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When operating with sigma algebras, what does it mean when we talk about a countably infinite set? And, what are then closed countably infinite interesections?

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A set which is countably infinite. I don't follow. As for the second question, you should quote whatever it is you're having trouble following in full. –  Qiaochu Yuan Nov 13 '10 at 12:17
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These definitions would be near the front of any textbook on measure theory. –  Carl Mummert Nov 13 '10 at 13:13

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I expect that you have read that a sigma algebra $\mathcal{A}$ has to be "closed under countably infinite intersections". What this means is that if you have a family $(A_n)$ of elements of $\mathcal{A}$ indexed by the natural numbers $\mathbb{N}$, then the intersection $\bigcap_{n\in\mathbb{N}}A_n$ of all the $A_n$ is also an element of $\mathcal{A}$.

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