You are here
Homemeasurable function
Primary tabs
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.^{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 $\displaystyle{\mathcal{L}^{0}\big(X,\mathcal{B}(X)\big)}$.
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:

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.
Mathematics Subject Classification
28A20 no label found Forums
 Planetary Bugs
 HS/Secondary
 University/Tertiary
 Graduate/Advanced
 Industry/Practice
 Research Topics
 LaTeX help
 Math Comptetitions
 Math History
 Math Humor
 PlanetMath Comments
 PlanetMath System Updates and News
 PlanetMath help
 PlanetMath.ORG
 Strategic Communications Development
 The Math Pub
 Testing messages (ignore)
 Other useful stuff
 Corrections