7 - Kolmogorov's measure theoretic probabilities

Summary

FOUNDATIONS AND PHILOSOPHY OF MATHEMATICS

Grundbegriffe der Wahrscheinlichkeitsrechnung by Andrei Kolmogorov is the book which has become the symbol of modern probability theory, its year of appearance 1933 being seen as a turning point that made earlier studies redundant. In mathematics, it is fairly common to take a field of study as given, as being defined by a set of commonly accepted postulates. Kolmogorov's presentation of probability in terms of measure theory serves well to illustrate this supposedly ahistorical character of mathematical research: With some knowledge of set theory, one can take the book, and learn and start doing probability theory. In such an approach, the concepts and the structure of probability theory appear fixed, whereas the experience of those who built up modern probability must have been very different. There were many kinds of approaches to the foundations of the subject. The idea of a measure theoretic foundation was almost as old as measure theory itself, and it had been repeatedly presented and used in the literature. Therefore the mere idea was not the reason for the acceptance of Kolmogorov's measure theoretic approach, but rather what he achieved by the use of measure theoretic probabilities. The change brought about by Kolmogorov was a big step, but not the kind of dramatic revelation some later comments suggest. The two essential mathematical novelties of Grundbegriffe were the theory of conditional probabilities when the condition has probability 0, and the general theory of random or stochastic processes.