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It is well known that for any set A in R^d there exists a measurable set E such that E contains A and m*(A)=m*(E). Is it possible to go the other direction? In other words, is it true that for any measurable set E (such that m(E)>0) there is a non-measurable subset A such that m*(A)=m*(E)?

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2 Answers 2

up vote 13 down vote accepted

A set $E$ with positive Lebesgue measure can be decomposed as a union $E = A \cup B$ where each of $A$ and $B$ have zero inner measure, and therefore each of $A$ and $B$ are nonmeasurable with $m^\*(A) = m^\*(B) = m(E)$.

An example for this construction is a Bernstein set.

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Thanks. This is what I was looking for. –  Haim Jan 25 '10 at 18:02

Yes, I believe so - since subsets of a null set $A$ (i.e., $m(A)=0$) are not necessarily measurable, but will obviously still have outer measure 0, given any measurable set $E$ you "should" (i.e., I think so, but not sure) be able to find a non-measurable subset $S$ of a null set $A$ inside $E$, remove $S$, and since $m(E)=m(E-S)+m(S)$ and $m(S)$ isn't anything, we must have that $m(E-S)$ isn't anything, while we still have $m^{\ast}(E)=m^{\ast}(E-S)+m^{\ast}(S)=m^{\ast}(E-S)$.

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Perhaps I'm missing something here, but isn't the Lebesgue measure complete? i.e. for every measurable set A with m(A)=0, each subset of A is Lebesgue-measurable with measure zero. –  Haim Jan 25 '10 at 14:31
    
Subsets of null sets are always Lebesgue measurable, but not necessarily Borel measurable. Perhaps you should clarify what you mean by "measurable" in your answer. –  François G. Dorais Jan 25 '10 at 14:31
    
Ah, right - I meant Lebesgue measure in my answer, but got mixed up. Clarifying which measure you mean is a good idea, as it will help avoid confusions such as mine :) –  Zev Chonoles Jan 25 '10 at 14:41

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