Developments in Nonstandard Mathematics

Nigel J Cutland, Vitor Neves, A F Oliveira, Jose Sousa-Pinto

• Publisher: Chapman and Hall/CRC
• Year: 2020
• eBook ISBN: 9781000724646
• Topic: Mathematics
• Subtopic: Differential Equations
• Index: Mathematics

This book contains expository papers and articles reporting on recent research by leading world experts in nonstandard mathematics, arising from the International Colloquium on Nonstandard Mathematics held at the University of Aveiro, Portugal in July 1994. Nonstandard mathematics originated with Abraham Robinson, and the body of ideas that have developed from this theory of nonstandard analysis now vastly extends Robinson's work with infinitesimals. The range of applications includes measure and probability theory, stochastic analysis, differential equations, generalised functions, mathematical physics and differential geometry, moreover, the theory has implicaitons for the teaching of calculus and analysis. This volume contains papers touching on all of the abovbe topics, as well as a biographical note about Abraham Robinson based on the opening address given by W.A>J> Luxemburg - who knew Robinson - to the Aveiro conference which marked the 20th anniversary of Robinson's death. This book will be of particular interest to students and researchers in nonstandard analysis, measure theory, generalised functions and mathematical physics.

NIGEL CUTLAND
Loeb measure theory

1. Introduction

Loeb measures, discovered by Peter Loeb in 1975 [32], are very rich yet tractable measure spaces, which play a central and powerful rôle in many applications of nonstandard analysis – in measure and probability theory, stochastic analysis, functional analysis, mathematical physics, economics and mathematical finance theory. At the same time their richness and simple construction makes them a new and fascinating class of structures worthy of study in their own right.

In this article we give an introduction to the basics of Loeb measure and integration theory, designed to make the extensive literature that uses these notions accessible. In particular we will illustrate the way in which Loeb measures can be used to construct or represent standard measures (such as Lebesgue measure or Wiener measure) in a very simple way – often by means of counting measures. This is in many cases the key to their power and usefulness. In the final section we give a sample of applications to differential equations.

We work with a nonstandard universe that is ℵ₁-saturated, and where necessary assume extra saturation. To expand on this slightly, we suppose given a superstructure

$V\left(S\right)={\displaystyle \underset{n\in ℝ}{\cup }{V}_n\left(S\right)}$

where S is a non-empty set (usually S = ℝ is sufficient)...

Table of contents

• Cover
• Pitman Research Notes in Mathematics Series
• Title Page
• Copyright Page
• Dedication Page
• Abraham Robinson a Biographical Note
• Table of Contents
• Preface
• The infinitesimal rule of three
• Nonstandard methods in the precalculus curriculum
• Difference quotients and smoothness
• Continuous maps with special properties
• Some nonstandard methods in geometric topology
• Delayed bifurcations in perturbed systems analysis of slow passage of Suhl-threshold
• Functional analysis and NSA
• Near-Standard compact internal linear operators
• Discrete Fredholm’s equations
• Nonstandard theory of generalized functions
• Representing distributions by nonstandard polynomials
• Contributions of nonstandard analysis to partial differential equations
• Loeb measure theory
• Unions of Loeb nullsets: the context
• Gradient lines and distributions of functionals in infinite dimensional Euclidean spaces
• Nonstandard flat integral representation of the free Euclidean field and a large deviation bound for the exponential interaction
• Nonstandard analysis in selective universes
• Lattices and monads
• A neometric survey
• Long sequences and neocompact sets