Problem : [Let $(X, \mathcal{A})$ and $(Y, \mathcal{B})$ be two measurable spaces and let $f\geq 0$ be measurable with respect to $\mathcal{A} \times \mathcal{B}$. Let $g(x)=\sup_{y\in Y} f(x, y)$ and suppose $g(x)<\infty$ for each $x$. Is $g$ necessarily measurable with respect to $\mathcal{A}$? If not, find a counterexample.]
I tried to find the counterexample. I set $X=Y=[0, 1]$ and $$f(x, y)=\begin{cases}\mathbf{1}_{y} (x) \quad y\in E_0 \\ 0 \quad \text{otherwise}\end{cases}$$ for some nonmeasurable set $E_0\in [0, 1]$. Then $g(x)=\sup_y f(x, y)$ becomes $g(x)=\mathbf{1}_{E_0} (x)$, which is non-measurable, but I'm confused whether I can say two-variable function $f$ is measurable. Could you give me some hints?? Thanks.
The function you specified is not measurable (it's the characteristic function of $E_0$ seen as a subset of the diagonal), a counterexample needs to depend on both coordinates in a substantial way.
To construct a counterexample, the following fact may be useful: there is a Borel subset of $[0,1]^2$ such that its projection onto the first interval is not Borel (the projection is a non-Borel analytic set). The rest is straightforward.