Joint Measurability from Weak Measurability and Continuity

Question

Suppose I have a function $f:X\times H_1\to H_2$, where $X$ is a measurable space with some associated $\sigma$-algebra, $\mathcal{X}$, and $H_1$ and $H_2$ are both separable Hilbert spaces (not necessarily the same). Assume I know that there is a constant $C$ such that $f$ is Lipschitz in the $h$ argument, $$ |f(x,u) - f(x,v)|_{H_2}\leq C |u-v|_{H_1} $$ and that for all $u$ and $h$, the real valued mapping $x\mapsto (f(x,u),h)_{H_2}$ is measurable.

I would like to conclude something about the joint measurability of $f$, in particular that $f$ is $(X\times H_1, \mathcal{X}\times \mathcal{B}(H_1))\to (H_2, \mathcal{B}(H_2))$, but don't quite see how to do it.

I think this can reduced to the problem to proving measurability of $$ (x,u)\mapsto(f(x,u),h_n)_{H_2} $$ where $\{h_n\}$ form an orthonormal basis of $H_2$.

Answer 1

I realize now this is really just an instance of a Caratheodory functions. Once we define $$ f_n(x,u) = (f(x,u),h_n)_{H_2} $$ we have a function which is continuous in one argument and measurable in the other on the separable and measurable spaces, so it is jointly measurable. Then using separability again, $$ |f(x,u)-y|_{H_2} < r\Leftrightarrow \sup_n |(f(x,u)-y,h_n)_{H_2}|<r $$ and the result follows by the countability of the $h_n$.