US Eco PHD Mathematics measure theory

This book arose out of two graduate courses that the authors have taught
during the past several years; the first one being on measure theory followed
by the second one on advanced probability theory.
The traditional approach to a first course in measure theory, such as in
Royden (1988), is to teach the Lebesgue measure on the real line, then the
differentation theorems of Lebesgue, Lp-spaces on R, and do general measure
at the end of the course with one main application to the construction
of product measures. This approach does have the pedagogic advantage
of seeing one concrete case first before going to the general one. But this
also has the disadvantage in making many students’ perspective on measure
theory somewhat narrow. It leads them to think only in terms of the
Lebesgue measure on the real line and to believe that measure theory is
intimately tied to the topology of the real line. As students of statistics,
probability, physics, engineering, economics, and biology know very well,
there are mass distributions that are typically nonuniform, and hence it is
useful to gain a general perspective.
This book attempts to provide that general perspective right from the
beginning. The opening chapter gives an informal introduction to measure
and integration theory. It shows that the notions of σ-algebra of sets and
countable additivity of a set function are dictated by certain very natural
approximation procedures from practical applications and that they
are not just some abstract ideas. Next, the general extension theorem of
Carathedory is presented in Chapter 1. As immediate examples, the construction
of the large class of Lebesgue-Stieltjes measures on the real line
and Euclidean spaces is discussed, as are measures on finite and countable




Preface vii
Measures and Integration: An Informal Introduction 1
1 Measures 9
1.1 Classes of sets . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2 Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.3 The extension theorems and Lebesgue-Stieltjes measures . . 19
1.3.1 Caratheodory extension of measures . . . . . . . . . 19
1.3.2 Lebesgue-Stieltjes measures on R . . . . . . . . . . . 25
1.3.3 Lebesgue-Stieltjes measures on R2 . . . . . . . . . . 27
1.3.4 More on extension of measures . . . . . . . . . . . . 28
1.4 Completeness of measures . . . . . . . . . . . . . . . . . . . 30
1.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2 Integration 39
2.1 Measurable transformations . . . . . . . . . . . . . . . . . . 39
2.2 Induced measures, distribution functions . . . . . . . . . . . 44
2.2.1 Generalizations to higher dimensions . . . . . . . . . 47
2.3 Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.4 Riemann and Lebesgue integrals . . . . . . . . . . . . . . . 59
2.5 More on convergence . . . . . . . . . . . . . . . . . . . . . . 61
2.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71