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Nonstandard Analysis for the Working Mathematician

  • Book
  • Jun 2000

Overview

Editors:
  1. Peter A. Loeb

    1. Department of Mathematics, University of Illinois at Champaign-Urbana, Champaign-Urbana, USA

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  2. Manfred Wolff

    1. Faculty of Mathematics, Universität Tübingen, Tübingen, Germany

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    You can also search for this editor in PubMed   Google Scholar

Part of the book series: Mathematics and Its Applications (MAIA, volume 510)

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Table of contents (8 chapters)

  1. Front Matter

    Pages i-xiv
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  2. An Introduction to Nonstandard Analysis by Peter A. Loeb

    1. Front Matter

      Pages 1-1
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    2. A Simple Introduction to Nonstandard Analysis

      • Peter A. Loeb
      Pages 3-33

    3. An Introduction to General Nonstandard Analysis

      • Peter A. Loeb
      Pages 35-72

    4. Topology and Measure Theory

      • Peter A. Loeb
      Pages 73-95

  3. Functional Analysis by Manfred P. H. Wolff

    1. Front Matter

      Pages 97-97
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    2. Functional Analysis

      • Manfred P. H. Wolff
      Pages 99-136

  4. Measure and Probability Theory and Applications by Horst Osswald and Yeneng Sun

    1. Front Matter

      Pages 137-137
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    2. Measure Theory and Integration

      • Horst Osswald
      Pages 139-169

    3. Probability Theory

      • Horst Osswald
      Pages 171-234

    4. Conventional Operations on Nonstandard Constructions

      • Yeneng Sun
      Pages 235-257

  5. Economics and Nonstandard Analysis by Yeneng Sun

    1. Front Matter

      Pages 259-259
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    2. Nonstandard Analysis in Mathematical Economics

      • Yeneng Sun
      Pages 261-305

  6. Back Matter

    Pages 306-311
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Keywords

About this book

This book is addressed to mathematicians working in analysis and its applications. The aim is to provide an understandable introduction to the basic theory of nonstan­ dard analysis in Part I, and then to illuminate some of its most striking applications. Much of the book, in particular Part I, can be used in a graduate course; problems are posed in all chapters. After Part I, each chapter takes up a different field for the application of nonstandard analysis, beginning with a gentle introduction that even non-experts can read with profit. The remainder of each chapter is then addressed to experts, showing how to use nonstandard analysis in the search for solutions of open problems and how to obtain rich new structures that produce deep insight into the field under consideration. The applications discussed here are in functional analysis including operator theory, probability theory including stochastic processes, and economics including game theory and financial mathematics. In all of these areas, the intuitive notion of an infinitely small or infinitely large quantity plays an essential and helpful role in the creative process. For example, Brownian motion is often thought of as a random walk with infinitesimal increments; the spectrum of a selfadjoint operator is viewed as the set of "almost eigenvalues"; an ideal economy consists of an infinite number of agents each having an infinitesimal influence on the economy. Already at the level of calculus, one often views the integral as an infinitely large sum of infinitesimal quantities.

Editors and Affiliations

  • Department of Mathematics, University of Illinois at Champaign-Urbana, Champaign-Urbana, USA

    Peter A. Loeb

  • Faculty of Mathematics, Universität Tübingen, Tübingen, Germany

    Manfred Wolff

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