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Richa Nandra, Lovely Professional University Unit 22: Introduction to Riemann-Stieltjes Integration, using Riemann Sums

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

CONTENTS
Objectives
Introduction
22.1 Riemann-Stieltjes Sums
22.2 More Notation: The Mesh (Size) of a Partition
22.3 The Riemann-Stieltjes – sum Definition of the Riemann-Stieltjes Integral
22.4 A Difficulty with the Definition; The Cauchy Criterion for Riemann-Stieltjes
Integrability
22.5 Functions of Bounded Variation: Definition and Properties
22.6 Some Properties of Functions of Bounded Variation
22.7 Summary
22.8 Keywords
22.9 Review Questions
22.10 Further Readings

Objectives

After studying this unit, you will be able to:
 Discuss Riemann–Stieltjes sums

 Know the Cauchy Criterion for Riemann–Stieltjes Integrability
Introduction

We will approach Riemann-Stieltjes integrals using Riemann-Stieltjes sums instead of the upper
and lower sums. The main reasons are to study Riemann-Stieltjes integrals with “integrators”
(x) that are not monotone, but are “of bounded variation,” and (most important) here you are
able to define Riemann-Stieltjes integrals when the values of my functions belong to an infinite
dimensional vector space, where upper and lower sums don’t make sense. This makes little
difference in the case of real-valued functions, since functions of bounded variation can always
be expressed as the difference of two monotone functions. At first, we don’t need “bounded
variation,” so that concept’s development will wait until it is needed.
Throughout this note, our functions f(x) will be “finite-valued.” They may be real, complex, or
vector-valued. Their values will thus lie in a vector space. They can thus be added pointwise, and
multiplied by scalars, and their values always have finite “distance from zero,” denoted |f(x)|,
which can denote absolute value or norm, such as the length of a vector, or the “L norm” and the
p
q
“L norm”. In case f(x) is actually a function of t for each x... We always assume that the ”absolute
value” is complete; Cauchy sequences converge.

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