Let $X$ and $Y$ be uncountable Polish spaces, $\Delta(Y)$ be the space of Borel probability measures on $Y$ endowed with the Borel $\sigma$-algebra induced by the variation distance, and let $g:X\to \Delta(Y)$ be Borel measurable. Is $g(X)$ separable? I'm also interested in the weaker question of whether we can find for every Borel probability measure on $X$ a null set $N$ such that $g(X\setminus N)$ is separable.

Separable subspaces of $\Delta(Y)$ have a nice structure; they correspond exactly to subspaces of $L_1(\mu)$ for some Borel probability measure $\mu$ on $Y$. But the topology on all of $\Delta(Y)$ is extremely fine. For example, every set of Dirac measures is closed. My vague intuition is that it is so fine that functions with a nonseparable range cannot be Borel measurable.

under the continuum hypothesisthe range of a Borel measurable map from a separable metric space to a metric space is separable. So, indeed, it seems that without that kind of additional assumption (or something alike) problem 10 on page 99 cannot be established. $\endgroup$1more comment