This problem is in Real analysis for graduate students, Richard F. Bass. Problem is the following
Let $(X,\mathcal A)$, $(Y,\mathcal B)$ be two measurable spaces and let $f \ge 0 $ be measurable with respect to $\thinspace$ $\mathcal A\times \mathcal B$ Let $g(x) = \sup_{y \in Y} f(x,y)$ and suppose $g(x) < \infty$ for each $x$.
I think that $g$ is not necessarily measurable with respect to $\mathcal A$. I have tried to find counter-example but nothing can be applied here.