In measure theory, measurable functions are defined by the property of their pre-images. The definition of measurable functions is similar to the definition of continuous functions in topology. In this chapter, we introduce measurable functions and relevant theorems.

5.1 Basic Concepts and Facts

Definition 5.1 (∑-Measurable Function). Let (S, ∑) be a measurable space. A function h : SR is called ∑-measurable, or measurable relative to the σ-algebra ∑, if and only if


where is the Borel σ-algebra on R (see Definition 2.6) and h−1(A) is defined as


The set of all ∑-measurable functions is denoted by m∑.

Definition 5.2 (Borel Function). Let S be a topological space (see Definition 2.5). A function h : SR is called a Borel function if h is B(S)-measurable or Borel measurable, where (S) is the σ-algebra generated by the collection of all open sets in S.

Definition 5.3 (∑1/∑2-measurable Function). Let (S1, ∑1) and (S2, ∑2) be two measurable spaces and h : S1S2. h is considered ∑1/∑2-measurable if and only if

where h−1(A) = {s S1 : h(s) A}.

Definition ...

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