Page 276 - DMTH401_REAL ANALYSIS

Real Analysis

Notes 22.1 Riemann-Stieltjes Sums

A Riemann-Stieltjes sum for a function f(x) defined on an interval [a, b] is formed with the help
of

1. A partition  of [a, b], namely an ordered, finite set of points x, with a = x < x < < x = b
i 0 1 n
(where n is a positive integer that can be any positive integer, and one that we will often
write as n = n ),

2. A selection vector  = ( , ...,  ) that has n components that must satisfy x    x , for
1 n  i – 1 i i
i = 1, 2, ..., n.
and
3. An integrator (x), which is a function defined on [a, b] that plays the role of the x in dx ...
A Riemann-Stieltjes sum for f over [a, b] with respect to the partition , using the selection
vector , and integrator , may be denoted (in greatest detail!) as follows, and it is given
by the value of the sum following it:

n 
4. RS (f, , [a, b], , ):= å f( )( (x ) -  (x i 1 )) .


i
i
-
=
i 1
22.2 More Notation: The Mesh (Size) of a Partition
In this definition, as in the Riemann-sums definition, we can write x := x —x or  := (x ) –
i i i – 1 i i
(x ). These are convenient because they are short and suggest the dx or d in an integral. But
i – 1
they can cause confusion because they leave out the dependence they have on x . The x is used
i – 1 i
in the Riemann-Stieltjes context.
A partition  can be thought of as “dividing” the interval [a, b] into subintervals. We may write
|[a, b] and read this as “ divides [a, b],” or “partitions [a, b].” We will denote the intervals of
 by I := [x , x ]. When we wish to work with 2 partitions at the same time we will have to
i i – 1 i
distinguish between them somehow, for example we can use y to denote the other’s points and
j
J to denote its intervals, etc.
j
We measure the fineness of a partition using the length of the longest interval in the partition.
This number is written
mesh():= max (x – x ) = max x .
 
 
1 i n  i i – 1 1 i n  i
This definition of mesh size is used and not max ((x ) –(x )) even in the Riemann-Stieltjes
 
1 i n  i i – 1
context.
22.3 The Riemann-Stieltjes – sum Definition of the
Riemann-Stieltjes Integral
Definition: A real-valued function f(x) defined on the bounded and closed interval [a, b] is
Riemann-Stieltjes integrable on [a, b] with respect to  if there exists a number RSI such that for
all  > 0 there exists  > 0 such that for every partition  of [a, b],
mesh() <   |RS(f, ) – RSI| < .

We write
b b

a ò f d = ò a f(x) d (x) := RSI

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