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Let $(X, \mathcal{A})$ and $(Y, \mathcal{B})$ be measurable spaces, and $f:X \to Y$. We call $f$ is measurable if and only if $$\forall B \in \mathcal B: f^{-1} (B) \in \mathcal A$$

I would like to ask if $f(X)$ is countable implies $f$ is measurable. Thank you so much!

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    $\begingroup$ Why would it? Unless $\mathcal A=\mathcal P(X)$, take $A\in\mathcal P(X)\setminus\mathcal A$ (i.e. a non-measurable set) and define $f$ to have one value on $A$ and a different value on $X\setminus A$... $\endgroup$ Feb 9, 2020 at 17:49

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Take a non measurable subset $A$ of $X$ and consider its indicator function $\chi_A$.

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