## 7.3. Measures

### Examples 7.3.10(b): Properties of Measure

What is the measure of the set

We have already shown that a countable set
has outer measure zero, and that sets with outer
measure zero are measurable. Therefore *of all rational numbers and the set***Q***of all irrational numbers inside***I***[0, 1]*.*is measurable with*

**Q***m(*.

*) = 0***Q**
The set *[0, 1]* is an interval, hence it is measurable with
*m([0, 1]) = 1 - 0 = 1*. Also, the set
*I = [0, 1] - Q*, so that

*is also measurable. Since*

**I***and*

**Q***are disjoint, we can use additivity of measure:*

**I**Therefore1 = m( [0, 1] ) = m(Q) = = m(I) + m(Q) = 0 + m(I)I

*m(*.

*) = 1***I**