Analysis: Measure and Integration

This lecture introduces the course on analysis four, focusing on measure and integration. The instructor, Maria Colombo, explains the motivation behind developing a new theory of integration, highlighting the limitations of the Riemann integral. She discusses the need for a more comprehensive approach to integration, leading to the concept of measurable sets and the Lebesgue measure. The lecture covers properties of measurable sets, the construction of the Lebesgue measure, and the fundamental theorem of Lebesgue measure existence. The goal is to establish a satisfactory theory of integration that overcomes the shortcomings of the Riemann integral and provides a robust framework for mathematical analysis.