Foundations of the Theory of Probability

Dover has reprinted one of the classics of twentieth century mathematics. (Indeed, it is one of Grattan-Guinness’s Landmark Writings in Mathematics.) Kolmogorov’s Grundbegriffe der Wahrscheinlichkeitrechnung was originally published in 1933; this English translation appeared in 1950, with a second edition in 1956. By that time, everyone had accepted Kolmogorov’s axiomatization of probability theory.

In the introduction, Kolmogorov says that once abstract measure theory was established it was clear to the experts how to use it to formalize probability theory. That probably understates the originality of the book, which goes well beyond the basic observation that a probability is a measure and that expectation is an integral. It includes, for example, Kolmogorov’s famous construction of an infinite-dimensional probability space and the definition of conditional probabilities and expectations when the conditioning event has zero probability. It also highlights the crucial notion of independence as the point where probability theory becomes different from real analysis.

One of the interesting aspects of the book is Kolmogorov’s attempt to justify his choice of axioms. He explains how the observation of finite probability spaces leads to his choice of axioms except for the axiom of continuity. Since the latter deals with infinite probability spaces, Kolmogorov claims that it does not refer to anything of which he have (or can have) direct experience, and so it is chosen as a mere convenience. Similarly, Kolmogorov emphasizes the extension theorem, which shows that any probability measure on an algebra of sets can be extended to the sigma-algebra it generates. He claims that the sets obtained when we do this (e.g., general Borel sets) do not correspond to anything real, but reads the extension theorem as saying that reasoning with such sets does not lead to false results about sets which do arise in real-world contexts.

I believe this little book is still worth reading, especially after one has learned the basics of measure-theoretic probability theory. (It is not, of course, intended as an introductory textbook.) Seeing Kolmogorov set everything up in 70 pages is certainly impressive.

Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College.