The 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.

lgrsnf/Hurd A.E., Loeb P.A. An introduction to nonstandard real analysis (AP, 1985)(ISBN 0123624401)(T)(O)(247s)_MC_.djvu

Hurd, Albert E., Loeb, Peter A.

edited by A.E. Hurd, P.A. Loeb

Albert Emerson Hurd

Morgan Kaufmann Publishers

Brooks/Cole

Pure and applied mathematics -- v.118, Orlando, United States, 1985

United States, United States of America

Elsevier Ltd., Orlando, Fla, 1985

Orlando, Florida, 1985

Bibliography: p. 222-224.
Includes index.

Includes bibliographical references and index.

Cover
Title Page
Contents
Preface
I Infinitesimals and The Calculus
I.1 The Hyperreal Number System as an Ultrapower
I.2 *-Transforms of Relations
I.3 Simple Languages for Relational Systems
I.4 Interpretation of Simple Sentences
I.5 The Transfer Principle for Simple Sentences
I.6 Infinite Numbers, lnfinitesimals, and the Standard Part Map
I.7 The Hyperintegers
I.8 Sequences and Series
I.9 Topology on the Reals
I.10 Limits and Continuity
I.11 Differentiation
I.12 Riemann Integration
I.13 Sequences of Functions
I.14 Two Applications to Differential Equations
I.15 Proof of the Transfer Principle
II Nonstandard Analysis on Superstructures
II.1 Superstructures
II.2 Languages and Interpretation for Superstructures
II.3 Monomorphisms between Superstructures: The Transfer Principle
II.4 The Ultrapower Construction for Superstructures
II.5 Hyperfinite Sets, Enlargements, and Concurrent Relations
II.6 Internal and External Entities; Comprehensiveness
II.7 The Permanence Principle
II.8
III Nonstandard Theory of Topological Spaces
III.1 Basic Definitions and Results
III.2 Compactness
III.3 Metric Spaces
III.4 Normed Vector Spaces and Banach Spaces
III.5 Inner-Product Spaces and Hilbert Space
III.6 Nonstandard Hulls of Metric Spaces
III.7 Compactifications
III.8 Function Spaces
IV Nonstandard Integration Theory
IV.1 Standardizations of Internal Integration Structures
IV.2 Measure Theory for Complete Integration Structures
IV.4 Basic Convergence Theorems
IV.5 The Fubini Theorem
IV.6 Applications to Stochastic Processes
Appendix: Ultrafilters
References
List of Symbols
Index

2024-07-31