This book, the first of a projected two volume series, is designed for a graduate course in modern probability. The first four chapters, along with the Appendix: On General Topology, provide the background in analysis needed for the study of probability. This material is available as a separate book called" Measure, Integration, and Functional Analysis."
Author(s): Robert B. Ash
Series: Probability and Mathematical Statistics
Publisher: Academic Press
Year: 1972
Language: English
Pages: C, xvi, 476, B
Tags: Mathematics Applied Geometry Topology History Infinity Mathematical Analysis Matrices Number Systems Popular Elementary Pure Reference Research Study Teaching Transformations Trigonometry Science Math
Preface
Summary of Notation
1 Sets
2 Real Numbers
3 Functions
4 Topology
S Vector Spaces
6 Zorn's Lemma
1 Fundamentals of Measure and Integration Theory
1.1 Introduction
Problems
1.2 Fields, O"-Fields, and Measures
Problems
1.3 Extension of Measures
Problems
1.4 Lebesgue-Stieltjes Measures and Distribution Functions
Problems
1.5 Measurable Functions and Integration
Problems
1.6 Basic Integration Theorems
Problems
1.7 Comparison of Lebesgue and Riemann Integrals
Problems
2 Further Results in Measure and Integration Theory
2.1 Introduction
Problems
2.2 Radon-Nikodym Theorem and Related Results
Problems
2.3 Applications to Real Analysis
Problems
2.4 L^p Spaces
Problems
2.5 Convergence of Sequences of Measurable Functions
Problems
2.6 Product Measures and Fubini's Theorem
Problems
2.7 Measures on Infinite Product Spaces
Problems
2.8 References
3 Introduction to Functional Analysis
3.1 Introduction
3.2 Basic Properties of Hilbert Spaces
Problems
3.3 Linear Operators on Normed Linear Spaces
Problems
3.4 Basic Theorems of Functional Analysis
Problems
3.5 Some Properties of Topological Vector Spaces
Problems
3.6 References
4 The Interplay between Measure Theory and Topology
4.1 Introduction
4.2 The Daniell Integral
Problems
4.3 Measures on Topological Spaces
Problems
4.4 Measures on Uncountably Infinite Product Spaces
Problems
4.5 Weak Convergence of Measures
Problems
4.6 References
5 Basic Concepts of Probability
5.1 Introduction
5.2 Discrete Probability Spaces
5.3 Independence
5.4 Bernoulli Trials
5.5 Conditional Probability
5.6 Random Variables
5.7 Random Vectors
5.8 Independent Random Variables
Problems
5.9 Some Examples from Basic Probability
Problems
5.10 Expectation
Problems
5.11 Infinite Sequences of Random Variables
Problems
5.12 References
6 Conditional Probability and Expectation
6.1 Introduction
6.2 Applications
6.3 The General Concept of Conditional Probability and Expectation
Problems
6.4 Conditional Expectation Given a \sigma-field
Problem
6.5 Properties of Conditional Expectation
Problems
6.6 Regular Conditional Probabilities
Problems
6.7 References
7 Strong Laws of Large Numbers and Martingale Theory
7.1 Introduction
Problems
7.2 Convergence Theorems
Problems
7.3 Martingales
Problems
7.4 Martingale Convergence Theorems
Problems
7.5 Uniform Integrability
Problems
7.6 Uniform Integrability and Martingale Theory
Problems
7.7 Optional Sampling Theorems
Problems
7.8 Applications of Martingale Theory
Problems
7.9 Applications to Markov Chains
7.10 References
8 The Central Liinit Theorem
8.1 Introduction
Problems
8.2 The Fundamental Weak Compactness Theorem
Problems
8.3 Convergence to a Normal Distribution
Problems
8.4 Stable Distributions
Problem
8.5 Infinitely Divisible Distributions
Problems
8.6 Uniform Convergence in the Central Limit Theorem
8.7 Proof of the Inversion Formula (8.1.4)
8.8 Completion of the Proof of Theorem 8.3.2
8.9 Proof of the Convergence of Types Theorem (8.3.4)
8.10 References
Appendix on General Topology
A1 Introduction
Al Convergence
A3 Product and Quotient Topologies
A4 Separation Properties and Other Ways of Classifying Topological Spaces
A5 Compactness
A6 Semicontinuous Functions
A7 The Stone-Weierstrass Theorem
A8 Topologies on Function Spaces
A9 Complete Metric Spaces and Category Theorems
A10 Uniform Spaces
Solutions to Problems
Chapter 1
Chapter 2
Chapter 3
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Subject Index
