Nonstandard analysis
The content of this book goes way beyond its title. The authors show the use of non-standard methods in analysis, topology, and stochastics, as well as develop enough of the underlying logic. The text should be successful in an advanced undergraduate or graduate course, because it presumes little background, proceeds slowly and carefully, is well- written and easy to understand, and contains a fair number of exercises with hints or solutions at the end. Still, a surprisingly large amount of material is covered. The non-standard reals are constructed as an ultrapower of the standard reals; this approach is then contrasted at the end of the text with Nelson's axiomatic development of a non-standard universe. The section on analysis begins with the most elementary properties of the hyperreals, and includes integration, differentiation, sequences of functions, and distributions. The topology part assumes no knowledge of the subject, and besides introducing point-set topology also develops metric and normed spaces. The material on stochastic processes begins with a lot of measure theory, both classical measure and Loeb measure, before culminating in Brownian motion.
scientific article; zbMATH DE number 542030