Measurable function: Difference between revisions

Revision as of 22:05, 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.

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

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	~~Theorem.~~ Let <math>(X,M)</math> and <math>(Y,N)</math> be measure spaces. Suppose that <math>N</math> is generated by a set <math>\varepsilon</math>. A map <math>f: X \to Y</math> is <math>(M,N)</math>-measurable if <math>f^{-1}(E) \in M</math> for all <math>E \in \varepsilon.</math>		==Basic theorems of measurable functions==
			* Let <math>(X,M)</math> and <math>(Y,N)</math> be measure spaces. Suppose that <math>N</math> is generated by a set <math>\varepsilon</math>. A map <math>f: X \to Y</math> is <math>(M,N)</math>-measurable if <math>f^{-1}(E) \in M</math> for all <math>E \in \varepsilon.</math>