I'm working on problem 10.G in Bartle's Elements of Integration and Lebesgue Measure:
Let $f$ and $g$ be real-valued functions on $X$ and $Y$, respectively; suppose that $f$ is $\mathcal{X}$-measurable and $g$ is $\mathcal{Y}$-measurable. If $h$ is defined on $X\times Y$ by $h(x,y) = f(x) g(y)$, show that $h$ is $(\mathcal{X} \times \mathcal{Y})$-measurable.
My problem is that I can't really think of a way to show that any function is $(\mathcal{X} \times \mathcal{Y})$-measurable. The only thing I can think to do is try to show that
$ h^{-1}(\alpha,\infty] \in \mathcal{X} \times \mathcal{Y}, $
but I can't make any progress in this direction. Any advice would be appreciated.