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I guess this is equivalent to say algebra that is monotone(increasing) class is a sigma-algebra.

However, can anyone tell me how to prove it? Till now I can't think of any construction or partition of sets to prove the the closure of countable unions.

Thanks!

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up vote 4 down vote accepted

HINT: Given $\{A_n\mid n\in\Bbb N\}$ in your algebra, consider $B_n=\bigcup_{k<n}A_k$.

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Should I reason it in this way? $B_n$ is in the algebra if $n$ is finite. Because of the monotone(increasing) property, the countable unions of $B_n$ is also in the algebra. so limit of $B_n$ as $n$ goes to infinity will also be in the algebra, which implies it's a sigma-algebra as well. –  Cancan Sep 11 '13 at 20:57
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Yeah, that sounds fine. –  Asaf Karagila Sep 11 '13 at 20:58
    
Thank you very much! –  Cancan Sep 11 '13 at 21:14
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