This course aims to cover all of the basics you'll need for studying measure theory and functional analysis. The goal of this module is to work our way up to the definition of the Lebesgue integral and its properties. To justify it, we begin with the Riemann integral, and show some cases where it is incapable of providing answers, thus motivating the need for a stronger theory.
Key to understanding the Lebesgue integral is understanding measure theory. A measure is a certain way to assign a subset a `size' - you can think of it as an abstraction of volume or area in Euclidean space. They are defined on σ -algebras of subsets, often not equal to the entire power set, which are necessary in cases where non-measurability is a concern.
Another thing we need to consider is how functions on measure spaces operate. When are they integrable, and what do we mean by convergence of functions? Formalising these notions properly is a goal of the second half of the course.