Notes on Measure and Integration

TL;DR

This paper provides an accessible introduction to Lebesgue measure and integration, focusing on key properties and convergence theorems, aimed at advanced undergraduates learning the subject.

Contribution

It offers a quick, intuitive approach to Lebesgue integration, emphasizing core concepts and properties without detailed measure construction, suitable for teaching purposes.

Findings

Introduces Lebesgue measure as a length generalization

Explains key properties: monotonicity, countable additivity, translation invariance

Discusses convergence theorems and $L^2$ space basics

Abstract

This text grew out of notes I have used in teaching a one quarter course on integration at the advanced undergraduate level. My intent is to introduce the Lebesgue integral in a quick, and hopefully painless, way and then go on to investigate the standard convergence theorems and a brief introduction to the Hilbert space of $L^2$ functions on the interval. The actual construction of Lebesgue measure and proofs of its key properties are relegated to an appendix. Instead the text introduces Lebesgue measure as a generalization of the concept of length and motivates its key properties: monotonicity, countable additivity, and translation invariance.