An Introduction to Nonstandard Real AnalysisThe aim of this book is to make Robinson's discovery, and some of the subsequent research, available to students with a background in undergraduate mathematics. In its various forms, the manuscript was used by the second author in several graduate courses at the University of Illinois at Urbana-Champaign. The first chapter and parts of the rest of the book can be used in an advanced undergraduate course. Research mathematicians who want a quick introduction to nonstandard analysis will also find it useful. The main addition of this book to the contributions of previous textbooks on nonstandard analysis (12,37,42,46) is the first chapter, which eases the reader into the subject with an elementary model suitable for the calculus, and the fourth chapter on measure theory in nonstandard models. |
Contents
| 1 | |
Chapter II Nonstandard Analysis on Superstructures | 70 |
Chapter III Nonstandard Theory of Topological Spaces | 109 |
Chapter IV Nonstandard Integration Theory | 164 |
Ultrafilters | 219 |
| 222 | |
| 225 | |
| 227 | |
Pure and Applied Mathematics | 233 |
Common terms and phrases
assume Banach space bounded called Cauchy sequence Chapter closed compact construction contains continuous functions Corollary countable defined Definition Let denote element entities equivalence classes example Exercise exists extension f is continuous finite number finite set Fubini property function f given Hausdorff hence Hilbert space hyperfinite hyperreal infinite infinitesimal inner-product space internal integration structure internal set interpretable k-saturated Lemma Let f linear mathematical measure metric space monad monomorphism monotone convergence theorem natural numbers near-standard nonstandard analysis normed vector space notion open sets proof of Theorem Proposition Prove reader real numbers real-valued functions relation satisfies Show simple sentence Skolem function ſº standard subspace superstructure Suppose symbols Theorem Let theory tion topological space topology transfer principle true ultrafilter ultrapower variables x e a,b x e H