Integration theory and general topology form the core of this textbook for a first-year graduate course in real analysis. After the foundational material in the first chapter (construction of the reals, cardinal and ordinal numbers, Zorn's lemma and transfinite induction), measure, integral and topology are introduced and developed as recurrent themes of increasing depth.
The treatment of integration theory is quite complete (including the convergence theorems, product measure, absolute continuity, the Radon-Nikodym theorem, and Lebesgue's theory of differentiation and primitive functions), while topology, predominantly metric, plays a supporting role. In the later chapters, integral and topology coalesce in topics such as function spaces, the Riesz representation theorem, existence theorems for an ordinary differential equation, and integral operators with continuous kernel function. In particular, the material on function spaces lays a firm foundation for the study of functional analysis.
Author(s): Sterling K. Berberian
Series: Universitext
Edition: 1
Publisher: Springer
Year: 1998
Language: German
Pages: 494
cover......Page 1
Universitext......Page 3
Fundamentals of Real Analysis......Page 4
Preface......Page 8
Contents......Page 10
CHAPTER 1 Foundations......Page 14
CHAPTER 2 Lebesgue Measure......Page 99
CHAPTER 3 Topology......Page 128
CHAPTER 4 Lebesgue Integral......Page 161
CHAPTER 5 Differentiation......Page 212
CHAPTER 6 Function Spaces......Page 286
CHAPTER 7 Product Measure......Page 377
CHAPTER 8 The Differential Equation y' = f(x, y)......Page 412
CHAPTER 9 Topics in Measure and Integration......Page 436
Bibliography......Page 483
Index......Page 487
