Measurable Functions - Core Definitions

Measurable functions are the natural objects of study in measure theory, generalizing the notion of continuous functions in topology. They form the foundation for integration theory.

DefinitionMeasurable Function

Let $(X, \mathcal{F})$ and $(Y, \mathcal{G})$ be measurable spaces. A function $f: X \to Y$ is measurable (or $\mathcal{F}/\mathcal{G}$ -measurable) if for every $E \in \mathcal{G}$ : $f^{-1}(E) = \{x \in X : f(x) \in E\} \in \mathcal{F}$

In other words, the preimage of every measurable set in $Y$ is a measurable set in $X$ .

For real-valued functions, we typically use the Borel sigma-algebra on $\mathbb{R}$ . A function $f: X \to \mathbb{R}$ is measurable if $f^{-1}(B) \in \mathcal{F}$ for all Borel sets $B \subseteq \mathbb{R}$ .

ExampleCharacterization via Level Sets

For $f: X \to \mathbb{R}$ (or $\mathbb{R} \cup \{\pm\infty\}$ ), the following are equivalent:

$f$ is measurable
$\{x : f(x) > a\} \in \mathcal{F}$ for all $a \in \mathbb{R}$
$\{x : f(x) \geq a\} \in \mathcal{F}$ for all $a \in \mathbb{R}$
$\{x : f(x) < a\} \in \mathcal{F}$ for all $a \in \mathbb{R}$
$\{x : f(x) \leq a\} \in \mathcal{F}$ for all $a \in \mathbb{R}$

This characterization is extremely useful because it reduces checking measurability to verifying a single condition on level sets.

DefinitionSimple Function

A simple function is a measurable function $s: X \to \mathbb{R}$ that takes only finitely many values. If $s$ takes values $a_1, \ldots, a_n$ , we can write: $s = \sum_{i=1}^{n} a_i \chi_{A_i}$ where $A_i = \{x : s(x) = a_i\}$ and $\chi_{A_i}$ is the characteristic function of $A_i$ .

Simple functions are the building blocks for integration, as any measurable function can be approximated by simple functions.

Remark

Key properties of measurable functions include:

Continuous functions are measurable: If $f: X \to Y$ is continuous and $X, Y$ are metric spaces with Borel sigma-algebras, then $f$ is measurable.
Composition: If $f: X \to Y$ is measurable and $g: Y \to Z$ is measurable, then $g \circ f: X \to Z$ is measurable.
Algebraic operations: If $f, g: X \to \mathbb{R}$ are measurable, then $f + g$ , $f - g$ , $fg$ , and $f/g$ (where $g \neq 0$ ) are measurable.