Last Friday we explained the change of variables formula for Riemann integrals by using Riemann-Stieltjes integrals. Today let’s push it a little further and prove a change of variables formula for Riemann-Stieltjes integrals.
We start with a function which we assume to be Riemann-Stieltjes integrable by the function . Now, instead of the full generality we used before, let’s just let be a strictly increasing continuous function with and . Define and to be the composite functions and . Then is Riemann-Stieltjes integrable by on , and we have the equality
For decreasing functions we get almost the exact same thing, so you should figure out the parallel statement and proof yourself.
Since is strictly increasing, it must be one-to-one, and it’s onto by assumption. In fact, is an explicit homeomorphism of the intervals and , and its inverse is also a strictly increasing continuous function. We can now use and its inverse to set up a bijection between partitions of and : if is a partition, then is a partition, and vice versa. Further, refinements of partitions of one side correspond to refinements of partitions on the other side.
So if we’re given an then there’s some partition of so that for any finer partition we have . Let be the corresponding partition of , and let be a partition of finer than it. Then it’s easily verified that the Riemann-Stieltjes sum is equal to the Riemann-Stieltjes sum . Everything else follows quickly from here.