measurable functionMeasurableFunctions2002-07-21 15:02:262006-12-09 02:33:55DefinitionBorel measurable functionmeasurable% this is the default PlanetMath preamble. as your knowledge
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% define commands hereLet $\big(X,\mathcal{B}(X)\big)$ and $\big(Y, \mathcal{B}(Y)\big)$ be two measurable spaces. Then a function $f\colon X\to Y$ is called a \emph{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 \emph{Borel measurable function}.{\footnote {More generally, a measurable function is called \emph{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. \\
\textbf{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.
\textbf{Example:}
\begin{itemize}
\item 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.
\end{itemize}