Measurable Sets: Countable Additivity

This lecture focuses on proving the countable additivity of measurable sets, demonstrating how to estimate the measure of intersections and unions of sets using subadditivity and finite additivity. The instructor explains the concept of measurable sets, the sigma algebra property, and the Borrell property, emphasizing the difference between Borel measurability and the broader notion of measurable sets. The lecture concludes with a discussion on the importance of understanding these concepts for defining measurable functions and compositions in analysis.