Preface
PART ONE Set Theory，Real Numbers，and Calculus
1 SET THEORY
Biography: Georg Cantor
1.1 Basic Definitions and Properties
1.2 Functions and Sets
1.3 Equivalence of Sets； Countability
1.4 Algebras，σ-Algebras，and Monotone Classes
2 THE REAL NUMBER SYSTEM AND CALCULUS
Biography: Georg Friedrich Bernhard Riemann
2.1 The Real Number System
2.2 Sequences of Real Numbers
2.3 Open and Closed Sets
2.4 Real-Valued Functions
2.5 The Cantor Set and Cantor Function
Preface
PART ONE Set Theory，Real Numbers，and Calculus
1 SET THEORY
Biography: Georg Cantor
1.1 Basic Definitions and Properties
1.2 Functions and Sets
1.3 Equivalence of Sets； Countability
1.4 Algebras，σ-Algebras，and Monotone Classes
2 THE REAL NUMBER SYSTEM AND CALCULUS
Biography: Georg Friedrich Bernhard Riemann
2.1 The Real Number System
2.2 Sequences of Real Numbers
2.3 Open and Closed Sets
2.4 Real-Valued Functions
2.5 The Cantor Set and Cantor Function
2.6 The Riemann Integral
PART TWO Measure，Integration，and DifFerentiation
3 LEBESGUE THEORY ON THE REAL LINE
Biography: Emile Felix-Edouard-Justin Borel
3.1 Borel Measurable Functions and Borel Sets
3.2 Lebesgue Outer Measure
3.3 Further Properties of Lebesgue Outer Measure
3.4 Lebesgue Measure
4 THE LEBESGUE INTEGRAL ON THE REAL LINE
Biography: Henri Leon Lebesgue
4.1 The Lebesgue Integral for Nonnegative Functions
4.2 Convergence Properties of the Lebesgue Integral for
Nonnegative Functions
4.3 The General Lebesgue Integral
4.4 Lebesgue Almost Everywhere
5 ELEMENTS OF MEASURE THEORY
Biography: Constantin Carath~odory
5.1 Measure Spaces
5.2 Measurable Functions
5.3 The Abstract Lebesgue Integral for Nonnegative Functior
5.4 The General Abstract Lebesgue Integral
5.5 Convergence in Measure
6 EXTENSIONS TO MEASURES AND PRODUCT MEASURE
Biography: Guido Fubini
6.1 Extensions to Measures
6.2 The Lebesgue-Stieltjes Integral
6.3 Product Measure Spaces
6.4 Iteration of Integrals in Product Measure Spaces
7 ELEMENTS OF PROBABILITY
Biography: Andrei Nikolaevich Kolmogorov
7.1 The Mathematical Model for Probability
7.2 Random Variables
7.3 Expectation of Random Variables
7.4 The Law of Large Numbers
8 DIFFERENTIATION AND ABSOLUTE CONTINUITY
Biography: Giuseppe Vitafi
8.1 Derivatives and Dini-Derivates
8.2 Functions of Bounded Variation
8.3 The Indefinite Lebesgne Integral
8.4 Absolutely Continuous Functions
9 SIGNED AND COMPLEX MEASURES
Biography: Johann Radon
9.1 Signed Measures
9.2 The Radon-Nikodym Theorem
9.3 Signed and Complex Measures
9.4 Decomposition of Measures
9.5 Measurable Transformati6ns and the General
Change-of-Variable Formula
PART THREE
Topological, Metric, and Normed Spaces
10 TOPOLOGIES, METRICS, AND NORMS
Biography: Felix Hausdorff
10.1 Introduction to Topological Spaces
10.2 Metrics and Norms
10.3 Weak Topologies
10.4 Closed Sets, Convergence, and Completeness
10.5 Nets and Continuity
10.5 Separation Properties
10.7 Connected Sets
11 SEPARABILITY AND COMPACTNESS
Biography: Maurice Frechet
11.1 Separability, Second Countability, andMetrizability
11.2 Compact Metric Spaces
11.3 Compact Topological Spaces
11.4 Locally Compact Spaces
11.5 Function Spaces
12 COMPLETE AND COMPACT SPACES
Biography: Marshall Harvey Stone
12.1 The Baire Category Theorem
12.2 Contractions of Complete Metric Spaces
12.3 Compactness in the Space C(□, A)
12.4 Compactness of Product Spaces
12.5 Approximation by Functions from a Lattice
12.5 Approximation by Functions from an Algebra
13 HILBERT SPACES AND BANACH SPACES
Biography: David Hilbert
13.1 Preliminaries on Normed Spaces
13.2 Hilbert Spaces
13.3 Bases and Duality in Hilbert Spaces
13.4 □-Spaces
13.5 Nonnegative Linear Functionals on C(□)
13.5 The Dual Spaces of C(□) and C0(□)
14 NORMED SPACES AND LOCALLY CONVEX SPACES
Biography: Stefan Banach
14.1 The Hahn-Banach Theorem
14.2 Linear Operators on Banach Spaces
14.3 Compact Self-Adjoint Operators
14.4 Topological Linear Spaces
14.5 Weak and Weak* Topologies
14.5 Compact Convex Sets
PART FOUR
Harmonic Analysis, Dynamical Systems, and Hausdorff Measure
15 ELEMENTS OF HARMONIC ANALYSIS
Biography: Ingrid Daubechies
15.1 Introduction to Fourier Series
15.2 Convergence of Fourier Series
15.3 The Fourier Transform
15.4 Fourier Transforms of Measures
15.5 □-Theory of the Fourier Transform
15.5 Introduction to Wavelets
15.7 Orthonormal Wavelet Bases; The Wavelet Transform
15 MEASURABLE DYNAMICAL SYSTEMS Biography: Claude E/woodShannon
16.1 Introduction and Examples
16.2 Ergodic Theory
16.3 Isomorphism of Measurable Dynamical Systems;Entropy
16.4 The Kolmogorov-Sinai Theorem; Calculation of Entropy
17 HAUSDORFF MEASURE AND FRACTALS Biography: Benoit B.Mandelbrot
17.1 Outer Measure and Measurability
17.2 Hausdorff Measure
17.3 Hausdorff Dimension and Topological Dimension
17.4 Fractals
Index
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