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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.


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up vote 5 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
Yeah, that sounds fine. – Asaf Karagila Sep 11 '13 at 20:58
Thank you very much! – Cancan Sep 11 '13 at 21:14
Why does the limit should be in the algebra? – user162343 Nov 19 '15 at 2:45
@user162343: Yes, that is the reason. – Asaf Karagila Nov 19 '15 at 13:42

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