7.3. Measures

Examples 7.3.7(b): Measurable Sets

Show that every set with outer measure 0 is Lebesgue measurable.
We need to prove that
m*(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.
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