Interactive Real Analysis
- part of Example 7.3.3(a): Outer Measure of Intervals
Find the outer measure of the empty set O, and prove that
m*(A)
m*(B)
for all
A 
B.
Context
Since outer measure should be related to 'length' we expect that the outer
measure of the empty set is zero. Indeed, that is the case.
m*(B)
for all
A B.
ContextTake the open interval (-1/n, 1/n), whose length is 2/n. It covers the empty set, because the empty set is a subset of every set. Therefore m*(O) < 2/n for all n which implies that m*(O) = 0.
Now assume that A B.
Then every cover of B is also a cover of A, but not every
cover of A covers B. That means that there are more collections
to consider when computing m*(A) instead of
m*(B), so that the infimum in the first case is
smaller than in the second case.