Increasing Integrators and Order
For what we’re about to do, I’m going to need a couple results about increasing integrators, and how Riemann-Stieltjes integrals with respect to them play nicely with order properties of the real numbers.
When we consider an increasing integrator we have a certain positivity result: if the integrand is nonnegative and the integral exists, then it is nonnegative as well. That is, for increasing and on we have as long as it exists. This should be clear, since every Riemann-Stieltjes sum takes the form
where the inequality follows because each value and each difference is nonnegative. Thus the limit of the sums must be nonnegative as well. From this and the linearity of the integral we see that if is increasing and on , then we have the inequality
as long as both integrals exist.
Now, when we talked about absolute values — the metric for the real numbers — we saw that the absolute value of a sum was always less than the sum of the absolute values. That is, . And since an integral is just a limit of sums, it stands to reason that a similar result would hold here. Specifically, if is increasing and is integrable with respect to on , then so is the function , and further we have the inequality
Indeed, given a partition of the difference between the supremum and infimum of on the th subinterval is the supremum of , where and range across . Then, adapting the above inequality we see that
and so we conclude that
Then we can multiply by and sum over a partition to find
Riemann’s condition then tells us that is integrable, and the inequality follows by the previous result.
We might hope to extend these results to integrators of bounded variation, but it won’t work right. This is because we go from increasing functions to functions of bounded variation by subtracting, and this operation will break the order properties.