I'm pretty sure I can prove the "Theorem" given further below (without very much difficulty), but it seems way too basic not to have been noticed before.
So I'm wondering if there are any papers/textbooks which state part or all of this "Theorem" - or at the least, which define the $\sigma$-algebra being considered.
Two random variables over a probability space $(\Omega,\mathcal{F},\mathbb{P})$ are said to be $\mathbb{P}$-equivalent if they agree $\mathbb{P}$-almost everywhere. Given a probability space $(\Omega,\mathcal{F},\mathbb{P})$ and a measurable space $(S,\mathcal{S})$, we write $L^0(\mathbb{P},\mathcal{S})$ for the set of all $\mathbb{P}$-equivalence classes of $(\mathcal{F},\mathcal{S})$-measurable functions.
Now define on $L^0(\mathbb{P},\mathcal{S})$ the $\sigma$-algebra $$ \mathfrak{L}(\mathbb{P},\mathcal{S}) := \sigma( \, [X] \mapsto \mathbb{P}(E \cap X^{-1}(A)) \, : \, E \in \mathcal{F}, A \in \mathcal{S} \, ) $$ where $[X]$ denotes the $\mathbb{P}$-equivalence class of $X$.
Remark 1. The map $\,[X] \mapsto \int_\Omega h(\omega) g(X(\omega)) \, d\mathbb{P}\,$ is measurable with respect to $\mathfrak{L}(\mathbb{P},\mathcal{S})$ for every $h \in L^1(\mathbb{P})$ and every bounded measurable $g \colon S \to \mathbb{R}$, and likewise for all measurable $h \colon \Omega \to [0,\infty]$ and $g \colon S \to [0,\infty]$.
Remark 2. Just as with $L^\infty(\mathbb{P})$, the set $L^0(\mathbb{P},\mathcal{S})$ does not really depend fully on $\mathbb{P}$, but only on the equivalence class of $\mathbb{P}$ (under the standard notion of equivalence of measures). Furthermore, due to Remark 1, the $\sigma$-algebra $\mathfrak{L}(\mathbb{P},\mathcal{S})$ also only depends on the equivalence class of $\mathbb{P}$.
Are there any papers or textbooks that define the above $\sigma$-algebra $\mathfrak{L}(\mathbb{P},\mathcal{S})$? And are there any that state (either as a proved result or as an exercise) any parts of the following?
Theorem. Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space, and let $S$ be a separable metrisable topological space.
A sequence of $S$-valued random variables $X_n$ converges in probability to an $S$-valued random variable $X$ if and only if $\int_E g(X_n) \, d\mathbb{P} \to \int_E g(X) \, d\mathbb{P}$ for every $E \in \mathcal{F}$ and every bounded continuous $g \colon X \to \mathbb{R}$.
As a consequence: Suppose $\mathcal{F}$ is countably generated. Then the Borel $\sigma$-algebra of the topology of convergence in probability, defined on $L^0(\mathbb{P},\mathcal{B}(S))$ with reference to the topology of $S$, is precisely $\mathfrak{L}(\mathbb{P},\mathcal{B}(S))$.
In the case that $S=\mathbb{R}$, if $\mathcal{F}$ is countably generated, then the Borel $\sigma$-algebra of $L^p(\mathbb{P})$ is precisely $\mathfrak{L}(\mathbb{P},\mathcal{B}(\mathbb{R}))$ for any $p \in [1,\infty]$.
If $(\Omega,\mathcal{F})$ and $(S,\mathcal{B}(S))$ are both standard Borel spaces, then there exists an $(\mathfrak{L}(\mathbb{P},\mathcal{B}(S))\otimes\mathcal{F},\mathcal{B}(S))$-measurable function $e \colon L^0(\mathbb{P},\mathcal{B}(S)) \times \Omega \to S$ such that for every $x\in L^0(\mathbb{P},\mathcal{B}(S))$, the function $\omega \mapsto e(x,\omega)$ is a representative of $x$.
[I have proved the first statement on Math.SE, here. For the final statement, taking $\Omega=\mathbb{R}$ and $S=[0,1]$ or a countable subset thereof, just define $e$ by the Lebesgue differentiation theorem (which holds for arbitrary probability measures on $\mathbb{R}$).]
Remark 3. The $\sigma$-algebra $\mathfrak{L}(\mathbb{P},\mathcal{S})$ seems to me a very natural $\sigma$-algebra on $L^0(\mathbb{P},\mathcal{S})$. In particular, I think it serves as a way of way of overcoming the problem of the lack of a meaningful measurable structure on the set of all $(\mathcal{F},\mathcal{S})$-measurable functions (see the accepted answer of this MO question), if you're working in a setting where there is a natural equivalence class of measures on $(\Omega,\mathcal{F})$. This is especially justified by the final statement of the above "Theorem", which stands nicely in contrast to the negative result cited in the accepted answer to the linked MO question.
PS. I think Hans Crauel's Random Probability Measures on Polish Spaces has a lot of results addressing these kinds of questions, but in view of what Google will let me access, I don't think it addresses the particular question of the Borel $\sigma$-algebra of the topology of convergence in probability. (However, I don't currently have access to the full book, and so I can't say this with complete confidence.)
