This is a most inspired and inspiring book.

When it was written two of Wiener's major papers had just appeared: Generalized Harmonic Analysis, and Tauberian Theorems. His previous works on potential theory, Brownian motion, Fourier analysis were highly appreciated by a few dozens of mathematicians in the world. It was a happy time in the life of Norbert Wiener. He was thirty-seven years old, he had been married for six years, he had just been promoted to a professorship at the Massachusetts Institute of Technology, and he had spent a pleasant year in Cambridge, England, with his wife Margaret and their small daughters, Barbara and Peggy.

Cambridge (I mean, Cambridge, England) plays a special role in Wiener's life. The beginning of his mathematical career coincided with his meeting with Bertrand Russell in Cambridge in 1913. He had just graduated from Harvard in mathematical logic – at only eighteen – and was supposed to go on in logic with Russell. Actually Russell's advice was to learn more mathematics and physics. Accordingly, Norbert Wiener took courses with Hardy, Littlewood, Mercer, and read Einstein's papers of 1905, and Niels Bohr's recent works. Without any doubt Einstein's theory of Brownian motion had a decisive role in Wiener's inspiration.

The academic year 1931–32 in Cambridge was an opportunity for Wiener to meet Hardy and Littlewood again, to lecture on Fourier integrals (in Hardy's chair) and to discover an ideal collaborator, the young R. E. A. C. Paley.

The present book is in substance an elaboration of a course of fifteen lectures on the Fourier Integral and its Applications, given at the University of Cambridge during the Lent Term of 1932. When I arrived in Cambridge during the Michaelmas Term of 1931, on leave of absence from the Massachusetts Institute of Technology, I had vague plans of writing up certain topics in the theory of harmonic analysis into a book on the subject. My original idea was of a rather comprehensive treatise, proceeding from the elements of Lebesgue integration through the L2 theory of the Fourier series to the Plancherel theorem, the Fourier Integral, the periodogram, and lastly, to theorems of Tauberian type. My impulse to write a book of this type arose from a dissatisfaction with the preponderant rôle of convergence theory in existing textbooks on the subject, and from the need for a treatment more in line with the extensive periodical literature.

As far as my desire to write a book sprang from the need for a textbook to use in my course at the Massachusetts Institute of Technology, it has largely been dissipated by the recent appearance of a book on the Theory of Functions by Professor Titchmarsh. Several chapters of his book are devoted to the treatment of Fourier series from the modern point of view. Unfortunately—from my standpoint—he does not allot a great deal of space to the Fourier Integral and related matters.

INTRODUCTION

Nonstandard Analysis – or the Theory of Infinitesimals as some prefer to call it – is now a little more than 25 years old (see Robinson (1961)). In its early days it was often presented as a surprising solution to the old and – it had seemed – impossible problem of providing infinitesimal methods in analysis with a logical foundation. It soon became clear, however, that the theory was much more than just a reformulation of the Calculus, when Bernstein and Robinson (1966) gave the first indication of its powers as a research tool by proving that all polynomially compact operators on Hilbert spaces have nontrivial invariant subspaces. Since then nonstandard techniques have been used to obtain new results in such diverse fields as Banach spaces, differential equations, probability theory, algebraic number theory, economics, and mathematical physics just to mention a few. Despite the wide variety of topics involved, these applications have enough themes in common that it is natural to regard them as examples of the same general method.

This paper is intended as an exposition of these recurrent themes and the theory uniting them. I have called it “An invititation to nonstandard analysis” because it is meant as an invitation – a friendly welcome requiring no other background than a smattering of general mathematical culture. My point of view is that of applied nonstandard analysis; I'm interested in the theory as a tool for studying and creating standard mathematical structures.

The present book has been written so as to necessitate as little consultation by the reader as reasonably possible of other published material. I have hoped to thereby make it accessible to people far from large research centres or any ‘good library”, and to those who have only their summer vacations to work on mathematics. It is for the same reason that references, where unavoidable, have been made to books rather than periodicals whenever that could be done.

In general, I consider the developments leading up to the various results in the book to be more important than the latter taken by themselves; that is why those developments are set out in more detail than is now customary. My aim has been to enable one to follow them by mostly just reading the text, without having to work on the side to fill in gaps. The reader's active participation is nevertheless solicited, and problems have been given. These are usually accompanied by hints (sometimes copious), so that one may be encouraged to work them out fully rather than feeling stymied by them. It is assumed that the reader's background includes, beyond ordinary undergraduate mathematics, the material which, in North America, is called graduate real and complex variable theory (with a bit of functional analysis). Practically everything needed of this is contained in Rudin's well-known manual.

INTRODUCTION

Nonstandard analysis is a specific mathematical technique as well as a way of thinking: both aspects are also represented in the interaction between nonstandard analysis and mathematical physics, which is the subject of this paper. As in other domains of application of nonstandard analysis, mathematical physics (or the mathematical study of problems of physics) has particular aspects that make some of the nonstandard methods most natural to use. Often in mathematical physics one has to study systems with many interacting components, idealized as systems with infinitely many degrees of freedom. To look upon a fluid or a gas as a composed of infinitely many particles might seem at first sight to be a very rough abstraction, but is a useful one for mathematical purposes, being in some sense easier to handle than the more realistic case of finitely many particles. On the other hand, in quantum field theory, for example, the abstraction itself creates its own problems, like the famous ones connected with divergences, about which we will say a few more words below; sometimes also it is only in a limit, like that of infinitely many degrees of freedom, that one “sees” some specific phenomenon, raising challenging problems, like phase transitions in thermodynamic systems (only perceived in the so called ”infinite volume“ or “thermodynamic limit”),or exact invariance properties (under a continuous group of symmetries), in systems idealized as “continua” (as in field theories).

The methods of Abraham Robinson's Nonstandard (or Infinitesimal) Analysis (NSA) are currently being used across the whole spectrum of mathematics – from ‘pure’ mathematics through to mathematical physics. This book is designed as an introduction to NSA and to some of its many applications, with the working mathematician or student particularly in mind. It has emerged from a conference with the same title held at the University of Hull in 1986, which had the aim of making NSA more widely known in the mathematical community through a series of introductory lecture courses and lectures on current research. The first part of this book consists of papers based on the introductory lectures given at the conference by Tom Lindstrøm, Ward Henson, Jerry Keisler and Sergio Albeverio. The latter part of the book contains papers that present a sample of recent developments in the more advanced applications of NSA.

Lindstrøm's An Invitation to Nonstandard Analysis expounds the foundations of the theory. It is designed to be “a friendly welcome requiring no other background than a smattering of general mathematical culture”, offered in the belief that NSA “is of greater interest to the analyst than to the logician”. Lindstrøm writes “I have tried to make the subject look the way it would had it been developed by analysts or topologists and not logicians.” To this end, his presentation of NSA is somewhat different from others in the literature, in that he builds a nonstandard universe and shows how to practice NSA without any use of logic.

The aim of this article is to explain how and why the Loeb measure construction can be applied to problems which arise outside of nonstandard analysis. The Loeb measure and the necessary background from nonstandard analysis are presented in the paper of Lindstrøm in this volume. Most of the applications of Loeb measures in probability theory fit the following pattern. First, lift the original classical problem to a hyperfinite setting. Second, make some hyperfinite computations. Third, take standard parts of everything in sight to obtain the desired classical result. In keeping with this pattern, we shall concentrate on hyperfinite Loeb spaces, that is, Loeb spaces on hyperfinite sets. As we go along, we shall present several examples of applications of hyperfinite Loeb spaces to probability theory. However, our main emphasis will be on the general theory which underlies the applications.

We shall begin with a study of random variables on probability spaces, and then pass to a deeper parallel theory of stochastic processes on adapted spaces. The hyperfinite adapted spaces are counterparts of the hyperfinite Loeb spaces. Corresponding to the distribution of a random variable, we introduce the adapted distribution of a stochastic process. We shall focus on three features of hyperfinite adapted spaces: liftings, universality, and homogeneity. These features help to explain why the applications work. The adapted lifting theorem shows that standard stochastic processes can be lifted to hyperfinite processes with almost the same adapted distribution.