A Course in Analysis: Vol. III: Measure and Integration Theory, Complex-Valued Functions of a Complex Variable

✍ Scribed by Niels Jacob, Kristian P Evans

• Publisher: WSPC
• Year: 2017
• Tongue: English
• Leaves: 783
• Category: Library

✦ Synopsis

In this third volume of "A Course in Analysis", two topics indispensible for every mathematician are treated: Measure and Integration Theory; and Complex Function Theory.

In the first part measurable spaces and measure spaces are introduced and Caratheodory's extension theorem is proved. This is followed by the construction of the integral with respect to a measure, in particular with respect to the Lebesgue measure in the Euclidean space. The Radon–Nikodym theorem and the transformation theorem are discussed and much care is taken to handle convergence theorems with applications, as well as L^p-spaces.

Integration on product spaces and Fubini's theorem is a further topic as is the discussion of the relation between the Lebesgue integral and the Riemann integral. In addition to these standard topics we deal with the Hausdorff measure, convolutions of functions and measures including the Friedrichs mollifier, absolutely continuous functions and functions of bounded variation. The fundamental theorem of calculus is revisited, and we also look at Sard's theorem or the Riesz–Kolmogorov theorem on pre-compact sets in L^p-spaces.

The text can serve as a companion to lectures, but it can also be used for self-studying. This volume includes more than 275 problems solved completely in detail which should help the student further.

✦ Table of Contents

Preface
Introduction
Contents
List of Symbols
Part 6: Measure and Integration Theory
1 A First Look at ˙-Fields and Measures
2 Extending Pre-Measures. Carath´eodory’s Theorem
3 The Lebesgue-Borel Measure and Hausdorff Measures
4 Measurable Mappings
5 Integration with Respect to a Measure — The Lebesgue Integral
6 The Radon-Nikodym Theorem and the Transformation Theorem
7 Almost Everywhere Statements, Convergence Theorems
8 Applications of the Convergence Theorems and More
9 Integration on Product Spaces and Applications
10 Convolutions of Functions and Measures
11 Differentiation Revisited
12 Selected Topics
Part 7: Complex-valued Functions of a Complex Variable
13 The Complex Numbers as a Complete Field
14 A Short Digression: Complex-valued Mappings
15 Complex Numbers and Geometry
16 Complex-Valued Functions of a Complex Variable
17 Complex Differentiation
18 Some Important Functions
19 Some More Topology
20 Line Integrals of Complex-valued Functions
21 The Cauchy Integral Theorem and Integral Formula
22 Power Series, Holomorphy and Differential Equations
23 Further Properties of Holomorphic Functions
24 Meromorphic Functions
25 The Residue Theorem
26 The 􀀀-function, the -function and Dirichlet Series
27 Elliptic Integrals and Elliptic Functions
28 The Riemann Mapping Theorem
29 Power Series in Several Variables
Appendix I: More on Point Set Topology
Appendix II: Measure Theory, Topology and Set Theory
Appendix III: More on M¨obius Transformations
Appendix IV: Bernoulli Numbers
Solutions to Problems of Part 6
Solutions to Problems of Part 7
References
Mathematicians Contributing to Analysis (Continued)
Subject Index