Examples 7.3.7(b): Measurable Sets
We need to prove thatm*(A) = m*(A E) + m*(A comp(E))or because of the subadditivity of outer measure
m*(A) m*(A E) + m*(A comp(E))But A E E so that we know m*(A E) = 0. On the other hand, A comp(E)) A so that m*(A comp(E)) m*(A). But that right away implies that
m*(A) m*(A comp(E)) = m*(A comp(E)) + m*(A E)which finishes the proof.