## Example 7.3.5(a): Properties of Outer Measure

Show that the outer measure of a single point is 0, and the outer measure
of a countable set is also 0.

To find the measure of the set *is easy: since*

*= {a}***A***(a-1/n, a+1/n)*covers

*we know that*

**A**for all integersm^{*}() mA^{*}(a-1/n, a+1/n) = 2/n

*n*. But then

*m*.

^{*}(*) = 0***A**
Now let *A = { a_{n} }*.
Using subadditivity we know that

But each outer measure on the right is zero by the first part, so thatm^{*}() mA^{*}(a_{n})

*m*.

^{*}(*) = 0***A**