MTH644 Quiz

PREFACE

These are lecture notes on integration theory for a eight-week course at the Chalmers University of Technology and the Göteborg University. The parts de…ning the course essentially lead to the same results as the …rst three chapters in the Folland book [F ] ; which is used as a text book on the course. The proofs in the lecture notes sometimes di¤er from those given in [F ] : Here is a brief description of the di¤erences to simplify for the reader. In Chapter 1 we introduce so called -systems and -additive classes, which are substitutes for monotone classes of sets [F ]. Besides we prefer to emphasize metric outer measures instead of so called premeasures. Throughout the course, a variety of important measures are obtained as image measures of the linear measure on the real line. In Section 1.6 positive measures in R induced by increasing right continuous mappings are constructed in this way. Chapter 2 deals with integration and is very similar to [F ] and most other texts. Chapter 3 starts with some standard facts about metric spaces and relates the concepts to measure theory. For example Ulam’s Theorem is included. The existence of product measures is based on properties of -systems and -additive classes. Chapter 4 deals with di¤erent modes of convergence and is mostly close to [F ] : Here we include a section about orthogonality since many students have seen parts of this theory before. The Lebesgue Decomposition Theorem and Radon-Nikodym Theorem in Chapter 5 are proved using the von Neumann beautiful L2-proof. To illustrate the power of abstract integration these notes contain several sections, which do not belong to the course but may help the student to a better understanding of measure theory. The corresponding parts are set between the symbols

and

respectively.

CONTENT

1 Measures 1.1 -Algebras and Measures 1.2 Measure Determining Classes 1.3 Lebesgue Measure 1.4 Carathéodory’s Theorem 1.5 Existence of Linear Measure

2 Integration 2.1 Integration of Functions with Values in [0; 1] 2.2 Integration of Functions with Arbitrary Sign 2.3 Comparison of Riemann and Lebesgue Integrals

3 Further Construction Methods of Measures 3.1 Metric Spaces 3.2 Linear Functionals and Measures 3.3 q-Adic Expansions of Numbers in the Unit Interval 3.4 Product Measures 3.5 Change of Variables in Volume Integrals 3.6 Independence in Probability

4 Modes of Convergence 4.1 Convergence in Measure, in L1(); and in L2() 4.2 Orthogonality 4.3 The Haar Basis and Wiener Measure

5 Decomposition of Measures 5.1 Complex Measures 5.2 The Lebesgue Decomposition and the Radon-Nikodym Theorem 5.3 The Wiener Maximal Theorem and Lebesgue Di¤erentiation Theorem

CHAPTER 1

MEASURES

Introduction

The Riemann integral, dealt with in calculus courses, is well suited for computations but less suited for dealing with limit processes. In this course we will introduce the so called Lebesgue integral, which keeps the advantages of the Riemann integral and eliminates its drawbacks. At the same time we will develop a general measure theory which serves as the basis of contemporary analysis and probability. In this introductory chapter we set forth some basic concepts of measure theory, which will open for abstract Lebesgue integration.

1. 1. -Algebras and Measures

Throughout this course

N = f0; 1; 2; :::g (the set of natural numbers) Z = f:::; �2; �1; 0; 1; ; 2; :::g (the set of integers) Q = the set of rational numbers R = the set of real numbers C = the set of complex numbers.

If A  R; A+ is the set of all strictly positive elements in A: If f is a function from a set A into a set B; this means that to every x 2 A there corresponds a point f (x) 2 B and we write f : A ! B: A function is often called a map or a mapping. The function f is injective if

(x 6= y) ) (f (x) 6= f (y))