Some Skorohod-type results

Let $S$ be a metric space, $g:S\rightarrow\mathbb{R}$ a Borel function, and $(\mu_n:n\ge 0)$ a sequence of tight probability measures on $\mathcal{B}(S)$. If $\mu_n=\mu_0$ on $\sigma(g)$, there are $S$-valued random variables $X_n$, all defined on the same probability space, such that $X_n\sim\mu_n$ and $g(X_n)=g(X_0)$ for all $n\ge 0$. Moreover, $X_n\overset{a.s.}\longrightarrow X_0$ if and only if $E_{\mu_n}(f\mid g)\,\overset{\mu_0-a.s.}\longrightarrow\,E_{\mu_0}(f\mid g)$ for each $f\in C_b(S)$. This result, proved in \cite{PR2023}, is the starting point of this paper. Three types of contributions are provided. First, $\sigma(g)$ is replaced by an arbitrary sub-$\sigma$-field $\mathcal{G}\subset\mathcal{B}(S)$. Second, the result is applied to some specific frameworks, including equivalence couplings, total variation distances, and the decomposition of cadlag processes with finite activity. Third, following \cite{HMMS}, the result is extended to models and kernels. This extension has a fairly natural interpretation in terms of decision theory, mass transportation and statistics.