A course in probability theory

In recent years books on probability theory have mushroomed. The two books under review have this in common : they are concerned with probability theory proper, rather than with its foundations (like the recent elegant but austere book by Neveu), and they give a balanced view of the theory rather than dwelling at length on some specialized topics. What is probability theory? Chung's spirited preface at first tells us what it is not: the "specious utterance" that "probability is just a chapter of measure theory" finds no more favor with him than the use of probability "as a front for certain types of analysis such as combinatorial, Fourier, functional and whatnot." Elsewhere he says about the same utterance that it is "not so much false as it is fatuous, as it would be to say that number theory is just a chapter of algebra." Thus over the years Professor Chung continues his disagreement with Professor Doob who in his Stochastic processes writes: "Probability is simply a branch of measure theory, with its own special emphasis and field of application." On the positive side Professor Chung remarks in the preface of his book: "The random walks in Chapter 8 illustrate well the way probability theory transforms other parts of mathematics. I t does so by introducing the trajectories of the process, thereby turning what was static into a dynamic structure." As for Dr. Breiman, without defining probability he tells us about its right and left hand. "On the right is the rigorous foundational work using the tools of measure theory. The left hand i thinks probabilistically/ reduces problems to gambling situations, coin-tossing, motions of a physical particle." Breiman acknowledges M. Loève as his teacher of the first aspect of probability (and D. Blackwell as the teacher of the second) and Professor Loève in his treatise on the subject (p. 170) defines a probabilistic property as one which can be described in terms of distribution functions. Professor Neveu (Preface, op. cit.) strongly disagrees: "Attempts have been made, it is true, to treat convergence problems of probability theory within the restricted framework of the study of distribution functions; but this 1 J. Neveu, Mathematical foundations of the calculus of probability, Holden-Day, San Francisco, Calif., 1965. 2 In the review of A. Rényi's textbook, Bull. Amer. Math. Soc. 70 (1964), 225.