Let $\mathcal{M}_1(\mathbb R)$ denote the space of Borel probability measures on $\mathbb R$. The space is a Polish space (a space which admits a complete, separable, metric) using, say the Levy-Prokhorov metric. For $\mu \in \mathcal{M}_1(\mathbb R)$, let $L^1(\mu)$ denote the Banach space of $\mu$-integrable functions (mod $\mu$-null). Again, for each $\mu$, $L^1(\mu)$ is a Polish space.

Is the space $$\mathbb X = \left\{(\mu,f) \ \middle|\ \mu \in \mathcal{M}_1(\mathbb R),\ f\in L^1(\mu)\right\}$$ a Polish space?

Of course we need to specify a topology. I am assume the most natural topology. (I believe this is the topology generated by sets of the form
$$\left\{(\nu,g) \in \mathbb X \ \middle|\ d(\mu,\nu)<\varepsilon \text{ and } \|g-f\|_{L^1(\nu)} < \varepsilon \right\}$$

where $f \in C_0(\mathbb R)$, $\mu \in \mathcal{M}_1(\mathbb R)$, $\varepsilon > 0$, and $d$ is a metric on $\mathcal{M}_1(\mathbb R)$ which makes it a Polish space.)

If $\mathbb X$ is a Polish space, is there a nice metric?

If $\mathbb X$ is not a Polish space, how nice is this space?

The motivation for my question is whether $\mathbb X$ is a computable metric space, so feel free to answer that question as well? :)

**Terminology:** Feel free to correct me if I am mistaken on the terminology. I originally called $\mathbb X$ a "coproduct" and wrote it as
$$\coprod_{\mu \in \mathcal{M}_1(\mathbb R)} L^1(\mu),$$
but I was corrected in the comments. (However, the underlying set is a direct sum of sets, and I think similar notation $\sum_{x:A} B(x)$ is used in homotopy type theory to express the sum of dependent types.)

After doing some research I think I should say the map $\mathbb X \to \mathcal{M}_1(\mathbb R)$ given by $(\mu,f)\mapsto \mu$ is a *fiber bundle* where $\mathbb X$ is the *total space* and $\mathcal{M}_1(\mathbb R)$ is the *base space*. (Although, I could be mistaken.)

**Topology:** In case I gave the wrong basis for the topology above, I want the following maps to be continuous:
$$(\mu,f)\mapsto \mu,\quad (\mu,f)\mapsto \|f\|_{L^1(\mu)},\quad (\mu,f)\mapsto \int fg\,d\mu$$
where $g$ is a bounded continuous function.