Find all Lebesgue measurable sets $A ⊂ \mathbb{R}$ with the following property: All subsets $B ⊂ A$ are measurable.
2026-03-25 05:05:22.1774415122
Find all measurable sets whose subsets are measurable
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If $m(A)=0$, then every subset of $A$ is measurable, since the Lebesgue measure is complete.
If $A$ is not measurable, then as it is a subset of itself, it cannot have this property.
If $m(A)>0$, imitate the construction of the Vitali Set to produce a $B\subset A$ which is not measurable.
The conclusion is: the only subsets with that property are the measure-zero sets.