## 515 – Caratheodory’s characterization of measurability (Homework 3)

April 12, 2012

This set is due Friday, April 27.

The goal of these problems is to prove Carathéodory‘s theorem that “extracts” a measure from any outer measure. In particular, when applied to Lebesgue outer measure, this construction recovers Lebesgue measure.

## 515 – Homework 2

February 14, 2012

This set is due Feb. 29 at the beginning of lecture. Let me know if more time is needed or anything like that. Problem 4 was incorrect as stated; I have fixed it now. Thanks to Tara Sheehan for bringing the problem to my attention.

## 515 – Plotting Féjer’s kernels

February 6, 2012

Féjer’s kernels are the functions that play the role of “approximations to the Dirac delta” in the computations we will use to obtain Weierstrass approximation theorem. The $n$th approximation is given by $\displaystyle K_n(s)= \frac1{n+1}\left(\frac{\sin\frac{(n+1)s}2}{\sin\frac s2}\right)^2$ for $s\ne0$, $K_n(0)=n+1$.

## 515 – The fundamental theorem of calculus

February 6, 2012

Suppose that $f\in{\mathcal R}[a,b]$ and $f$ has an antiderivative $G$. Then $\displaystyle G(b)-G(a)=\int_a^b f(t)dt.$

## 515 – Homework 1

January 25, 2012

This set is due Feb. 8 at the beginning of lecture. Of course, let me know if more time is needed or anything like that.

## 515 – The Dehn-Sydler theorem

January 23, 2012

As mentioned in lecture, Hilbert’s third problem was an attempt to understand whether the Bolyai-Gerwien theorem could generalize to ${\mathbb R}^3:$ January 13, 2012

Syllabus for Math 515: Advanced calculus AKA Analysis II.

Instructor: Andrés E. Caicedo.
Contact Information: See here.
Time: MWF 9:40-10:30 am.
Place: MG 124.
Office Hours: MF 11-12.

Text: An introduction to measure theory“, by Terence Tao. AMS, Graduate studies in mathematics, vol 126, 2011. ISBN-10: 0-8218-6919-1. ISBN-13: 978-0-8218-6919-2. Errata.

Mathematicians find it easier to understand and enjoy ideas which are clever rather than subtle. Measure theory is subtle rather than clever and so requires hard work to master.

Thomas W. Körner, Fourier Analysis, p. 572.

Contents: From the Course Description on the Department’s site:

Introduction to the fundamental elements of real analysis and a foundation for writing graduate level proofs. Topics may include: Banach spaces, Lebesgue measure and integration, metric and topological spaces.

We will emphasize measure theory, paying particular attention to the Lebesgue integral. Additional topics, depending on time, may include the Banach-Tarski paradox, and an introduction to Functional Analysis. Grading: Based on homework. No late homework is allowed. Collaboration is encouraged, although you must turn in your own version of the solutions, and give credit to books/websites/… you consulted and people you talked/emailed/… to.

There will be no exams in this course. However, an important component of being proficient in mathematics is a certain amount of mental agility in recalling notions and basic arguments. I plan to assess these by requesting oral presentations of solutions to some of the homework problems throughout the term.