outer measure zero sets are measurable

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Proposition

If $m^{*} (E) = 0$ then $E$ is measurable.

Proof

Let $A \subset R$ . Then $A \cap E \subset E$ means

m^{*} (A \cap E) \leq m^{*} (E) = 0 ⟹ m^{*} (A \cap E) = 0

But then we see

m^{*} (A \cap E) + m^{*} (A \cap E^{c}) = m^{*} (A \cap E^{c}) \leq m^{*} (A)

because $A \cap E^{c} \subset A$ .

outer measure zero sets are measurable

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