measurable function

Let $\big{(}X,\mathcal{B}(X)\big{)}$ and $\big{(}Y,\mathcal{B}(Y)\big{)}$ be two measurable spaces. Then a function $f\colon X\to Y$ is called a measurable function if:

$f^{-1}\big{(}\mathcal{B}(Y)\big{)}\subseteq\mathcal{B}(X)$

where $f^{-1}\big{(}\mathcal{B}(Y)\big{)}=\{f^{-1}(E)\mid E\in\mathcal{B}(Y)\}$ .

In other words, the inverse image of every $\mathcal{B}(Y)$ -measurable set is $\mathcal{B}(X)$ -measurable. The space of all measurable functions $f\colon X\to Y$ is denoted as

$\mathcal{M}\big{(}\big{(}X,\mathcal{B}(X)\big{)},\big{(}Y,\mathcal{B}(Y)\big{)% }\big{)}.$

Any measurable function into $(\mathbb{R},\mathcal{B}(\mathbb{R}))$ , where $\mathcal{B}(\mathbb{R})$ is the Borel sigma algebra of the real numbers $\mathbb{R}$ , is called a Borel measurable function.¹¹More generally, a measurable function is called Borel measurable if the range space $Y$ is a topological space with $\mathcal{B}(Y)$ the sigma algebra generated by all open sets of $Y$ . The space of all Borel measurable functions from a measurable space $(X,\mathcal{B}(X))$ is denoted by $\displaystyle{\mathcal{L}^{0}\big{(}X,\mathcal{B}(X)\big{)}}$ .

Similarly, we write $\displaystyle{\bar{\mathcal{L}}^{0}\big{(}X,\mathcal{B}(X)\big{)}}$ for $\displaystyle{\mathcal{M}\big{(}\big{(}X,\mathcal{B}(X)),(\bar{\mathbb{R}},% \mathcal{B}(\bar{\mathbb{R}})\big{)}\big{)}}$ , where $\mathcal{B}(\bar{\mathbb{R}})$ is the Borel sigma algebra of $\bar{\mathbb{R}}$ , the set of extended real numbers.

Remark. If $f:X\to Y$ and $g:Y\to Z$ are measurable functions, then so is $g\circ f:X\to Z$ , for if $E$ is $\mathcal{B}(Z)$ -measurable, then $g^{-1}(E)$ is $\mathcal{B}(Y)$ -measurable, and $f^{-1}\big{(}g^{-1}(E)\big{)}$ is $\mathcal{B}(X)$ -measurable. But $f^{-1}\big{(}g^{-1}(E)\big{)}=(g\circ f)^{-1}(E)$ , which implies that $g\circ f$ is a measurable function.

Example:

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Let $E$ be a subset of a measurable space $X$ . Then the characteristic function $\chi_{E}$ is a measurable function if and only if $E$ is measurable.

Title	measurable function
Canonical name	MeasurableFunction
Date of creation	2013-03-22 12:50:50
Last modified on	2013-03-22 12:50:50
Owner	CWoo (3771)
Last modified by	CWoo (3771)
Numerical id	18
Author	CWoo (3771)
Entry type	Definition
Classification	msc 28A20
Synonym	Borel measurable
Related topic	ExampleOfFunctionNotLebesgueMeasurableWithMeasurableLevelSets
Related topic	LusinsTheorem2
Related topic	BorelGroupoid
Related topic	BorelMorphism
Defines	Borel measurable function