Kolmogorov: Foundations of the Theory of Probability

Click at a thumbnail to start reading from there. DJVU file Back to Mathematik.com.

The book "Kolmogorov: Foundations of the Theory of Probability" by Andrey Nikolaevich Kolmogorov is historically very important. It is the foundation of modern probability theory. The monograph appeared as "Grundbegriffe der Wahrscheinlichkeitsrechnung" in 1933 and build up probability theory in a rigorous way similar as Euclid did with geometry. Today, it is mainly a historical document and can hardly be used as a textbook any more. The book is still readable and its structure survived in many modern probability books. Still, there are changes. The distribution function F for example is defined as F(s) = P[X < s], with an inquality, not "smaller equal" as today. The book is out of print and can only be purchased on the (now often electronic) flea markets.
 Foundations of the Theory of Probability, By A.N. Kolmogorov, Chelsea Publishing Company, New Yori, 1956
 Editors note: english translation
 Preface by Kolmogorov, Easter 1933
 Contents
 Chapter 1: Elementary Theory of Probability
 Axioms
 Notes on Terminology
 Corollaries of the Axioms
 Independence
 Theorem I and II
 Conditional probabilities as Random Variables, Markov Chains
 Chapter II, Infinite Probability Fields, Axioms of Continuity
 Borel Fields of Probability, Extension Theorem
 Examples of Infinite Fields of Probability
 Chapter III: Random Variables
 Definition of Random Variables and of Distribution Functions
 Multi-dimensional Distribution functions
 Probabilities in Infinite-dimensional Spaces
 Borel cylinder sets, Fundamental theorem
 Proof of the fundamental theorem
 Equivalent Random Variables, Various kinds of Convergence
 Convergence in probability implies convergence in distribution functions
 Chapter IV: Mathematical Expectations
 Absolute and Conditional Mathematical Expectations
 Conditional mathematical expectation with respect to an event
 Chebychev inequality, some criteria for convergence
 Chapter V: Conditional probabilities and mathematical expectations
 Explanation of a Borel Paradox, conditional probabilities
 Conditional mathematical expectations
 Conditional expectation and conditonal probability
 Chapter IV: independence: the law of large numbers
 Independent random variables
 The law of large numbers
 Theorem of Tchebychev
 Notes on the concept of mathematical expectation
 Strong law of large numbers, convergence of series
 Appendx: Zero or one law
 End of text
 Bibliography
 Supplementary bibliography
 Notes on supplementary bibliography Notes
 Supplementary bibliography
 End of supplementary bibliography
 Last update: 6/23/2006. Back to Mathematik.com page page