Timeline for Can all uncountable (but small) families of sets with positive measure have an uncountable subfamily with an intersection of positive measure?

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Feb 28, 2021 at 13:28	history	edited	mtg	CC BY-SA 4.0	modified typos
Feb 26, 2021 at 19:03	comment	added	Dave L Renfro		is to let $E_n$ be those numbers having a nonzero $n$'th decimal digit—each $E_n$ has measure $0.9,$ and the intersection of any $k$-many of these sets has measure $(0.9)^k,$ and thus the measure of the intersection of any infinite subcollection of these sets is zero.)
Feb 26, 2021 at 19:03	comment	added	Dave L Renfro		Of possible related interest is this 13 May 2005 sci.math post which gives of survey of ZFC results inspired by the following: Let $E_1,$ $E_2,$ $\ldots$ be Lebesgue measurable subsets of $[0,1],$ and let $C>0.$ Does there exist a subsequence ${E_{n_k}}$ such that $\bigcap_{k=1}^{\infty}E_{n_k}$ is nonempty? (Yes, by Fatou's lemma for appropriate characteristic functions.) Does there exist a subsequence ${E_{n_k}}$ such that $\bigcap_{k=1}^{\infty}E_{n_k}$ has positive measure? (No, and a common counterexample (continued)
Feb 26, 2021 at 17:34	answer	added	Will Brian		timeline score: 5
Feb 26, 2021 at 17:12	comment	added	mtg		Thanks! Actually, the Łuzin-Sierpiński construction answers the question w.r.t. the families of sets of positive outer measure. I added the restriction to measurable sets.
Feb 26, 2021 at 17:11	history	edited	mtg	CC BY-SA 4.0	reformulated the question in light of the comment
Feb 26, 2021 at 17:00	history	edited	mtg	CC BY-SA 4.0	added a sub-question
Feb 26, 2021 at 16:44	comment	added	Wojowu		This answer mentions a continuum-sized family of pairwise disjoint sets such that each has full outer measure. I haven't read your question fully so I'm not sure if this family (or some uncountable subfamily of it) leads to a complete answer.
Feb 26, 2021 at 16:31	history	asked	mtg	CC BY-SA 4.0