目錄
Introduction
1 Measures
1.1 Algebras and Sigma-Algebras
1.2 Measures
1.3 Outer Measures
1.4 Lebesgue Measure
1.5 Completeness and Regularity
1.6 Dynkin Classes
2 Functions and Integrals
2.1 Measurable Functions
2.2 Properties That Hold Almost Everywhere
2.3 The Integral
2.4 Limit Theorems
2.5 The Riemann Integral
2.6 Measurable Functions Again, Complex-Valued
Functions, and Image Measures
3 Convergence
3.1 Modes of Convergence
3.2 Normed Spaces
3.3 Definition of LP and LP
3.4 Properties of LP and LP
3.5 Dual Spaces
4 Signed and Complex Measures
4.1 Signed and Complex Measures
4.2 Absolute Continuity
4.3 Singularity
4.4 Functions of Finite Variation
4.5 The Duals of the LP Spaces
5 Product Measures
5.1 Constructions
5.2 Fubini's Theorem
5.3 Applications
6 Differentiation
6.1 Change of Variable in Rd
6.2 Differentiation of Measures
6.3 Differentiation of Functions
7 Measures on Locally Compact Spaces
7.1 Locally Compact Spaces
7.2 The Riesz Representation Theorem
7.3 Signed and Complex Measures; Duality
7.4 Additional Properties of Regular Measures
7.5 The μ*-Measurable Sets and the Dual ofL1
7.6 Products of Locally Compact Spaces
7.7 The Daniell-Stone Integral
8 Polish Spaces and Analytic Sets
8.1 Polish Spaces
8.2 Analytic Sets
8.3 The Separation Theorem and Its Consequences
8.4 The Measurability of Analytic Sets
8.5 Cross Sections
8.6 Standard, Analytic, Lusin, and Souslin Spaces
9 Haar Measure
9.1 Topological Groups
9.2 The Existence and Uniqueness of Haar Measure
9.3 Properties of Haar Measure
9.4 The Algebras L1 (G) and M(G)
10 Probability
10.1 Basics
10.2 Laws of Large Numbers
10.3 Convergence in Distribution and the Central Limit Theorem
10.4 Conditional Distributions and Martingales
10.5 Brownian Motion
10.6 Construction of Probability Measures
A Notation and Set Theory
B Algebra and Basic Facts About R and C
C Calculus and Topology in Rd
D Topological Spaces and Metric Spaces
E The Bochner Integral
F Liftings
G The Banach-Tarski Paradox
H The Henstock-Kurzweii and McShane Integrals
References
Index of notation
Index
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