgue-measurable sets appears to lie close to foundational issues in set theory. Googling “non-measurable set” will provide a wealth of links, but there is in particular a Math Overflow page on the axiom of choice and non-measurable sets,
    which touches on some of the subtleties involved.