Measures, Integrals and Martingales

Cover
Half-Title
Title Page
Imprints Page
Contents
List of Symbols
Prelude
Dependence Chart
1 Prologue
	Problems
2 The Pleasures of Counting
	Problems
3 σ-Algebras
	Problems
4 Measures
	Problems
5 Uniqueness of Measures
	Problems
6 Existence of Measures
	Existence of Lebesgue measure in R
	Existence of Lebesgue measure in R[sup(n)]
	Problems
7 Measurable Mappings
	Problems
8 Measurable Functions
	Problems
9 Integration of Positive Functions
	Problems
10 Integrals of Measurable Functions
	Problems
11 Null Sets and the 'Almost Everywhere'
	Problems
12 Convergence Theorems and Their Applications
	Application 1: Parameter-Dependent Integrals
	Application 2: Riemann vs. Lebesgue Integration
	Improper Riemann Integrals
	Examples
	Problems
13 The Function Spaces L[sup(p)]
	Convergence in L[sup(p)] and Completeness
	Convexity and Jensen's Inequality
	Convexity Inequalities in R[sup(2)][sub(+)]
	Problems
14 Product Measures and Fubini's Theorem
	Integration by Parts and Two Interesting Integrals
	Distribution Functions
	Minkowski's Inequality for Integrals
	More on Measurable Functions
	Problems
15 Integrals with Respect to Image Measures
	Convolutions
	Regularization
	Problems
16 Jacobi's Transformation Theorem
	A useful Generalization of the Transformation Theorem
	Images of Borel Sets
	Polar Coordinates and the Volume of the Unit Ball
	Surface Measure on the Sphere
	Problems
17 Dense and Determining Sets
	Dense Sets
	Determining Sets
	Problems
18 Hausdorff Measure
	Constructing (Outer) Measures
	Hausdorff Measures
	Hausdorff Dimension
	Problems
19 The Fourier Transform
	Injectivity and Existence of the Inverse Transform
	The Convolution Theorem
	The Riemann–Lebesgue Lemma
	The Wiener Algebra, Weak Convergence and Plancherel
	The Fourier Transform in S(R[sup(n)])
	Problems
20 The Radon–Nikodým Theorem
	Problems
21 Riesz Representation Theorems
	Bounded and Positive Linear Functionals
	Duality of the Spaces L[sup(p)](μ), 1 ≤ p < ∞
	The Riesz Representation Theorem for C[sub(c)](X)
	Vague and Weak Convergence of Measures
	Problems
22 Uniform Integrability and Vitali's Convergence Theorem
	Different Forms of Uniform Integrability
	Problems
23 Martingales
	Problems
24 Martingale Convergence Theorems
	Problems
25 Martingales in Action
	The Radon–Nikodým Theorem
	Martingale Inequalities
	The Hardy–Littlewood Maximal Theorem
	Lebesgue's Differentiation Theorem
	The Calderón–Zygmund Lemma
	Problems
26 Abstract Hilbert Spaces
	Convergence and Completeness
	Problems
27 Conditional Expectations
	Extension from L[sup(2)] to L[sup(p)]
	Monotone Extensions
	Properties of Conditional Expectations
	Conditional Expectations and Martingales
	On the Structure of Subspaces of L[sup(2)]
	Separability Criteria for the Spaces L[sup(p)] (X, A, μ)
	Problems
28 Orthonormal Systems and Their Convergence Behaviour
	Orthogonal Polynomials
	The Trigonometric System and Fourier Series
	The Haar System
	The Haar Wavelet
	The Rademacher Functions
	Well-Behaved Orthonormal Systems
	Problems
Appendix A: lim inf and lim sup
Appendix B: Some Facts from Topology
	Continuity in Euclidean Spaces
	Metric Spaces
Appendix C: The Volume of a Parallelepiped
Appendix D: The Integral of Complex-Valued Functions
Appendix E: Measurability of the Continuity Points of a Function
Appendix F: Vitali's Covering Theorem
Appendix G: Non-measurable Sets
Appendix H: Regularity of Measures
Appendix I: A Summary of the Riemann Integral
	The (Proper) Riemann Integral
	The Fundamental Theorem of Integral Calculus
	Integrals and Limits
	Improper Riemann Integrals
References
Index