ABSTRACT
This chapter introduces one of the most important concepts in measure theory, namely, the integral of non-negative measurable functions, and proves many important results, including the most celebrated monotone convergence theorem. The definition of the concept of the integral is motivated in a natural manner; first defining the integral of a simple measurable function motivated by the Riemann integral of a step function, and then using the fact that any non-negative measurable function f is a pointwise limit of a monotonically increasing sequence of non-negative simple measurable functions. The method used for the motivation of the definition is replicated in proving the monotone convergence theorem as well. The chapter also discusses the Radon-Nikodym theorem which is important in probability theory.
