There are some remaining topics to clean up in the theory of the Riemann-Stieltjes integral. First up is a question that seems natural from the perspective of iterated integrals: what can be said about the continuity of the inner integrals?
More explicitly, let be a continuous function on the rectangle , and let be of bounded variation on . Define the function on by
Then is continuous on . Similar statements can be made about other “partial integrals”, where and are each vector variables and we use in place of the Stieltjes integrator .
Specifically, this is a statement about interchanging limit operations. The Riemann-Stieltjes integral involves taking the limit over the collection of tagged partitions of , while to ask if is continuous asks whether
As we knew back when we originally discussed integrators of bounded variation, we can write our integrator as the difference of two increasing functions. It’s no loss of generality, then, to assume that is increasing. We also remember that the Heine-Cantor theorem tells us that since is compact, is actually uniformly continuous.
Uniform continuity tells us that for every there is a (depending only on so that for every pair of points and , with we have .
So now let’s take two points and with and consider the difference
where we’ve used the integral mean value theorem. Clearly by choosing the right we can find a to make the right hand side as small as we want, proving the continuity of .