I'm currently considering the following map: Let $\mathcal{F}$ be a subset of some $L^2$ space with respect to a probability measure on $\mathbb{R}$ and let $X=(X_1,X_2,\ldots,X_n)$ be points on the real line. I would like to prove joint measurability of the map
\begin{equation} \mathbb{R}^n \times \mathcal{F}\to \mathbb{R}, \end{equation}
given by
\begin{equation} (X,f)\longmapsto \int f d\sum^n_{i=1}\delta_{X_i} = \sum^n_{i=1}f(X_i). \end{equation}
My intuition tells me that joint measurability holds, but I can't seem to quite get there. Is this true in my case, or do we need more assumptions?
We can assume that $\mathcal{F}$ is a Donsker class.
Edit: I have not been able to show this in the general case, but it does hold when you assume some sort of continuity. Both continuity of each function $f\in\mathcal{F}$ or right/left continuity of each function $f\in\mathcal{F}$. It also holds for right/left continuity in higher dimensions in the sense of Neuhaus 1971.