Ebook: A Short Course on the Lebesgue Integral and Measure Theory

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.