Indefinite Integrals and Convergence I
If is a mean Cauchy sequence of integrable simple functions, and if each has indefinite integral , then the limit exists for all measurable sets . Indeed, for each we have a sequence of real numbers . We compare
and find that since the sequence of simple functions is mean Cauchy the sequence of real numbers is Cauchy. And thus it must converge to a limiting value, which we define to be . In fact, the convergence is uniform, since the last step of our inequality had nothing to do with the particular set !
Now, this set function is finite-valued as the uniform limit of a sequence of finite-valued functions. Since limits commute with finite sums, and since each is finitely additive, we see that is finitely additive as well; it turns out that it’s actually countable additive.
If is a disjoint sequence of measurable sets whose (countable) union is , then for every pair of positive integers and the triangle inequality tells us that
Choosing a large enough we can make the first and third terms arbitrarily small, and then we can choose a large enough to make the second term arbitrarily small. And thus we establish that
We can say something about the sequence of set functions : each of them is — as an indefinite integral — absolutely continuous, but in fact the sequence is uniformly absolutely continuous. That is, for every there is a independent of so that for every measurable set with .
Let be a sufficiently large integer so that for we have
which exists by the fact that is mean Cauchy. Then we can pick a so that
for all and with . We know that such a exists for each by absolute continuity, and so we just pick the smallest of them for .
This will then work for all , but what if ? Well, then we can write
and so the same works for all as well.