Let $(\Omega,\mathcal{F},P)$ be a probability space and $X$ be a Polish space. Let $\mu:\Omega\rightarrow Pr(X)$ be a random probability measure with marginal $P$ on $\Omega$ and $\mu_\omega$ be the integration of $\mu$. My question is: Can we pick up random closed set $\omega\rightarrow B(\omega)$ such that $$ 0<\mu_\omega(B(\omega))\equiv constant<1?$$
If can, how to construct this random closed set?
Thanks!