Let $(\Omega,\mathcal F)$ be a measurable space and $\mathcal P$ a weak*-compact set of the set of all probability measures $\mathcal M_1(\Omega)$. Does there always exist a probability measure $\mathbb Q\in\mathcal M_1(\Omega)$ such that every $\mathbb P\in\mathcal P$ is absolutely continuous to $\mathbb Q$, i.e. such that $\mathbb Q$ dominates all measures in $\mathcal P$?
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