## 7.3. Measures

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

Recall the definition of the Cantor set: letand define, for eachA_{0}= [0, 1]

*n*, the sets

*recursively as*

**A**_{n}Then the Cantor set is given as:A_{n}=A_{n-1}\

We have already shown that the Cantor set has "length" zero, but the concept of length does not really apply since the Cantor set is not an interval. Now we can redo our previous calculations, but under the mathematically correct name of measure.=CA_{n}

First, the Cantor set is expressed as a combination of unions and
intersections of intervals so that * C* is measurable.
Let's compute the measure of the sets

*:*

**A**_{n}

so thatA_{0}= [0, 1]m(A_{0}) = 1

so thatA_{1}=A_{0}- (1/3, 2/3)

m(A_{1}) = m(A_{0}) - m(1/3, 2/3) = 1 - 1/3

so thatA_{2}=A_{1}- (1/9, 2/9) - (7/9, 8/9)

m(A_{2}) = m(A_{1}) - m(1/9, 2/9) - m(7/9, 8/9) = 1 - 1/3 - 2*1/9

In general we haveso thatA_{3}=A_{2}- (1/27, 2/27) - (7/27, 8/27) - (19/27, 20/25) - (25/27, 26/27)

m(A_{3}) = 1 - 1/3 - 2*1/9 - 4*1/27

for all

*n*. Since

*for all*

**C****A**_{n}*n*we have that

for allm() (2/3)C^{n}

*n*. Therefore

*m(*.

*) = 0***C**