Let $(E,d),(F,d')$ be separable metric spaces endowed with their Borel algebra, $f:E\rightarrow F$ a continuous surjective function, and $Q$ a probability measure on $F$ with separable support.


**Question:** Does there exist a probability measure $P$ on $E$ such that $Q$ is the pushforward measure of $P$ by $f$?



Sufficient condition for a probability measure to be a pushforward measure
$\newcommand\C{\mathscr C}\newcommand\de{\delta}$A sufficient condition is that $E$ be compact. Indeed, since $Q$ has a separable support, without loss of generality $F$ is separable.
So, for each natural $n$ there is an (at most) countable set $\C\_n$ of nonempty pairwise disjoint Borel subsets of $F$ of diameter $\le1/n$ such that $\bigcup\C\_n=F$. For each $C\in\C\_n$, take any $y\_{n,C}\in C$ and let
$$Q\_n:=\sum\_{C\in\C\_n}Q(C)\de\_{y\_{n,C}},$$
where $\de\_y$ is the Dirac probability measure supported on the set $\{y\}$. Then $Q\_n\to Q$ (weakly).


Next, for each $n$ and each $C\in\C\_n$, take any $x\_{n,C}$ such that $f(x\_{n,C})=y\_{n,C}$, and let
$$P\_n:=\sum\_{C\in\C\_n}Q(C)\de\_{x\_{n,C}}.$$
Then $Q\_n=P\_nf^{-1}$, the pushforward measure of $P\_n$ by $f$.


By the compactness of $E$, passing to a subsequence if needed, without loss of generality $P\_n\to P$ for some probability measure $P$ on $E$. Since $f$ is continuous, it follows that $Q\_n=P\_nf^{-1}\to Pf^{-1}$. But $Q\_n\to Q$. So, $Pf^{-1}=Q$. That is, $Q$ is the pushforward measure of $P$ by $f$, as desired.


