Prove that $\mathbb{R}$ can be partitioned into continuum disjoint sets with positive outer measure.
2026-03-25 17:39:29.1774460369
Prove that $\mathbb{R}=\sqcup_{i\in \mathfrak{c}}A_i, \mu^*(A_i)>0$
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Partition $\mathbb R$ into continuum many disjoint sets (so-called Bernstein sets), each of which has nonempty intersection with every uncountable closed set. (A set which meets every uncountable closed set while containing no uncountable closed set is called a Bernstein set.) This is easily done by transfinite induction, since there are just continuum many uncountable closed sets, and every uncountable closed set contains continuum many points.