The Mathematical Theory of Probability has lately become of growing importance owing to the great variety of its applications, and also to its purely mathematical interest. The subject of this tract is the development of the purely mathematical side of the theory, without any reference to the applications. The axiomatic foundations of the theory have been chosen in agreement with the theory given by A. Kolmogoroff in his work Grundbegriffe der Wahrscheinlichkeitsrechnung, to which I am greatly indebted. In accordance with this theory, the subject has been treated as a branch of the theory of completely additive set functions. The method principally used has been that of characteristic functions (or Fourier-Stieltjes transforms).

The limitation of space has made it necessary to restrict the programme somewhat severely. Thus in the first place it has proved necessary to consider exclusively probability distributions in spaces of a finite number of dimensions. With respect to the advanced part of the theory, I have found it convenient to confine myself almost entirely to problems connected with the so-called Central Limit Theorem for sums of independent variables, and with some of its generalizations and modifications in various directions. This limitation permits a certain uniformity of method, but obviously a great number of important and interesting problems will remain unmentioned.

My most sincere thanks are due to my friends W. Feller, O. Lundberg and H. Wold for valuable help with the preparation of this work.