ABSTRACT
This chapter considers the basic properties and structures of measurable functions as well as the limit operations on them and focuses on measurable and measure spaces. It proves the theorems of Egorov, Luzin and Riesz. The chapter includes some results on image measures under measurable mappings. It presents a few results on the behavior of measures and their images under measurable mappings. These image measures play an important role in certain applications, especially in probability and statistics, but also in classical harmonic analysis among others. The chapter also includes multiple exercises that help students try themselves and perform measurable functions in measure theory and integration.
