## Examples 7.3.7(b): Measurable Sets

We need to prove thator because of the subadditivity of outer measurem^{*}() = mA^{*}(A) + mE^{*}(comp(A))E

Butm^{*}() mA^{*}(A) + mE^{*}(comp(A))E

*so that we know*

**A****E****E***m*. On the other hand,

^{*}(**A***) = 0***E***so that*

*comp(***A***))***E****A***m*. But that right away implies that

^{*}(*comp(***A***)) m***E**^{*}(A)which finishes the proof.m^{*}() mA^{*}(comp(A)) = mE^{*}(comp(A)) + mE^{*}(E)A