Notes on Lebesgue Integration

These notes record the lectures on Lebesgue integration, which is a topic not covered in the textbook. It is largely inspired by the approach of Leon Simon when he previously taught this course. (All mistakes are my own of course!) The notes will be posted after each lecture. Any comments or corrections, even very minor ones, are very much appreciated! Our goal is to define the notion of Lebesgue integration on a compact interval [a, b] ⊂ R. The main purpose is to show that Lebesgue integration has a “completeness property” and this will eventually allow us to view (appropriately defined) L and L spaces as Banach spaces.