The question is from Stein, "Analysis 2", Chapter 1, Problem 5:
Suppose E is measurable with m(E) < ∞, and E = E1 ∪ E2, E1 ∩ E2 = ∅.
a) If m(E) = m∗(E1) + m∗(E2), then E1 and E2 are measurable.
b) In particular, if E ⊂ 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 ε > 0 there exists an open set O with E ⊂ O and m∗(O − E) ≤ ε, so I'm looking for a set of implications that lead me back to this definition.
The Attempt at a Solution
The problem seems suspiciously similar to the definition of a measurable set as one that satisfies the 'caratheodory criterion'. My attempt at a solution has been to try to show that what we are given in the problem must imply that the caratheodory criterion holds and from there show that if the caratheodory criterion holds then the set is measurable in the above sense. I'm having trouble knowing where to start filling in the details.
I also wonder though, if there is a simpler and neater way to solve the problem?
Thanks in advance for any help you can give me - it's very much appreciated. This one is doing my head in!