Does there exist a function $h:X\to\mathbb{R}$ that is Lebesgue measurable on its support, that is, $\{x\in X:h(x)\neq0\}$ but that's not Lebesgue measurable?
I don't think constructing some indicator function on a non measurable set $V\subset X$ would work, because even the complement of a non measurable subset in a measurable set can be non measurable (I think).
If it matters, the question I saw about this had $X=[0,1]$.