In this post, we will see the book Foundations of the Theory of Probability by A. N. Kolmogorov.

# About the book

The purpose of this monograph is to give an axiomatic foundation for the theory of probability. The author set himself the task of putting in their natural place, among the general notions of modern mathematics, the basic concepts of probability theory—concepts which until recently were considered to be quite peculiar.

This task would have been a rather hopeless one before the introduction of Lebesgue’s theories of measure and integration. However, after Lebesgue’s publication of his investigations, the analogies between measure of a set and probability of an event, and between integral of a function and mathematical expectation of a random variable, became apparent. These analogies allowed of further extensions; thus, for example, various properties of independent random variables were seen to be in complete analogy with the corresponding properties of orthogonal functions. But if probability theory was to be based on the above analogies, it still was necessary to make the theories of measure and integration independent of the geometric elements which were in the foreground with Lebesgue. This has been done by Frechet.

While a conception of probability theory based on the above general viewpoints has been current for some time among certain mathematicians, there was lacking a complete exposition of the whole system, free of extraneous complications.

I wish to call attention to those points of the present exposition which are outside the above-mentioned range of ideas familiar to the specialist. They are the following: Probability distributions in infinite-dimensional spaces (Chapter III, § 4) ; differentiation and integration of mathematical expectations with respect to a parameter (Chapter IV, § 5) ; and especially the theory of condi tional probabilities and conditional expectations (Chapter V). It should be emphasized that these new problems arose, of necessity, from some perfectly concrete physical problems.

The sixth chapter contains a survey, without proofs, of some results of A. Khinchine and the author of the limitations on the applicability of the ordinary and of the strong law of large numbers. The bibliography contains some recent works which should be of interest from the point of view of the foundations of the subject.

The book was translated from Russian by Nathan Morrison and was published in 1950.

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You can get the book here.

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# Contents

EDITOR’S NOTE iii

PREFACE v

## I. ELEMENTARY THEORY OF PROBABILITY

§ 1. Axioms 2

§ 2. The relation to experimental data 3

§ 3. Notes on terminology 5

§ 4. Immediate corollaries of the axioms; conditional probabilities; Theorem of Bayes 6

§5. Independence 8

§6. Conditional probabilities as random variables; Markov Chains 12

## II. INFINITE PROBABILITY FIELDS

§ 1. Axiom of Continuity 14

§ 2. Borel fields of probability 16

§ 3. Examples of infinite fields of probability 18

## III. RANDOM VARIABLES

§ 1. Probability functions 21

§ 2. Definition of random variables and of distribution functions 22

§ 3. Multi-dimensional distribution functions 24

§ 4. Probabilities in infinite-dimensional spaces 27

§ 5. Equivalent random variables; various kinds of convergence 33

## IV. MATHEMATICAL EXPECTATIONS

§1. Abstract Lebesgue integrals 37

§ 2. Absolute and conditional mathematical expectations 39

§ 3. The Tchebycheff inequality 42

§ 4. Some criteria for convergence 43

§ 5. Differentiation and integration of mathematical expectations with respect to a parameter 44

## V. CONDITIONAL PROBABILITIES AND MATHEMATICAL EXPECTATIONS

§ 1. Conditional probabilities 47

§ 2. Explanation of a Borel paradox 50

§ 8. Conditional probabilities with respect to a random variables 51

§ 4. Conditional mathematical expectations 52

## VI. INDEPENDENCE THE LAW OF LARGE NUMBERS

§ 1. Independence 57

§ 2. Independent random variables 58

§ 3. The Law of Large Numbers 61

§ 4. Notes on the concept of mathematical expectation 64

§ 5. The Strong Law of Large Numbers; convergence of a series 66

APPENDIX Zero-or-one law in the theory of probability 69

BIBLIOGRAPHY 71

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