Page 277 - DMTH401_REAL ANALYSIS

Unit 22: Introduction to Riemann-Stieltjes Integration, using Riemann Sums

and we call this the Riemann-Stieltjes integral of f over [a, b] with respect to . Notes
If f and  are real-valued and we imagine the set of all numbers RS(f, , ) that can be formed
(using all possible appropriate selection vectors and all possible partitions whose mesh sizes
are less than ), the definition demands that they all lie in the open interval (RSI – , RSI + ).
When we had (x) = x this led to a Theorem.

Theorem: If f is Riemann integrable on [a, b] then f is bounded on [a, b].
This Theorem has to be modified in the Riemann-Stieltjes context! A simple example: suppose
that [a, b] is [0, 1] and that (x) = 0 if 0  x  c, where 0 < c < 1, and (x) = 1 if c < x  1. Then every
function f(x) that is continuous at c is Riemann-Stieltjes integrable on [0, 1] with respect to this .
In particular the function that is 1/x except at zero, where we define it to be zero, is Riemann-
Stieltjes integrable on [0, 1] with respect to this , but f is not bounded. The difference is that
when (x) was just x, we had x > 0 for every i. In our example,  = 0 unless I contains c and
i i i
some d with c < d. What we need is that on the set where the function  “really” varies, f must be
bounded. To make a definition, we will extend the definitions of f and  beyond the interval [a,
b] by setting them equal to their values at the endpoints. Thus we think of f(x) = f(a) if x < a and
f(x) = f(b) if x > b, with the same idea used to extend . We now define the oscillation of f on an
interval U by
(f, U) := sup |f(x) – f(y)|.
Î
x,y U
We allow the interval to be open or half-open now!
As before, we will let  =  (f) = (f, I ) when I is an interval (closed!) of a partition . But now we
i i i i
need to use oscillations of  as well.
Definition: If (x) is defined for x Î [a, b], we denote by  = (, [a, b]) the set of all c Î [a, b] such
that every open interval U that contains c contains x < c < x with |(x ) –(x )| > 0.
1 2 1 2

Notes Here c can be a or b because of our extension beyond [a, b]! For instance, if for all
 > 0 there exists x such that a < x < a +  and |(a) –(x )| > 0, then a Î (, [a, b]) because
2 2 2
for every x < a we have |(x ) –(x )| = |(a) –(x )| > 0.
1 1 2 2

Task Prove that (, [a, b]) is closed.

Theorem: If f is Riemann-Stieltjes integrable on [a, b] with respect to  then f is bounded on
(, [a, b]).
Proof: There exists a sequence {x } in := (, [a, b]) such that |f(x )| > n. Since f(x) is finite at
n n
every point x in , there are infinitely many distinct x , and so some subsequence (that we will
n
still denote {x }) converges to a point x* in . We now choose  = 1 in the definition of Riemann-
n
Stieltjes integrability, and obtain a corresponding  > 0. We can then construct a partition  with
o
mesh size less than  in such a way that x* is contained in the interior of some interval I of 
i o o
(unless x* is an endpoint of [a, b]; in that case, we can, by the Note, still use the following
argument, with I = I or I = I ). We know that every neighbourhood of x* contains infinitely
i o 1 i o n 
many of the x . Now we will refine  . We know that Int(I ) contains points ˆ x < x* < ˆ x with
n o i o 1 2
|( ˆ x )—( ˆ x )| > 0. We add these points to  , giving us a new partition , and mesh() < . We
1
2
o
ˆ
will now call [ ˆ x , ˆ x ], which is an interval of , I . Next we pick the components  of a selection
1
2
i
ˆ
ˆ
ˆ
vector  in an arbitrary way when I  I , and we let  be some x Î I . Then |RS(f, , ) – RSI| <
i N
ˆ
ˆ
ˆ
1. We next modify  by changing only  = x to ¢ := x , where x Î I , and we call the new
N M M
LOVELY PROFESSIONAL UNIVERSITY 271