I'm wondering if this pigeonhole-like statement holds. Let $(S_i)_{i \in N}$ be a countable family of subsets of $[0,1]$ such that each $S_i$ is (either Jordan or Lebesgue) measurable and there is some $\delta$ s.t. each $S_i$ has measure at least $\delta$. Is it true that there is some infinite subset $I \subset N$ s.t $\cap_{i \in I}S_i$ is nontrivial? Would this depend on the measure used/in case this doesn't hold, could additional assumptions on the sets be added to make it true?
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2Various such results can be found in this 13 May 2005 sci.math post. – Dave L. Renfro May 27 '21 at 21:55
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brilliant! thank you – ham_ham01 May 27 '21 at 21:59
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2Maybe someone could post relevant parts of that sci.math post here, as an answer, so we'd still have it here when/as/if sci.math goes away? – Gerry Myerson May 28 '21 at 01:23
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2@Gerry Myerson: I'll try to do this in the next day or two, possibly also including some more references, as this is a topic I've been keeping references to as I encounter them (typically by random chance). Somewhat related is Can all uncountable (but small) families of sets with positive measure have an uncountable subfamily with an intersection of positive measure?, where about 3 months ago I mentioned one result mentioned that 2005 sci.math post. (However, it seems that I forgot to say what $C > 0$ is used for!) – Dave L. Renfro May 28 '21 at 07:38
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2@Gerry Myerson: This will take a few more days, as it seems like something worth doing well, like my answers example 1 and example 2 and Example 3 and Example 4 and Example 5 and Example 6, among others, although I don't anticipate it being as lengthy as some of these (maybe like #5, with 25-35 references appended). – Dave L. Renfro May 29 '21 at 18:52
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No worries, @Dave, take your time. – Gerry Myerson May 30 '21 at 03:25