Capinski, Kopp - Measure Integral & Probability.pdf

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m-i-p-2.dvi
Marek Capinski and Ekkehard Kopp
Measure, Integral and Probability
Springer-Verlag
Berlin Heidelberg NewYork
London Paris Tokyo
Hong Kong Barcelona
Budapest
To our children; grandchildren:
Piotr, Maciej, Jan, Anna; Lukasz
Anna, Emily
Preface
The central concepts in this book are Lebesgue measure and the Lebesgue
integral. Their role as standard fare in UK undergraduate mathematics courses
is not wholly secure; yet they provide the principal model for the development of
the abstract measure spaces which underpin modern probability theory, while
the Lebesgue function spaces remain the main source of examples on which
to test the methods of functional analysis and its many applications, such as
Fourier analysis and the theory of partial dierential equations.
It follows that not only budding analysts have need of a clear understanding
of the construction and properties of measures and integrals, but also that those
who wish to contribute seriously to the applications of analytical methods in
a wide variety of areas of mathematics, physics, electronics, engineering and,
most recently, nance, need to study the underlying theory with some care.
We have found remarkably few texts in the current literature which aim
explicitly to provide for these needs, at a level accessible to current under-
graduates. There are many good books on modern probability theory, and
increasingly they recognize the need for a strong grounding in the tools we
develop in this book, but all too often the treatment is either too advanced for
an undergraduate audience or else somewhat perfunctory. We hope therefore
that the current text will not be regarded as one which lls a much-needed gap
in the literature!
One fundamental decision in developing a treatment of integration is
whether to begin with measures or integrals, i.e. whether to start with sets or
with functions. Functional analysts have tended to favour the latter approach,
while the former is clearly necessary for the development of probability. We
have decided to side with the probabilists in this argument, and to use the
(reasonably) systematic development of basic concepts and results in proba-
bility theory as the principal eld of application { the order of topics and the
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viii
Preface
terminology we use reect this choice, and each chapter concludes with further
development of the relevant probabilistic concepts. At times this approach may
seem less `ecient' than the alternative, but we have opted for direct proofs
and explicit constructions, sometimes at the cost of elegance. We hope that it
will increase understanding.
The treatment of measure and integration is as self-contained as we could
make it within the space and time constraints: some sections may seem too
pedestrian for nal-year undergraduates, but experience in testing much of the
material over a number of years at Hull University teaches us that familiar-
ity and condence with basic concepts in analysis can frequently seem some-
what shaky among these audiences. Hence the preliminaries include a review
of Riemann integration, as well as a reminder of some fundamental concepts of
elementary real analysis.
While probability theory is chosen here as the principal area of application
of measure and integral, this is not a text on elementary probability, of which
many can be found in the literature.
Though this is not an advanced text, it is intended to be studied (not
skimmed lightly) and it has been designed to be useful for directed self-study
as well as for a lecture course. Thus a signicant proportion of results, labelled
`Proposition', are not proved immediately, but left for the reader to attempt
before proceeding further (often with a hint on how to begin), and there is
a generous helping of Exercises. To aid self-study, proofs of the Propositions
are given at the end of each chapter, and outline solutions of the Exercises are
given at the end of the book. Thus few mysteries should remain for the diligent.
After an introductory chapter, motivating and preparing for the principal
denitions of measure and integral, Chapter 2 provides a detailed construction
of Lebesgue measure and its properties, and proceeds to abstract the axioms ap-
propriate for probability spaces. This sets a pattern for the remaining chapters,
where the concept of independence is pursued in ever more general contexts,
as a distinguishing feature of probability theory.
Chapter 3 develops the integral for non-negative measurable functions, and
introduces random variables and their induced probability distributions, while
Chapter 4 develops the main limit theorems for the Lebesgue integral and com-
pares this with Riemann integration. The applications in probability lead to a
discussion of expectations, with a focus on densities and the role of character-
istic functions.
In Chapter 5 the motivation is more functional-analytic: the focus is on the
Lebesgue function spaces, including a discussion of the special role of the space
L 2 of square-integrable functions. Chapter 6 sees a return to measure theory,
with the detailed development of product measure and Fubini's theorem, now
leading to the role of joint distributions and conditioning in probability. Finally,

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