Joint Measurability from Weak Measurability and Continuity

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 $$H_2$$.

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, $$|f(x,u)-y|_{H_2} < r\Leftrightarrow \sup_n |(f(x,u)-y,h_n)_{H_2}| and the result follows by the countability of the $$h_n$$.