7.3. Measures

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 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 = { an }. Using subadditivity we know that

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