Functions of Bounded Variation I
In our coverage of the Riemann-Stieltjes integral, we have to talk about Riemann-Stieltjes sums, which are of the form
How can we choose a function to make sums like this get really big? Well, we could make the function get big, but that’s sort of a cheap trick. Let’s put a cap that . Now how can we make a Riemann-Stieltjes sum big?
Well, sometimes and sometimes . In the first case, we can try to make , and in the second we can try to make . In either case, we’re adding as much as we possibly can, subject to our restriction.
We’re saying a lot of words here, but what it boils down to is this: given a partition we can get up to in our sum. As we choose finer and finer partitions, it’s not hard to see that this number can only go up. So, if there’s an upper bound for our , then we’re never going to get a Riemann-Stieltjes sum bigger than . That will certainly make it easier to get nets of sums to converge.
So let’s make this definition: a function is said to be of “bounded variation” on the interval if there is some so that for any partition we have
That is, is an upper bound for the set of variations as we look at different partitions of . Then, by the Dedekind completeness of the real numbers, we will have a least upper bound , which we call the “total variation” of on
Let’s make sure that we’ve got some interesting examples of these things. If is monotonic increasing — if implies that — then we can just drop the absolute values here. The sum collapses, and we’re just left with for every partition. Similarly, if is monotonic decreasing then we always get . Either way, is clearly of bounded variation.
An easy consequence of this condition is that functions of bounded variation are bounded! In fact, if has variation on , then for all . Indeed, we can just use the partition and find that
from which the bound on easily follows.