Kolmogorov: Foundations of the Theory of Probability

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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 Today, it is mainly a historical document and can hardly be used as a textbook any more.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
Chapter 1: Elementary Theory of Probability
Notes on Terminology
Corollaries of the Axioms
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
Supplementary bibliography
Notes on supplementary bibliography Notes
Supplementary bibliography
End of supplementary bibliography
Last update: 6/23/2006. Back to Mathematik.com page page