  It is well known that any pair of random variables $(X,Y)$ with values in
Polish spaces, provided that $Y$ is nonatomic, can be approximated in joint law
by random variables of the form $(X',Y)$ where $X'$ is $Y$-measurable and $X'
\stackrel{d}{=} X$. This article surveys and extends some recent dynamic
analogues of this result. For example, if $X$ and $Y$ are stochastic processes
in discrete or continuous time, then, under a nonatomic assumption as well as a
necessary and sufficient causality (or compatibility) condition, one can
approximate $(X,Y)$ in law in path space by processes of the form $(X',Y)$,
where $X'$ is adapted to the filtration generated by $Y$. In addition, in
finite discrete time, we can take $X'$ to have the same law as $X$. A similar
approximation is valid for randomized stopping times, without the first
marginal fixed. Natural applications include relaxations of (mean field)
stochastic control and causal optimal transport problems as well as new
characterizations of the immersion property for progressively enlarged
filtrations.
