ABSTRACT
This chapter presents a rigorous treatment of measure and integration theory. There is a plethora of texts on measure theory that the reader can turn to for additional results and exposition with Halmos being the standard reference for decades. The chapter describes an important component of measure theory, that of a measurable mapping. Any random object is essentially described by a mapping that is measurable. A very important class of measurable functions are the so-called simple functions. Decomposition theorems in measure theory provide results that connect different types of measures and have great applications in all manner of proofs. The construction of the Lebesgue measure that follows is a special case of the extension of measure by Caratheodory. The chapter describes how to use the Monotone convergence theorem and integration to the limit in order to prove statements about a nonnegative measurable function.
