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?
What's a closed countably infinite intersection?
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measure-theory
elementary-set-theory
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1A 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. – 2010-11-13
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1These definitions would be near the front of any textbook on measure theory. – 2010-11-13
1 Answers
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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 $\mathcal{A}$.