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If you have two infinite product measures, $P$ and $Q$, and a single $B^{\infty }$-measurable random variable $X$ (where $B^{\infty}$ is the tail $\sigma$-field) such that $P(X=c)=1$ and $Q(X=d)=1$, then must $c=d$?

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    Boris, this infinite product measure can have "factor" probability measures of any sort ? I mean, you allow $$P=\prod_{i\in \mathbb{N}}\mu_i,$$ where $\mu_i$ is not same measure for any $i\in\mathbb{N}$, for example ?2011-10-31

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