## 7.3. Measures

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

Find the outer measure of the set

First, let's find the outer measure of the set *of all rational numbers in***A***[0, 1]*. Also show that for any*finite*collection of intervals covering*we have that the sum of their lengths is greater or equal to 1.***A***. The rational numbers in*

**A***[0, 1]*are countable so we can write the set

*. For each*

*= { r***A**_{1}, r_{2}, r_{3}, ... }*r*define the set

_{n}Then the collectionR_{n}= (r_{n}- 2^{-n}/, r_{n}+ 2^{-n}/)

*{*is a countable cover of

**R**_{n}}*with open intervals, so it is part of the infimum for computing*

**A***m*. But

^{*}(*)***A**Thereforel(R_{n}) = 2^{-n}=

*m*for every positive . But that means that

^{*}(*)***A***m*.

^{*}(*) = 0***A**
Well, alright, the above proof is off by a factor *2* or so, but
it does not matter if
*m ^{*}(A) *
or

*m*, so the prove is valid (fix the constants, though).

^{*}(*) 2***A**As for the second part, it is left as an exercise. Compare with some of the previous examples as a hint.