- Strukturtheorie der Wahrscheinlichkeitsfelder und -Räume (1960), by Demetrios Kappos.
1.1. Review by: A A Grau.
The American Mathematical Monthly 68 (7) (1961), 683.
The author points out that since there already exist systematic treatises on probability theory from the measure-theoretic point of view, this book does not attempt such a presentation. The book thus confines itself, as the title implies, to the systematic presentation of the mathematical theory of the structure of probability fields and, in term of these, to that of the structure of probability spaces. The former are defined in terms of a probability function and the latter in terms of measure, but since it is shown that every probability space may be represented as a probability field, the former approach permits a more direct formulation of problems that have been treated by the latter. The latter part of the book includes material on cartesian products and concepts of independence that play a dominant role in the characterisation of the structure of probability fields. A theory of the nonseparable invariant extensions of Lebesque measure related to concepts of independence of Kakatani and Oxtoby, and some treatment of a generalisation of the concept of probability space due to Renyi, appear at the end.
1.2. Kappos's Preface.
(1) The first terms and theorems of probability theory were formed in the mathematical treatment of gambling. In the beginning one could help oneself in this theory with elementary mathematical tools. However, the methods used were not satisfactory and accurate enough. In addition, the great importance that this theory gained in our century as an auxiliary science in many other sciences demanded new mathematical methods and concepts to deal with the newly arising problems. The mathematical apparatus was mainly provided by the simultaneously highly developed measure and integration theory. At the present time, the methods of measure theory have penetrated so far into probability theory that one can assert that probability theory is a branch of the abstract theory of measure and integration. However, this does not completely take away its character. Many of its concepts can be formulated and treated in terms of measure theory, but they were first shaped in probabilistic problems and thus opened up new fields and methods to the theory of measure and integration.
(2) A systematic introduction of measure-theoretical methods into probability theory first took place in Part 3 of the second volume of the Ergebnisse der Mathematik und ihrer Grenzgebiete, written by A Kolmogorov with the title: 'Basic Concepts of Probability Theory' in 1933. Kolmogorov's point of view on the mathematical foundations of probability theory was demanded from the statistical point of view by the critique of W Wald (The Consistency of the Collective Concept: Results of a Mathematical Colloquium, Issue 8, Vienna, 1937) on the von Mises' statistical justification of probability theory. Kolmogorov's book has formed the basis of mathematical research in probability theory over the past three decades. Recently, systematic textbooks have emerged which, in terms of theories of measure, deal with the theory of probability or certain sub-areas of it. We will therefore not repeat in our booklet what has already been done systematically, but limit ourselves to a specific question concerning both probability theory and measure theory, namely the question of the structure of probability fields. In the last two decades, much research has been done in this direction, which has not been systematically compiled to this day.
(3) The abstract measure theory introduced by C Caratheodory on Boolean rings allows a better overview of the structure of probability fields. With the help of this theory one can also remove the following deficiency, with which the set theoretic grounding of the probability theory is afflicted. In the explanation of events as measurable subsets of a set E whose elements (points) are also considered as elementary events, one is forced to assign the probability zero to certain events (zero sets, i.e. sets with the measure zero), although these events are different from the impossible event (empty set), that is realisable. Correspondingly one may have to assign the probability 1 to events which are different from certainty (basic set E). In addition, certain subsets of the base set (non-measurable subsets) are not considered to be events, although the points that make up those sets are considered to be viable events. In the first place, the concept of the elementary event (point) in base sets [characteristic spaces] of a power > ℵ0 is an artificial (ideal) invention for probability theory, which is introduced with the help of concrete events. In the set-theoretic justification of probability theory, however, one places this artificial concept in the foreground and hereby declares events that may have a concrete meaning as sets of such points. In addition, an important property of probability, namely alpha-additivity (total-additivity) in the set-theoretic justification of probability theory is axiomatically required but not sufficiently justified; it is well known not to follow finite additivity, which is a natural and empirically justified property of probability.
In order to avoid such shortcomings of the set-theoretic justification, we explain in Chapter I a probability field as a boolean ring with a strictly positive and additive probability. Probabilities that do not possess the property of strict positivity are called quasi-probabilities. In Chapter II we show how one can always obtain the important property of the alpha-additivity of probability by a metric extension of the probability field. It is only in Chapter III that we introduce the concept of the probability space of classical theory and show that every probability field can always be represented by a probability space such that the probability in this representation has the property of alpha-additivity. Conversely, however, one can pass from a probability space through the formation of residual classes modulo zero sets to a probability field. The two theories therefore do the same. However, it would be desirable to directly address all known problems of probability theory, i.e. without transition to a probability space, to formulate and to prove this, because we believe that not only known results will be clearer, but also the direct methods will lead to new insights. This shows, for example, the direct methods used by K Krickeberg in his investigations in the theory of stochastic processes. Such questions do not concern us in our booklet, but we hope that this point of view of the theory of probability, the importance of which Kolmogorov has recently emphasised, will stimulate interest in investigations of this kind.
(4) The concept of the Cartesian product with any number of factors, as well as the notions of the different types of independence have arisen from problems of probability theory and play a major role in the characterisation of the structure of the probability fields. In this direction, much has been done by Polish mathematicians. We dedicate the remaining chapters of our booklet to these terms. We shall report on the non-separable invariant extensions of the linear Lebesgue measure. This interesting theory, which is also closely related to the concept of independence, is due to S Kakutani and J C Oxtoby. Recently, A Renyi has introduced a generalisation of the concept of probability space, which is significant for many applications of probability theory. At the end of the booklet we explain a lot about it, especially about the structural theory of such spaces. A direct formulation of this theory would be desirable.
- Probability algebras and stochastic spaces (1969), by Demetrios Kappos.
2.1. Publisher's Description.
Probability Algebras and Stochastic Spaces explores the fundamental notions of probability theory in the so-called point-free way. The space of all elementary random variables defined over a probability algebra in a point-free way is a base for the stochastic space of all random variables, which can be obtained from it by lattice-theoretic extension processes. This book is composed of eight chapters and begins with discussions of the definition, properties, scope, and extension of probability algebras. The succeeding chapters deal with the Cartesian product of probability algebras and the principles of stochastic spaces. These topics are followed by surveys of the expectation, moments, and spaces of random variables. The final chapters define generalized random variables and the Boolean homomorphisms of these variables. This book will be of great value to mathematicians and advance mathematics students.
2.2. Extract from Preface.
The origin of this book is in a set of lectures which I gave in the academic year 1963- 64 at the Catholic University of America, Washington, D.C. The Statistical Laboratory there issued a mimeographed version of my notes under the title "Lattices and their Applications to Probability". The present text is a revised and expanded version of these notes, maintaining the central mathematical ideas of the lectures, namely probability algebras and stochastic spaces (i.e., spaces of random variables). The part of the notes devoted to pure lattice theory has been shortened and the most important concepts and theorems of this theory have been staled in two appendices, mostly without proofs. In addition to the material in the mimeographed edition, the present volume contains a general way to introduce the concept of random variables taking values in spaces endowed with any algebraic or topological structure. In particular, we study the cases in which the space of the values is a lattice group, or vector lattice. When the space of the values is a Banach space, a theory of expectation and moments is stated. A theory of expectation can be easily stated in more general cases of spaces of values: for example, locally convex vector spaces or topological vector spaces, for which an integration theory is known. We have restricted ourselves on the introduction and study only of the fundamental mathematical notions. We mention only a few facts about the concepts of independence and conditional probabilities and expectations. Certainly, it would be interesting to state the theory of stochastic processes and, especially, the theory of martingales. But this would go beyond the scope of the present monograph, or it would have to be published in a second volume. The author wishes to express his gratitude to Dr Eugene Lukacs of the Catholic University, who made it possible for him to give lectures and publish them. It is a pleasure to offer thanks to F Papangelou and G Anderson who read critically the manuscript of the lectures and made valuable suggestions during the mimeographed edition of them.