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Let $(\Omega,\mathcal{B})$ be a measurable space and let $\mu$ and $\nu$ be two probability measures on this space. It is well known that $\mathcal{A} = \{A\in \mathcal{B} \enspace | \enspace \mu (A) = \nu (A) \}$ is a Dynkin-system. However given a Dynkin system $\mathcal{D}$ on $\Omega$, with $\mathcal{D} \subset \mathcal{B}$, in general there do not exist two probability measures which agree exactly on $\mathcal{D}$. For example let $\Omega = \{1,2,3,4 \}$ and let $\mathcal{B} = 2^{\Omega}$. Let $\mathcal{D} =\{\phi, \Omega,\mbox{ all sets containing exactly 2 elements}\}$. It is clear that $\mathcal{D}$ is a Dynkin system, but any two probability measures which agree on $\mathcal{D}$ are equal.

So my question is what is the characterization of sets on which two probability measures coincide? In particular, I would be very happy if anyone could give an answer for the case when $\mathcal{B}$ is countably generated.

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