Chapter 75 - A.N. Kolmogorov, Grundbegriffe der wahrscheinlichkeitsrechnung (1933)

Abstract

This chapter describes A.N. Kolmogorov's book, Grundbegriffe der Wahrscheinlichkeitsrechnung (fundamental concepts of probability). In this short book, Kolmogorov laid out the foundations of probability theory in terms of set and measure theory, bringing into definitive form a line of thought among some probabilists of the past three decades. This book has become the symbol of modern measure-theoretic probability theory, its year of appearance 1933 being seen as a turning point that made earlier studies redundant. This book established that set-theoretic foundation. The book proper starts with the axiom system for a finite set of events. The sense of probability that Kolmogorov endorses is addressed, characteristically, in the chapter dealing with the theory of probability for a finite set of events. The strictly infinitary parts of the theory are purely mathematical. These are the theory of conditional probabilities and the construction of a random process as a probability measure over an infinite-dimensional product space. Paul Halmos's book of 1950, Measure theory, the standard treatise on its topic for a long time, devotes one chapter to probability measures.