Measurable function: Difference between revisions

Revision as of 21:56, 14 November 2020

Let $(X,M)$ and $(Y,N)$ be measure spaces. A map $f:X\to Y$ is $(M,N)$ -measurable if $f^{-1}(E)\in M$ for all $E\in N.$

A function $f:\mathbb {R} \to {\overline {\mathbb {R} }}$ is called a Lebesgue measurable function if $f$ is $(L,B_{\overline {\mathbb {R} }})$ - measurable, where $L$ is the class of Lebesgue measurable sets and $B_{\overline {\mathbb {R} }}$ is Borel $\sigma$ -algebra.
A function $f:X\to Y$ is called Borel measurable if $f$ is $(B_{X},B_{Y})$ -measurable.

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 ==Examples of measurable functions==
 * A function <math>f: \mathbb{R} \to \overline{\mathbb{R}}</math> is called a '''Lebesgue measurable function''' if <math>f</math> is <math>(L, B_{\overline{\mathbb{R}}})</math>- measurable, where <math>L</math> is the class of Lebesgue measurable sets and <math>B_{\overline{\mathbb{R}}}</math> is Borel <math>\sigma</math>-algebra.
+* A function <math>f: X \to Y</math> is called Borel measurable if <math>f</math> is <math>(B_X, B_Y)</math>-measurable.