The problem statement, all variables and given/known data
The question is from Stein and Shakarchi, Real Analysis 2, Chapter 1, Problem 5:
Suppose $E$ is measurable with $m(E) < \infty$, and $E=E_1\cup E_2$, $E_1\cap E_2=\emptyset$.
Prove:
a) If $m(E) = m^{*}(E_1) + m^{*}(E_2)$, then $E_1$ and $E_2$ are measurable.
b) In particular, if $E \subset Q$, where $Q$ is a finite cube, then $E$ is measurable if and only if $m(Q) = m^{*}(E) + m^{*}(Q − E)$.
The definition of a 'measurable set' given in the book is that for any $\epsilon > 0$ there exists an open set $O$ with $E \subset O$ and $m^{*}(O − E) \leq \epsilon$, so I'm looking for a set of implications that lead me back to this definition.
all i could prove is that if $E$ measurable from my definition up, iff $ m(A) = m( A \cap E) + m(A \cap E^{c}) $
Thanks in advance for any help you can give me - it's very much appreciated.