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][1].


  [1]: http://en.wikipedia.org/wiki/Bernstein_set