Pollard D. A User’s Guide to Measure Theoretic Probability

Cambridge: Cambridge University Press, 2001. — 366 p.This book grew from a one-semester course offered for many years to a mixed audience of graduate and undergraduate students who have not had the luxury of taking a course in measure theory The core of the book covers the basic topics of independence conditioning martingales convergence in distribution and Fourier transforms In addition there are numerous sections treating topics traditionally thought of as more advanced such as coupling and the KMT strong approximation option pricing via the equivalent martingale measure and the isoperimetric inequality for Gaussian processes The book is not just a presentation of mathematical theory but is also a discussion of why that theory takes its current form It will be a secure starting point for anyone who needs to invoke rigorous probabilistic arguments and understand what they meanMotivation
Why bother with measure theory?
The cost and benefit of rigor
Where to start: probabilities or expectations?
The de Finetti notation
Fair prices
Problems
Notes
A modicum of measure theory
Measures and sigma-fields
Measurable functions
Integrals
Construction of integrals from measures
Limit theorems
Negligible sets
LP spaces
Uniform integrability
Image measures and distributions
Generating classes of sets
Generating classes of functions
Problems
Densities and derivatives
Densities and absolute continuity
The Lebesgue decomposition
Distances and affinities between measures
The classical concept of absolute continuity
Vitali covering lemma
Densities as almost sure derivatives
Problems
Product spaces and independence
Independence
Independence of sigma-fields
Construction of measures on a product space
Product measures
Beyond sigma-fi niteness
SLLN via blocking
SLLN for identically distributed summands
Infinite product spaces
Problems
Notes
Conditioning
Conditional distributions: the elementary case
Conditional distributions: the general case
Integration and disintegration
Conditional densities
Invariance
Kolmogorov's abstract conditional expectation
Sufficiency
Problems
Notes
Martingale et al
What are they?
Stopping times
Convergence of positive supermartingales
Convergence of submartingales
Proof of the Krickeberg decomposition
Uniform integrability
Reversed martingales
Symmetry and exchangeability
Problems
Notes
Convergence in distribution
Definition and consequences
Lindeberg's method for the central limit theorem
Multivariate limit theorems
Stochastic order symbols
Weakly convergent subsequences
Problems
Fourier transforms
Definitions and basic properties
Inversion formula
A mystery?
Convergence in distribution
A martingale central limit theorem
Multivariate Fourier transforms
Cramer-Wold without Fourier transforms
The Levy-Cramer theorem
Problems
Brownian motion
Prerequisites
Brownian motion and Wiener measure
Existence of Brownian motion
Finer properties of sample paths
Strong Markov property
Martingale characterizations of Brownian motion
Functionals of Brownian motion
Option pricing
Problems
Representations and couplings
What is coupling?
Almost sure representations
Strassen's Theorem
The Yurinskii coupling
Quantile coupling of Binomial with normal
Haar coupling-the Hungarian construction
The Komlos-Major-Tusnady coupling
Problems
Exponential tails and the law of the iterated logarithm
LIL for normal summands
LIL for bounded summands
Kolmogorov's exponential lower bound
Identically distributed summands
Problems
Notes
Multivariate normal distributionsFernique's inequality
Proof of Fernique's inequality
Gaussian isoperimetric inequality
Proof of the isoperimetric inequality
Problems
Notes
Measures and integrals
Measures and inner measure
Tightness
Countable additivity
Extension to the nc-closure
Lebesgue measure
Integral representations
Problems
Notes
Hilbert spaces
Definitions
Orthogonal projections
Orthonormal bases
Series expansions of random processes
Problems
Notes
Convexity
Convex sets and functions
One-sided derivatives
Integral representations
Relative interior of a convex set
Separation of convex sets by linear functionals
Problems
Notes
Binomial and normal distributions
Martingales in continuous time
Filtrations sample paths and stopping times
Preservation of martingale properties at stopping times
Supermartingales from their rational skeletons
The Brownian filtration
Problems
Notes
Disintegration of measures
Representation of measures on product spaces
Disintegrations with respect to a measurable map
Problems
Notes