Exercise 22, § 4

Let X, Y be two locally compact spaces, $\mu$ a bounded measure on X, $\nu$ a bounded measure on Y, so that $\mu \otimes \nu$ is a bounded measure on $X \times Y$. Show that for every function $f \in \mathscr{C}^0(X \times Y; \mathbf{C})$, the function $x \mapsto \int f(x, y) d\nu(y)$ belongs to $\mathscr{C}^0(X; \mathbf{C})$ and