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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 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:
Footnotes
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- 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 $Y$ .
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