Measure Theory

## Introduction to the subject

In mathematics, measure theory is the study of measures, which generalize the notions of length, area, and volume. A measure on a set may be thought of as a function which assigns a number to subsets of a set, which gives a notion of size. Measure theory is incredibly important to the development of mathematical analysis, as the subject serves as a prerequisite to advanced topics like functional analysis, partial differential equations, advanced complex analysis, and geometric measure theory.

For a more detailed overview of the subject, click here.

## Syllabus

### Course Overview

This course in measure theory covers the same topics as one would encounter in a graduate course in real analysis and integration theory. Instead of studying the Riemann integral, which is done in undergraduate real analysis, measure theory generalizes the notion of an integral by introducing the concept of a measure, whereby one may do integration with respect to a measure. This gives way to an important type of integral - the Lebesgue integral, which becomes an important topic of focus.

Topics include the general study of measures, Lebesgue measure, measurable functions,

L
p

{\displaystyle L^p}

 spaces, signed and complex measures, product measures, differentiation of measures, locally compact spaces, and the Riesz representation theorem.

### Format and Scope

Lectures will be on an on-demand basis. Math discussion will be facilitated on the private Stack Exchange (for formal, on-topic discussion) and on the Element server (for informal on-topic discussion). Collaborative notes and on-topic information sharing (examples, explanations for concepts, etc.) will be facilitated here on the Fandom.

The scope of our study in measure theory with respect to the contents of the book is partitioned into three main phases:

#### Phase I - Basic measure theory

The first phase covers basis concepts in the theory of measures. Topics covered in phase I include

* Sigma-algebras
* (Outer) measures
* Lebesgue Measure
* Complete measure spaces
* Measurable functions
* L
p

{\displaystyle L^p}

 spaces

#### Phase II - Special measures and measure theory in several variables

The second phase focuses on expanding the topics in phase I by applying them to functions of several variables, in addition to discussing measures on spaces that are not

R

n

{\displaystyle \R^n}

, namely the complex plane. Topics include

* Signed measures
* Absolute continuity
* Functions of bounded variation
* The Radon-Nikodym theorem and derivative
* Product measures
* Fubini's theorem
* Change of variable for multivariable functions
* Differentiation

#### Phase III - Additional topics

The third phase of our study explores locally compact spaces and the Riesz representation theorem.

### Textbook

We will be using the textbook *Measure Theory*, 2ed by Donald L. Cohn (Amazon).

### Coursework

Those who take part in studying measure theory are highly encouraged to solve problems. Homework will be "assigned" each week, along with a reading "assignment". At the end of each chapter, a short, 75 minute chapter quiz will be assigned to be completed over the weekend. At the end of each phase, a long-form test will be assigned, which may require up to 2 hours of work.

### Prerequisites

It is highly recommended that the student be familiar with concepts in basic topology and introductory real analysis before embarking on a study in measure theory.