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An introduction to nonstandard real analysis. (English) Zbl 0583.26006

Pure and Applied Mathematics, 118. Orlando etc.: Academic Press, Inc. (Harcourt Brace Javanovich, Publishers). XII; 232 p. $ 35.00; £31.00 (1985).
Since the discovery in the early sixties by the late Abraham Robinson of non-standard analysis a number of textbooks and monographs on the subject have appeared.
As the authors state in the Preface, the aim of the present book is to present Robinson’s idea at the level of students with merely a background in undergraduate mathematics. Apparently the second named author has used with success some versions of the manuscript in several beginning graduate courses. Upon closer examination of the book it seems, however, that Chapter I and certain parts of the other chapters could very well be used in an advanced undergraduate course.
The book contains four chapters. In Chapter I, entitled ”Infinitesimals and the calculus” the authors carefully ease the reader into the new world of infinitesimals and infinitely large numbers and their usage in the calculus. The method used for this purpose is that of over the years proven approach via ultrapowers of the real number system. The parts of the theory of ultrafilters needed for this purpose are carefully explained in an Appendix at the end of the book.
Chapter II is devoted to the treatment of non-standard analysis of superstructures by means of non-standard models in the form of ultrapowers of such structures. The material in this chapter requires a bit more mathematical sophistication on the part of the reader. But because it is all so carefully explained it should not deter the reader to go on with the rest of the book. Particularly, because this chapter prepares the reader for applying Robinson’s ideas to other mathematical structures than the reals. How this can be done fruitfully is explained in the last two chapters of the book that are devoted to the non-standard theory of topological spaces and integration theory respectively.
The applications to point set topology includes the basic facts of non- standard Banach space theory which is playing an increasingly more important role in functional analysis.
The chapter dealing with integration theory contains a lot of material which so far has not yet been available in book form. As examples we may mention a non-standard treatment of the Riesz representation theorem, Fubini’s theorem Egoroff’s theorem and stochastic processes.
The book finishes with an adequate list of references, a useful list of symbols and an extensive useful index.
The book is a welcome addition to the non-standard literature. It is a well-written and excellently organized introduction to the theory of infinitesimals. We wish it many readers.
Reviewer: W.A.J.Luxemburg

MSC:

26E35 Nonstandard analysis
26-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to real functions
03H05 Nonstandard models in mathematics
46S20 Nonstandard functional analysis