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Unit 22: Introduction to Riemann-Stieltjes Integration, using Riemann Sums
Then, if n and n¢ are so large that (b – a)/n < and (b – a)/n¢ < , we have Notes
mesh( ) < and mesh( ) < | – | < /2.
n n¢ n n¢
This means (since was arbitrary) that { } is a Cauchy sequence in our space. Thus we define
n
RSI := lim
n®¥ n
and it remains to show that if |[a, b] then
mesh() < |RS() – RSI| <.
This is essentially done. We choose the first n such that mesh( ) < , and we suppose that
n
mesh() < . Then
|RS() – RI| |RS() – | + | – RSI| < /2 + /2 = ,
n n
since RS()— = RS() – RS(f, , , ). The proof is complete.
n n n
Notes Nothing is said at first about the functions f and , beside the demand that the
integrability definition hold.
If f and have a discontinuity at the same point, then the Riemann-Stieltjes integral does not
exist.
They also show that if the Riemann-Stieltjes integral exists, then this integration-by-parts formula
holds:
b b
a ò f d = ò a df + f(b) (b) - f(a) (a)
-
(in the applications have far, far back in my mind, the integral on the right would have to be
b
a ò df , in order to keep the order of “multiplication” the same). The proof amounts to rearranging
the Riemann-Stieltjes sums, adding and subtracting terms in such a way that the become
i
partition points and the x become selection-vector components when 1 < i < n . There are some
i
b
leftovers, and these turn out to be the “boundary” term f(x)(x)| .
a
Wheeden and Zygmund state several properties, routine to prove, about Riemann-Stieltjes
integrals:
b
a ò f d is linear in both f and
b c
as long as all the integrals involved exist, and if ò f d exists and a < c <b, then both of ò f d
a a
b c b b
and c ò f d exist, and ò a f d + ò c f d = ò a f d .
What has been covered applies to all Riemann-Stieltjes integrals. That continuity plays a role
has already been mentioned.
22.5 Functions of Bounded Variation: Definition and Properties
In what we do from now on, at least one of f and will be a function of bounded variation, unless
otherwise stated. We will begin by discussing real-valued functions of bounded variation. This
material can also be found in Measure and Integral, by Wheeden and Zygmund.
Definition: A function f: [a, b] ® is a function of bounded variation on [a, b] if
n
V(f, [a, b]) := sup å|f(x ) – f(x i – 1 )| < ¥ and we say that f Î BV[a, b].
i
|[a,b] 1
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