V. S. VARADARAJAN 1. Introduction It is nearly thirty years since A. N. Kolmogorov explicitly wrote down the axioms of modern probability theory in his celebrated monograph [lo]. During the intervening decades this theory has seen remarkable development, both in its theoretical and practical aspects. The diverse theories of mathematical statistics, the rapidly developing field of information theory, the applications to thermodynamics and statistical mechanics are only some of a long list of fields which are dominated to a substantial degree by probability theory. Moreover, many of the mathematical questions raised and answered by this theory have given deep and subtle insights into some difficult problems of analysis. One has only to mention the modern theory of Markov processes which has given new insights into such classical problems as boundary values and potential theory. The Kolmogorov axioms combine generality with simplicity. Probability becomes a part of measure theory, with its own special emphasis. In his basic monograph Kolmogorov himself proved many theorems indicating clearly the scope of this new discipline: the extension theorem which proves the existence of random variables with preassigned joint distributions, conditional probabilities and expectations, sequences of independent 'random variablesone can multiply these examples almost
Communications on Pure & Applied Mathematics – Wiley
Published: Jan 1, 1962
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