Principles of Real Analysis: Measure, Integration, Functional Analysis, and Applications

Content: Cover
 Half Title
 Title
 Copyright
 Dedication
 Contents
 Preface
 0 Preliminaries
 0.1 Sets
 Set Operations
 Number Systems
 Relations
 Functions
 Cardinality
 0.2 Algebraic Structures
 Semigroups and Groups
 Linear Spaces
 Linear Transformations
 Quotient Linear Spaces
 Algebras
 0.3 Metric Spaces
 Open and Closed Sets
 Interior, Closure, and Boundary
 Sequential Convergence. Completeness
 Continuity
 Category
 0.4 Normed Linear Spaces
 Norms and Seminorms
 Banach Spaces
 Completion of a Normed Space
 In nite Series in Normed Spaces
 Unordered Sums in Normed Spaces Bounded Linear TransformationsBanach Algebras
 0.5 Topological Spaces
 Open and Closed Sets
 Neighborhood Systems
 Neighborhood Bases
 Relative Topology
 Nets
 0.6 Continuity in Topological Spaces
 De nition and General Properties
 Initial Topologies
 Product Topology
 Final Topologies
 Quotient Topology
 The Space of Continuous Functions
 F-sigma and G-delta Sets
 0.7 Normal Topological Spaces
 Urysohn's Lemma
 Tietze Extension Theorem
 0.8 Compact Topological Spaces
 Convergence in Compact Spaces
 Compactness of Cartesian Products
 Continuity and Compactness 0.9 Totally Bounded Metric Spaces0.10 Equicontinuity
 0.11 The Stone-Weierstrass Theorem
 0.12 Locally Compact Topological Spaces
 General Properties
 Functions with Compact Support
 Functions That Vanish at In nity
 The One-Point Compacti cation
 0.13 Spaces of Di erentiable Functions
 0.14 Partitions of Unity
 0.15 Connectedness
 I Measure and Integration
 1 Measurable Sets
 1.1 Introduction
 1.2 Measurable Spaces
 Fields and Sigma Fields
 Generated Sigma Fields
 Borel Sets
 Extended Borel Sets
 Product Sigma Fields
 Pi-Systems and Lambda-Systems
 Exercises
 1.3 Measures
 Set Functions Properties and Examples of MeasuresExercises
 1.4 Complete Measure Spaces
 Completion Theorem
 Null Sets
 Exercises
 1.5 Outer Measure and Measurability
 Construction of an Outer Measure
 Carath eodory's Theorem
 Exercises
 1.6 Extension of a Measure
 The Measure Extension Theorem
 Approximation Property of the Extension
 Completeness of the Extension
 Uniqueness of the Extension
 Exercises
 1.7 Lebesgue Measure
 The Volume Set Function
 Construction of the Measure
 Exercises
 1.8 Lebesgue-Stieltjes Measures
 Regularity
 One-Dimensional Distribution Functions Higher Dimensional Distribution FunctionsExercises
 *1.9 Some Special Sets
 An Uncountable Set with Lebesgue Measure Zero
 Non-Lebesgue-Measurable Sets
 A Lebesgue Measurable, Non-Borel Set
 Exercises
 2 Measurable Functions
 2.1 Measurable Transformations
 General Properties
 Exercises
 2.2 Measurable Numerical Functions
 Criteria for Measurability
 Almost Everywhere Properties
 Combinatorial and Limit Properties of Measurable Functions
 Exercises
 2.3 Simple Functions
 A Fundamental Convergence Theorem
 Applications
 Exercises
 2.4 Convergence of Measurable Functions
 Modes of Convergence