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I saw a question where we have $\overline{\mathbb{R}}:=\mathbb{R}\cup\{\pm\infty\}$ and $(X,S)$ is a measurable space, $f:\, X\to\overline{\mathbb{R}}$

In one part I was told to assume that $f$ is measurable as a function $f^{-1}(\mathbb{R})\to\mathbb{R}$

What does it mean ?

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up vote 2 down vote accepted

Note that $Y=f^{-1}(\mathbb R)$ is a subset of $X$ and that $T=\{A\cap Y\mid A\in S\}$ is a sigma-algebra on $Y$. The question is to determine whether the function $g:Y\to\mathbb R$ defined by $g(x)=f(x)$ for every $x$ in $Y$, is measurable or not, seen as a function from the measurable space $(Y,T)$ to the measurable space $(\mathbb R,\mathcal B(\mathbb R))$.

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