## 7.3. Measures

### Examples 7.3.7(a): Measurable Sets

Show that the empty set, the set

We have shown that the outer measure of the empty set *, and the complement of a measurable set are all measurable.***R***is zero, and sets with outer measure zero are automatically measurable.*

**0**
For the set * R* of all real numbers we have:

which shows thatm^{*}(A) + mR^{*}(comp(A)) = mR^{*}() + mA^{*}(A) = m0^{*}()A

*is measurable.*

**R**
If a set * E* is measurable we have:

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

*comp(*we then have:

*)***E**which shows thatm^{*}(comp(A)) + mE^{*}(comp(comp(A))) =E

= m^{*}(comp(A)) + mE^{*}(A) =E

= m^{*}()A

*comp(*is measurable.

*)***E**