## Indefinite Integrals and Convergence I

Let’s see how the notion of an indefinite integral plays with sequences of simple functions in the norm.

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.