Extending a Measure to an Outer Measure
That is, can be covered by a countable collection of sets in . For every such cover, sum up the -measures of all the sets in the cover, and define to be the greatest lower bound of such sums. Then is an outer measure, which extends to all of . Further, if is -finite, then will be too. We call the outer measure “induced by” .
First off, if itself, then we can cover it with itself and an infinite sequence of empty sets. That is, . Thus we must have . On the other hand, if is contained in the union of a sequence , then monotonicity tells us that , and thus . That is, must be equal to for sets ; as a set function, indeed extends . In particular, we find that .
If and are sets in with and is a sequence covering , then it must cover as well. Thus can be at most , and may be even smaller. This establishes that is monotonic.
We must show that is countably subadditive. Let and be sets so that is contained in the union of the . Let be an arbitrarily small positive number, and for each choose some sequence that covers such that
This is possible because the definition of tells us that we can find a covering sequence whose measure-sum exceeds by an arbitrarily small amount. Then the collection of all the constitute a countable collection of sets in which together cover . Thus we conclude that
Since was arbitrary, we conclude that
and so is countably subadditive.
Finally, if , we can pick a cover . If is -finite, we can cover each of these sets by a sequence so that . The collection of all the is then a countable cover of by sets of finite measure; the extension is thus -finite as well.