Lebesgue Measure: Definition and Properties

This lecture covers the definition and properties of the Lebesgue outer measure, which is used to define the Lebesgue measure on sets. The instructor explains how the outer measure is calculated for different sets, such as boxes and unions of sets, and discusses the concept of countable subadditivity. The lecture also delves into the proof of properties like the measure of the empty set, the translation invariance of the measure, and the countable subadditivity property. Additionally, the instructor demonstrates the application of the outer measure to sets like intervals and real numbers, highlighting the importance of understanding the outer measure in measure theory.