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 nonmeasurable subset A such that m*(A)=m*(E)?
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.
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 nonmeasurable subset $S$ of a null set $A$ inside $E$, remove $S$, and since $m(E)=m(ES)+m(S)$ and $m(S)$ isn't anything, we must have that $m(ES)$ isn't anything, while we still have $m^{\ast}(E)=m^{\ast}(ES)+m^{\ast}(S)=m^{\ast}(ES)$.

$\begingroup$ 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 Lebesguemeasurable with measure zero. $\endgroup$– HaimJan 25 '10 at 14:31

$\begingroup$ 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. $\endgroup$ Jan 25 '10 at 14:31

$\begingroup$ 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 :) $\endgroup$ Jan 25 '10 at 14:41