Can we find uncountably many disjoint measurable subsets of $\mathbb{R}$ with strictly positive Lebesgue measure?
2026-04-06 12:41:14.1775479274
Can we find uncountably many disjoint measurable subsets of $\mathbb{R}$ with strictly postive Lebesgue measure?
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If it were possible, for some integer $n$ we would be able to find uncountably many of your sets that meet the interval $[n,n+1]$ in a set of positive measure. But then for some positive integer $m$, uncountably many of them would meet it in a set of measure strictly larger than $1/m$. This is impossible, since the union of any $m+1$ of them would be a measurable subset of $[n,n+1]$ of measure strictly larger than $1$.