ABSTRACT
This chapter builds the measure theory on an arbitrary set, by making use of the essential properties that measurable subsets of the real line satisfy. It discusses the measure on an arbitrary σ-algebra as well as generated σ-algebra and Borel σ-algebra. The Lebesgue measure m on R is obtained by restricting the Lebesgue outer measure m* to the σ-algebra M. The chapter describes a procedure of obtaining a measure on a set X by first defining a general outer measure μ* on all subsets of X and then restricting it to a general class of measurable sets constructed out of μ*. It also defines supremum, infimum, limit superior, and limit inferior of sequences of functions. The chapter examines a class of functions which are more general than characteristic functions, and which plays a very important role in the theory of measure and integration.
