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Am I correct in assuming that for a sigma algebra with only finitely many sets, the intersection of all its sets will be the empty set (because the sigma algebra contains complements); but if the sigma algebra contains uncountably many sets, this doesn't necessarily hold?

I'm trying to develop a better understanding, I would greatly appreciate your explanations

Edit: My confusion has arisen from the following question, in which a countable intersection is taken on the sets in a sigma algebra, but answers suggest that the result need not be the empty set: Proofs regarding measure of intersection of sets

Community
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user273860
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  • "answers suggest that the result need not be the empty set" No. The result needs not be the empty set because this is a countable intersection of specific subsets, not the intersection of all the subsets in the sigma-algebra. – Did Sep 25 '15 at 13:04
  • Thanks! I was confused about this. what would it look like (in notation) to do an intersection of all the subsets in a (say, uncountable) sigma algebra? – user273860 Sep 25 '15 at 13:12
  • $$\bigcap_{A\in\mathcal F}A$$ – Did Sep 25 '15 at 13:15
  • Thanks. Is that what you call an "arbitrary" intersection? – user273860 Sep 25 '15 at 13:16
  • You are welcome. I call this an intersection. – Did Sep 25 '15 at 13:17

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Regardless whether the sigma algebra is finite or infinite, the intersection between all of its elements will certainly be $\varnothing$; simply because $\varnothing$ is an element of it.

In response to your edit: as long as the s.a. is infinite (countably or uncountably), the intersection of a particular infinite collection of elements of it doesn't need to be empty, because the collection needs neither contain the empty set nor a set and its complement.

  • Will this intersection have to be an "uncountable" intersection? Or is it possible to do a countable intersection of all the elements in an uncountable sigma algebra? – user273860 Sep 25 '15 at 12:52
  • The reason I'm asking is that I have been very confused about this since posing this question: http://math.stackexchange.com/questions/1450220/proofs-regarding-measure-of-intersection-of-sets – user273860 Sep 25 '15 at 12:55
  • If the s.a. is uncountable, then we are finding the intersection of uncountably many sets; if this is what you mean by saying "uncountable" intersection, then yes. If you are having an additional confusion, please edit your post to include and make precise what you are looking after. –  Sep 25 '15 at 12:59
  • Thanks, your edit cleared things up for me! I see now that an intersection of infinitely many sets need not contain all the sets of the sigma algebra. If I wanted to make an intersection on ALL the sets of the sigma algebra, what would that look like in notation? How would I write it up? And is there a name for this sort of intersection? – user273860 Sep 25 '15 at 13:09
  • Also: is there such a thing as a sigma algebra with countably many sets? What I've been reading suggests that there isn't (that they're either finite or uncountably infinite) – user273860 Sep 25 '15 at 13:14
  • @user273860 If you know the answer why are you asking? For a proof, search on the site. – Did Sep 25 '15 at 13:18
  • "If you know the answer why are you asking?" I DON'T know the answer. I might think I know the answer, but when somone much more informed than me refers to something that I thought was impossible, I tend to doubt what I think I know! – user273860 Sep 25 '15 at 13:26
  • @user273860 you're welcome. No, there isn't such a thing as a countable (infinite) sigma algebra on a set. For a proof, see: http://math.stackexchange.com/questions/320035/if-s-is-an-infinite-sigma-algebra-on-x-then-s-is-not-countable –  Sep 25 '15 at 13:35
  • Thanks Ahmed! You've been of great help – user273860 Sep 25 '15 at 13:36
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The intersection of all sets in a sigma algebra is always the empty set, whether its finite, countable, or uncountable, because the empty set is always an element of the sigma algebra.

Paul
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  • Will this intersection have to be an "uncountable" intersection? Or is it possible to do a countable intersection of all the elements in an uncountable sigma algebra? Also, I thought a sigma algebra could only be either finite or uncountable? – user273860 Sep 25 '15 at 12:49
  • @user273860 If you know the answer why are you asking? For a proof, search on the site. – Did Sep 25 '15 at 13:18
  • I don't presume to know the answer of anything. – user273860 Sep 25 '15 at 13:29
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No, if $\Omega$ is a set, $\mathcal F$ an algebra for $\Omega$ and $A \in \mathcal F$ is arbitrary, then $$ \bigcap_{B \in \mathcal F} B \subseteq A \cap (\Omega \setminus A) = \emptyset. $$

Cloudscape
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