Checking the measurability of a function

Question

I have to check check the measurability of such a function. $f(x):\mathbb{R} \rightarrow \mathbb{R}$

$f(x) = \left\{ \begin{array}{ll} 4x-3 & \textrm{when $x \in V$}\\ 0 & \textrm{when $x \in \mathbb{R}\V$}\\ \end{array} \right. $.

When $V$ is vitaly set. I know how to do it when the whole graph of a function is above or below the OX axis, here the situation seems different and I do not know what to do.

Answer 1

To disprove that the function is measurable, it suffices to find a measurable set $B$, such that $f^{-1}(B)$ is not measurable. For instance we could take $B=\{0\}$, and note that $$f^{-1}(\{ 0 \})= (\mathbb{R}\setminus V) \cup \{\frac34\}$$ which is not a measurable set, since neither the vitali set $V$ nor the complement $\mathbb{R}\setminus V$ are measurable.