# Is it true that $f(X)$ is countable implies $f$ is measurable?

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!

• 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 $X\setminus A$... Feb 9, 2020 at 17:49

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