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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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  • $\begingroup$ Can you precisely state what you mean by weak$^*$-compact? $\endgroup$ – Davide Giraudo Apr 15 '13 at 19:55
  • $\begingroup$ the weak*-topology is usually taken to be the weakest topology such that the linear functionals $l_Z:\mathcal M_1(\Omega)\rightarrow\mathbb R$, defined by $l_Z(\mu)=\int_\mathbb R Zd\mu$ is continuous for every bounded and measurable function $Z:\Omega\rightarrow\mathbb R$. $\endgroup$ – Andy Teich Apr 15 '13 at 20:34
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The question is now answered at Math Overflow.

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