Lebesgue Integral: Definition and Properties

This lecture introduces the Lebesgue integral, a powerful concept in measure theory. It explains how Lebesgue allowed functions to choose their own partition, leading to measurable sets. The integral calculation involves assigning measures to sets, highlighting the complexity of non-measurable sets. The lecture covers the definition of o-algebras and measures, as well as the properties of Lebesgue integrals, including monotone convergence and dominated convergence theorems.