Consider the following situation:

$S, T$ are standard Borel spaces (say $S = [0,1]^k$, $T = [0,1]$ if it is helpful).

There is a a random variable $\zeta: \Omega \to S$.

$f_n(\zeta) \to^d \eta$, i.e. $f_n(\zeta)$ converges in distribution to some random variable $\eta$.

Will there always exist

A random variable $\zeta': \Omega' \to S$ (i.e. possibly with a different base space than $\Omega$), and

functions $g_n, g: S \to T$

such that

$\zeta$ and $\zeta'$ have the same distribution,

for all $n$, $g_n(\zeta')$ has the same distribution as $f_n(\zeta)$, and

$g_n(\zeta') \to g(\zeta')\ a.e$, i.e. $g_n(\zeta')$ converges to $g(\zeta')$ almost everywhere (and in particular $g(\zeta')$ has the same distribution as $\eta$).