measurable function
Let $(X,\mathcal{B}(X))$ and $(Y,\mathcal{B}(Y))$ be two measurable spaces^{}. Then a function $f:X\to Y$ is called a measurable function^{} if:
$${f}^{1}\left(\mathcal{B}(Y)\right)\subseteq \mathcal{B}(X)$$ 
where ${f}^{1}\left(\mathcal{B}(Y)\right)=\{{f}^{1}(E)\mid E\in \mathcal{B}(Y)\}$.
In other words, the inverse image of every $\mathcal{B}(Y)$measurable set is $\mathcal{B}(X)$measurable. The space of all measurable functions $f:X\to Y$ is denoted as
$$\mathcal{M}((X,\mathcal{B}(X)),(Y,\mathcal{B}(Y))).$$ 
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.^{1}^{1}More 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 ${\mathcal{L}}^{0}(X,\mathcal{B}(X))$.
Similarly, we write ${\overline{\mathcal{L}}}^{0}(X,\mathcal{B}(X))$ for $\mathcal{M}((X,\mathcal{B}(X)),(\overline{\mathbb{R}},\mathcal{B}(\overline{\mathbb{R}})))$, where $\mathcal{B}(\overline{\mathbb{R}})$ is the Borel sigma algebra of $\overline{\mathbb{R}}$, 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 ${f}^{1}\left({g}^{1}(E)\right)$ is $\mathcal{B}(X)$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:

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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  20130322 12:50:50 
Last modified on  20130322 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 