Let $(\Omega,\Sigma,\mu)$ be a probability space. A $\mu$-atom is an $A\in\Sigma$ such that $\mu(A)>0$ and for all $B\in\Sigma$ such that $B\subseteq A$, either $\mu(B)=\mu(A)$ or $\mu(B)=0$ holds.

Now let $(X,\mathcal{X})$ be a measurable space with the property that for all $x,y\in X$ there is $B\in\mathcal{X}$ such that $x\in B$ and $y\notin B$.

Is every measurable function $f:\Omega\to X$ almost surely constant? That is, is there for every $\mu$-atom $A\in\Sigma$ some point $x\in X$ such that $f^{-1}\big(\{x\}\big)$ has $\mu$-outer measure $\mu(A)$?

It is not hard to prove that the answer is yes if $\mathcal{X}$ is generated by a countable family of sets, but the approach does not generalize.

*Motivation:* I face the problem in economic modelling. I want the agents to be represented by a probability space, where the atomic part represents "big" players. The behavior is given by a measurable function and want to ensure that the behavior of each big player is well defined.