At last we can show that the set function we defined on semiclosed intervals is a measure. It’s clearly real-valued and non-negative. We already showed that it’s monotonic, and this will come in handy as we show that it’s countably additive.
So, if is a countable disjoint sequence of semiclosed intervals whose union is also a semiclosed interval , then our first monotonicity property shows that for any finite we have
and so in the limit we must still have
But the sequence covers , and so our other monotonicity property shows that
which gives us the equality we want.
But this still isn’t quite a measure. Why not? It’s only defined on the collection of semiclosed intervals, and not on the ring of finite disjoint unions. But we’re in luck: there is a unique finite measure on extending on . That is, if , then .
Every set in is a finite disjoint union of semiclosed intervals, but not necessarily uniquely. Let’s say we have both
Then for each we have
which represents as a finite disjoint union of other sets in . Since is finitely additive, we must have
But since these sums are finite we can switch their order with no trouble. Thus we can unambiguously define
which doesn’t depend on how we represent as a finite disjoin union of semiclosed intervals.
This function clearly extends , since if we can just use itself as our finite disjoint union. It’s also easily seen to be finitely additive, and that there’s not really any other way to define a finitely additive set function to extend . But we still need to show countable additivity.
So, let be a disjoint sequence of sets in whose union is also in . Then for each we have
If happens to be in , then the collection of all the is countable and disjoint, and we can use the countable additivity of we proved above to show
In general, though, is a finite disjoint union
and we can apply the previous result to each of the :
From here on out, we’ll just write instead of for this measure on .