Suppose I have compact Hausdorff spaces $X$, $Y$ and a function $g: X\times Y\to \mathbb{C}$ such that
- $g(x,\cdot)$ is continuous for all $x\in X$
- $g(\cdot, y)$ is Borel for all $y\in Y$.
Can I conclude that for a Radon measure $\mu$ on $X$, $$y\mapsto \int g(x,y) d\mu(x)$$ is measurable?
Any help is much appreciated.