# CHAPTER 5

# MEASURABLE FUNCTIONS

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* : *S* → **R** 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* : *S* → **R** 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 (*S*_{1}, ∑_{1}) and (*S*_{2}, ∑_{2}) be two measurable spaces and *h* : *S*_{1} → *S*_{2}. *h* is considered ∑_{1}/∑_{2}-measurable if and only if

where *h*^{−1}(*A*) = {*s* *S*_{1} : *h*(*s*) *A*}.

**Definition ...**

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