Find all Lebesgue measurable sets $A ⊂ \mathbb{R}$ with the following property: All subsets $B ⊂ A$ are measurable.

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    $\begingroup$ Only sets with measure zero, because Lebesgue measure is complete. $\endgroup$ Oct 27, 2014 at 14:03
  • $\begingroup$ @JonasGomes the completeness of Lebesgue measure gives you that this is true for sets of measure zero, but it doesn't prevent other sets from having the same property. $\endgroup$ Oct 27, 2014 at 14:13
  • $\begingroup$ Of course @Omnomnomnom. But if the measure were not complete, not even those sets would have this property. $\endgroup$ Oct 27, 2014 at 14:18

1 Answer 1


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.

  • $\begingroup$ If you need some more explanation in that third item, tell me. $\endgroup$ Oct 27, 2014 at 18:06

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