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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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I'm not sure an answer here would benefit the community so much since you have not yet accepted an answer to your previous questions. Kindly accept some answers if you feel they have helped answer your question. –  fdart17 Feb 16 '11 at 1:34
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sorry i was not aware of the customs of the forum, i will starting accepting answers from now on –  jack Feb 16 '11 at 1:47
    
Is this really true? –  Mariano Suárez-Alvarez Feb 16 '11 at 3:59
    
He says "algebra", not $\sigma$-algebra. –  Yuval Filmus Feb 16 '11 at 4:28
    
Ahhh. There you go then :) –  Mariano Suárez-Alvarez Feb 16 '11 at 4:32

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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.

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