A Short Course on the Lebesgue Integral and Measure Theory

Steve Cheng0-201-51035-9

This article develops the basics of the Lebesgue integral and measure theory. In terms of content, it adds nothing new to any of the existing textbooks on the subject. But our approach here will be to avoid unduly abstractness and absolute generality, instead focusing on producing proofs of useful results as quickly as possible. Much of the material here comes from lecture notes from a short real analysis course I had taken, and the rest are well-known results whose proofs I had worked out myself with hints from various sources. I typed this up mainly for my own benefit, but I hope it will be interesting for anyone curious about the Lebesgue integral (or higher mathematics in general). I will be providing proofs of every theorem. If you are bored reading them, you are invited to do your own proofs. The bibliography outlines the background you need to understand this article.

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Table of contents :
Motivation for the Lebesgue integral……Page 2
Basic measure theory……Page 4
Measurable functions……Page 8
Definition of the Lebesgue Integral……Page 11
Convergence theorems……Page 15
Some Results of Integration Theory……Page 18
Lp spaces……Page 23
Construction of Lebesgue Measure……Page 28
Lebesgue Measure in Rn……Page 32
Riemann integrability implies Lebesgue integrability……Page 34
Product measures and Fubini’s Theorem……Page 36
Change of variables in Rn……Page 39
Vector-valued integrals……Page 41
C0 functions are dense in Lp(Rn)……Page 42
Other examples of measures……Page 47
Egorov’s Theorem……Page 50
Exercises……Page 51
Bibliography……Page 52