measurable function

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

$$f^{-1}\left(ℬ(Y)\right)\subseteq ℬ(X)$$

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

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

$$ℳ((X,ℬ(X)),(Y,ℬ(Y))).$$

Any measurable function into $(ℝ,ℬ(ℝ))$, where $ℬ(ℝ)$ is the Borel sigma algebra of the real numbers $ℝ$, is called a Borel measurable function. The space of all Borel measurable functions from a measurable space $(X,ℬ(X))$ is denoted by ${ℒ}^0(X,ℬ(X))$.

Similarly, we write ${\overline{ℒ}}^0(X,ℬ(X))$ for $ℳ((X,ℬ(X)),(\overline{ℝ},ℬ(\overline{ℝ})))$, where $ℬ(\overline{ℝ})$ is the Borel sigma algebra of $\overline{ℝ}$, 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 $ℬ(Z)$-measurable, then $g^{-1}(E)$ is $ℬ(Y)$-measurable, and $f^{-1}\left(g^{-1}(E)\right)$ is $ℬ(X)$-measurable. But $f^{-1}\left(g^{-1}(E)\right)={(g\circ f)}^{-1}(E)$, which implies that $g\circ f$ is a measurable function.

Example:

• 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