# Is the range of a probability-valued random variable with the variation topology (almost) separable?

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.

• Your problem seems in a way a bit similar to the one in my first MO-question. I have not thought anything of the possible details but maybe a solution could be obtained along the same lines.
– TaQ
Commented Jul 8, 2023 at 20:51
• Thank you, that is very useful! Actually, there should be a simpler solutions since the domain here is Polish. I'll try to figure out the details. Commented Jul 8, 2023 at 21:10
• @TaQ The answer there shows that for some null set $N$, the image $g(X\setminus N)$ is separable. The other answer would follow from the problem in Dudley's textbook, but I'm not sure how that problem works. There might be an implicit assumption that if $T$ is uncountable, then $2^T>2^\mathbb{N}=\mathfrak{c}$. This need not be the case. Commented Jul 8, 2023 at 22:44
• Dudley has Problem 9 on page 387 where it is asked to show that under the continuum hypothesis the 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.
– TaQ
Commented Jul 10, 2023 at 0:53
• This MO-question and this MO-answer are related to the gap in the above mentioned problem of Dudley.
– TaQ
Commented Jul 13, 2023 at 9:00