Measurable function: Difference between revisions

Revision as of 22:32, 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 $E\in \varepsilon .$
Let $(X,M)$ , $(Y,N)$ , and $(Z,P)$ be measure spaces. If a map $f:X\to Y$ is $(M,N)$ -measurable and $g:Y\to Z$ is $(N,P)$ -measurable, then $g\circ f:X\to Z$ is $(M,P)$ -measurable. If $N_{1}$ is another $\sigma$ -algebra on $Y$ and a map $h:Y\to Z$ is $(N_{1},P)$ -measurable, then $h\circ f:X\to Z$ is $(M,P)$ -measurable when $N_{1}\subseteq N.$

@@ Line 8: / Line 8: @@
 ==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>
-* Let <math>(X,M)</math>, <math>(Y,N)</math>, and <math>(Z,P)</math> be measure spaces. If a map <math>f: X \to Y</math> is <math>(M,N)</math>-measurable and <math>g: Y \to Z</math> is <math>(N,P)</math>-measurable, then <math>g\circ f: X \to Z</math> is <math>(M,P)</math>-measurable.
+* Let <math>(X,M)</math>, <math>(Y,N)</math>, and <math>(Z,P)</math> be measure spaces. If a map <math>f: X \to Y</math> is <math>(M,N)</math>-measurable and <math>g: Y \to Z</math> is <math>(N,P)</math>-measurable, then <math>g\circ f: X \to Z</math> is <math>(M,P)</math>-measurable. If <math>N_1</math> is another <math>\sigma</math>-algebra on <math>Y</math> and a map <math>h: Y \to Z</math> is <math>(N_1,P)</math>-measurable, then <math>h \circ f: X \to Z</math> is <math>(M,P)</math>-measurable when <math>N_1 \subseteq N.</math>
-If <math>N_1</math> is another <math>\sigma</math>-algebra on <math>Y</math> and a map <math>h: Y \to Z</math> is <math>(N_1,P)</math>-measurable, then <math>h \circ f: X \to Z</math> is <math>(M,P)</math>-measurable when <math>N_1 \subseteq N.</math>