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Is it possible to partition $[0, 1] = A \sqcup B$ such that given any $0 \leq x < y \leq 1$, both $\mu([x,y] \cap A) > 0$ and $\mu([x,y] \cap B) > 0$?

bof
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Zach
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    Hi Zach, what have you tried so far? – Jair Taylor Nov 01 '20 at 06:56
  • I've tried a few different things, but nothing seems to work. I don't know if I have the necessary knowledge to figure this out yet, as I have just started learning measure theory on my own. – Zach Nov 01 '20 at 07:07
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    See the answers to: https://math.stackexchange.com/questions/57317/construction-of-a-borel-set-with-positive-but-not-full-measure-in-each-interval?lq=1 – quasi Nov 01 '20 at 07:10

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