Measure Theory

Subject classification: this is a mathematics resource.

Introduction

This course starts from the question "How do you measure sets of real numbers?" This leads into a conversation and analysis about the Lebesgue outer measure, and the Lebesgue measure. Once we understand these concepts, we are then in a position to ask a next question, "How can we approximate measurable sets? And how can we approximate measurable functions?" To have a satisfying answer to these questions, we look at Littlewood's three principles, state them rigorously as theorems, and investigate their answers and related concepts. This includes Egoroff's Theorem and Lusin's Theorem. From here we use these tools to develop and study the Lebesgue integral, culminating in the Dominated Convergence Theorem. From here we study differentiation and $L^{p}$ spaces, which culminates in the Riesz-Fischer Theorem. This basically tells us that $L^{p}$ spaces satisfy a completeness property, allowing us solve broad classes of problems by way of approximations. This concludes the portion of the course dealing specifically with Lebesgue measure.

The second half of the course is devoted to generalizing the concept of measure to general measure spaces. A particular (but not exclusive) interest will be in applying more general measures to probability spaces.

Unit 1: Measuring Sets of Reals

• Unit 1, Chapter 1: Measuring sets of real numbers

About the Construction of this Course

The aim of this course is to present a slightly new and re-organized curriculum for Measure Theory. More so than other popular resources on the subject, we aim for a natural and motivated progression. So rather that merely presenting a definition, claiming certain theorems, and proving them, we try to give a sense of why a definition is good or interesting. We try to explain what a theorem is really claiming, and why we might want to know that the claim is true. And rather than proofs which focus on ensuring the truth of the theorem, we try to have proofs which are both valid and use exposition. That is to say, we strive to make the steps taken in proofs not seem pulled from thin air. A good exposition should make each step of a proof seem like it comes from earlier considerations, is intuitive, and can sensibly apply to future exercises and theorems.