I have posted the following question also here a longer time ago, but due to no answers I thought it might fit better to MO.

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

The weak*-topology is the weakest topology such that all the linear functionals $L_Z:\mathbb P\mapsto \int_\Omega Zd\mathbb P$ for $Z:\Omega\rightarrow\mathbb R$ $\mathcal F$-measurable and bounded, are continuous.

Note that when the set $\mathcal P$ is countable one can easily find a dominating measure for $\mathcal P$.

I have posted the following question also here a longer time ago, but due to no answers I thought it might fit better to MO.

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

The weak*-topology is the weakest topology such that all the linear functionals $L_Z:\mathbb P\mapsto \int_\Omega Zd\mathbb P$ for $Z:\Omega\rightarrow\mathbb R$ $\mathcal F$-measurable and bounded, are continuous.

Note that when the set $\mathcal P$ is countable one can easily find a dominating measure for $\mathcal P$.

I have posted the following question also here a longer time ago, but due to no answers I thought it might fit better to MO.

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

The weak*-topology is the weakest topology such that all the linear functionals $L_Z:\mathbb P\mapsto \int_\Omega Zd\mathbb P$ for $Z:\Omega\rightarrow\mathbb R$ $\mathcal F$-measurable, are continuous.

Note that when the set $\mathcal P$ is countable one can easily find a dominating measure for $\mathcal P$.

I have posted the following question also here a longer time ago, but due to no answers I thought it might fit better to MO.

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

The weak*-topology is the weakest topology such that all the linear functionals $L_Z:\mathbb P\mapsto \int_\Omega Zd\mathbb P$ for $Z:\Omega\rightarrow\mathbb R$ $\mathcal F$-measurable and bounded, are continuous.

Note that when the set $\mathcal P$ is countable one can easily find a dominating measure for $\mathcal P$.