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Let $X_i$ be a sequence of probability spaces and define $\displaystyle X=\prod_{i=1}^\infty X_i$

Let $A$ be the algebra on $X$ generated by the sets of the form $\displaystyle \prod_{i=1}^{n-1} X_i \times E_n \times \prod_{i=n+1}^{\infty} X_i, E_n \in P(X_n)$

Show that every element in $A$ can be written as $a= \displaystyle \cup_{j=1}^m (\prod_{i=1}^{n_j} E_{j,i} \times \prod_{i=n_j}^{\infty} X_i)$ where $E_{j,i} \in P(X_i)$ and the union is a disjoint union

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    Ahhh. There you go then :)2011-02-16

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You can find this in Paul Halmos' Measure theory, Chapter I, §5. He talks about generated rings but in your case the distintion is irrelevant---this is the content of one of the exercises in that section, in fact.