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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 ? –  Leandro Oct 31 '11 at 4:04
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For example, if your sample space is $\Omega = \{0,1\}^{\mathbb N}$ and $\omega_j$ are the coordinate functions, take $X = 1$ if $\omega_j = \omega_{j+1}$ for all sufficiently large $j$, 0 otherwise. If $P$ is the product of the unit mass $\delta_0$ at 0 in each coordinate, $P(X = 1) = 1$. If $Q$ is the product of $(\delta_0 + \delta_1)/2$ in each coordinate, $Q(X=0) = 1$.

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