MathCS.org - Real Analysis: Example 7.3.5(a): Properties of Outer Measure

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.

Context

To find the measure of the set A = {a} is easy: since (a-1/n, a+1/n) covers A we know that

m^*(A) m^*(a-1/n, a+1/n) = 2/n

for all integers n. But then m^*(A) = 0.

Now let A = { a_n }. Using subadditivity we know that

m^*(A) m^*(a_n)

But each outer measure on the right is zero by the first part, so that m^*(A) = 0.

Next | Previous | Glossary | Map | Discussion