Integrals on Unbounded Domains

This lecture covers the concept of integrals on unbounded domains and unbounded functions, starting with the idea of integrals on closed and bounded sets. The instructor explains the importance of uniform continuity on closed bounded sets and how to cut up domains into small subsets for approximation. The lecture then delves into the challenges of unbounded domains and functions that are not uniformly continuous, emphasizing the impact of discontinuities at isolated points or boundaries. Different cases are explored, including the definition of integrals for positive functions and the decomposition of functions into positive and negative parts. Examples are provided to illustrate the calculation of integrals over infinite domains and the importance of proper definitions for integral values.