## 7.3. Measures

### Example 7.3.3(c): Outer Measure of Intervals

As before, we expect that*m*.

^{*}(a, b) = b - a
Set *J = (a, b)* and define *J _{n} = [a+1/n, b-1/n]*
for

*n*a positive integer. Since

*J*is closed, we know by the previous example that

_{n}*m*. Also

^{*}(J_{n}) = b - a - 2/nso that by an earlier exampleJ_{n}J closure(J)

Butm^{*}(J_{n}) m^{*}(J) m^{*}(closure(J))

Henceb - a - 2/n = m^{*}(J_{n}) m^{*}(J) m^{*}(closure(J)) = b - a

*b - a - 2/n m*for all

^{*}(J) b - a*n*, which implies that

*m*, as we suspected.

^{*}(J) = b - a