Taylor J.C. An Introduction to Measure and Probability

New York: Springer-Verlag, 1997. — 313 p.The original version of this book was an attempt to present the basics of measure theory, as it relates to probability theory, in a form that might be accessible to students without the usual background in analysis. This set of notes was intended to be a primer for measure theory and probability — a cross between a workbook and a text — rather than a standard manual. They were intended to give the reader a hands-on understanding of the basic techniques of measure theory together with an illustration of their use in probability as illustrated by an introduction to the laws of large numbers and martingale theory. The sections of the first five chapters not marked by an asterisk are the current version of the original set of notes and constitute the essential core of the book. Since the main goal of the core of this book is to be a fast-track, self-contained approach to martingale theory for the reader without a solid technical background, many details are put into exercises, which the reader is advised to do if a basic mastery of the subject is the goal. The present book retains the basic purpose and philosophy of the original set of notes but has been expanded to make it of more use to statisticians by adding a sixth chapter on weak convergence, which is an introduction to this very large circle of ideas as well as to the central limit theorem. There are six chapters: the first two chapters constitute the basic introduction; and the remaining four explain some uses of measure theory and integration in probability.List of Symbols
Probability Spaces
Integration
Independence and Product Measures
Convergence of Random Variables and Measurable Functions
Conditional Expectation and an Introduction to Martingales
An Introduction to Weak Convergence
Appendix