Example 7.3.3(a): Outer Measure of Intervals
Take 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.