Yesterday, we showed that — given an outer measure on a hereditary -ring — the collection of -measurable sets forms a ring. In fact, it forms a -ring. That is, given a countably infinite sequence of -measurable sets, their union is also -measurable. Even better if the are pairwise disjoint, then
To see this, we start with a disjoint sequence and an equation we came up with yesterday:
We can keep going like this, adding in more and more of the :
for every natural number . This finite union is -measurable. This, along the fact that , tells us that
This is true for every , so we may pass to the limit and use the countable subadditivity of
But this is enough to show that is -measurable, and so is closed under countable disjoint unions. And this shows that is closed under countable unions in general, by our trick of replacing a sequence by a disjoint sequence with the same partial unions.
Since the previous inequalities must then actually be equalities, we see that we must have
It’s tempting to simply subtract from both sides, but this might be an infinite quantity. Instead, we’ll simply replace with , which has the same effect of giving us
as we claimed.
If we replace by in this equation, we find that — when restricted to the -ring — is actually countably additive. That is, if we define for , then is actually a measure. Even better, it’s a complete measure.
Indeed, if and , then as well. We must show that is actually -measurable, and so exists and equals zero. But we can easily see that for any
and this is enough to show that is -measurable, and thus that is complete.