Introdction to Measure and Probability

Author： J. F. C. Kingman; S. J. Taylor

Publisher： Cambridge University Press‎

Publication year： 2008

E-ISBN: 9780511981777

P-ISBN(Paperback): 9780521090322

Subject： O211 probability (probability theory, probability theory)

Keyword：概率论与数理统计

Language： ENG

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Introdction to Measure and Probability

Description

The authors believe that a proper treatment of probability theory requires an adequate background in the theory of finite measures in general spaces. The first part of their book sets out this material in a form that not only provides an introduction for intending specialists in measure theory but also meets the needs of students of probability. The theory of measure and integration is presented for general spaces, with Lebesgue measure and the Lebesgue integral considered as important examples whose special properties are obtained. The introduction to functional analysis which follows covers the material (such as the various notions of convergence) which is relevant to probability theory and also the basic theory of L2-spaces, important in modern physics. The second part of the book is an account of the fundamental theoretical ideas which underlie the applications of probability in statistics and elsewhere, developed from the results obtained in the first part. A large number of examples is included; these form an essential part of the development.

Chapter

1.4 Operations on subsets

1.5 Classes of subsets

1.6 Axiom of choice

Point set topology

2.1 Metric space

2.2 Completeness and compactness

2.3 Functions

2.4 Cartesian products

2.5 Further types of subset

2.6 Normed linear space

2.7 Cantor set

Set functions

3.1 Types of set function

3.2 Hahn–Jordan decompositions

3.3 Additive set functions on a ring

3.4 Length, area and volume of elementary figures

Construction and properties of measures

4.1 Extension theorem; Lebesgue measure

4.2 Complete measures

4.3 Approximation theorems

4.4 Geometrical properties of Lebesgue measure

4.5 Lebesgue–Stieltjes measure

Definitions and properties of the integral

5.1 What is an integral?

5.2 Simple functions; measurable functions

5.3 Definition of the integral

5.4 Properties of the integral

5.5 Lebesgue integral; Lebesgue–Stieltjes integral

5.6 Conditions for integrability

Related spaces and measures

6.1 Classes of subsets in a product space

6.2 Product measures

6.3 Fubini's theorem

6.4 Radon–Nikodym theorem

6.5 Mappings of measure spaces

6.6 Measure in function space

6.7 Applications

The space of measurable functions

7.1 Point-wise convergence

7.2 Convergence in measure

7.3 Convergence in pth mean

7.4 Inequalities

7.5 Measure preserving transformations from a space to itself

Linear functional

8.1 Dependence of ℒ₂ on the underlying (Ω, ℱ, μ)

8.2 Orthogonal systems of functions

8.3 Biesz–Fischer theorem

8.4 Space of linear functionals

8.5 The space conjugate to ℒ_p

8.6 Mean ergodic theorem

Structure of measures in special spaces

9.1 Differentiating a monotone function

9.2 Differentiating the indefinite integral

9.3 Point-wise differentiation of measures

9.4 The Daniell integral

9.5 Representation of linear functionals

9.6 Haar measure

What is probability?

10.1 Probability statements

10.2 The algebra of events

10.3 Probability as measure

10.4 Conditional probability

10.5 Independent trials

Random variables

11.1 Random variables as measurable functions

11.2 Expectations

11.3 Distributions of random variables

11.4 Types of distribution function

11.5 Independent random variables

11.6 Discrete distributions

11.7 Continuous distributions

11.8 Convergence of random variables

Characteristic functions

12.1 The space of distribution functions

12.2 Characteristic functions

12.3 The inversion and continuity theorems

12.4 Generating functions

Independence

13.1 Sequences of independent trials

13.2 The Borel–Cantelli lemmas and the zero-one law

13.3 Sums of independent random variables

13.4 The central limit theorem

13.5 The law of the iterated logarithm

Finite collections of random variables

14.1 Joint distributions

14.2 Conditioning with respect to a random variable

14.3 Conditioning with respect to a σ-field

14.4 Moments

14.5 The multinomial distribution

14.6 The multinormal distribution

Stochastic processes

15.1 Renewal processes

15.2 The general theory of stochastic processes

15.3 Gaussian processes

15.4 Stationary processes

Index of notation

General index

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