In classical analysis, the approach to general integration is either through measure theory or through functional analysis with a lattice formulation of the integral. Starting from the notion of “length”, for example, one constructs Lebesgue measure and the class of Lebesgue measurable sets. Lebesgue measurable functions form an extension of the class of simple functions (linear combinations of characteristic functions of sets of finite measure), and the Lebesgue integral extends the obvious calculation for simple functions. Long before seeing this development, however, each student of mathematics has used the notion of length to obtain the Riemann integral. We wisely refrain from telling our calculus students that the Riemann integral is a positive linear functional on the space of continuous functions with compact support and is, therefore, represented by a measure. That measure is, of course, Lebesgue measure; it is obtained by extending the Riemann integral from the continuous functions with compact support to the class of measurable functions and then noting the action on the characteristic functions of measurable sets. Here, the Lebesgue integral is constructed before Lebesgue measure.

The major theorem generalizing the extension of “length” to Lebesgue measure is the Caratheodory Extension Theorem, and the major theorem associated with constructing a measure representing the Riemann integral (or any positive linear functional on continuous functions) is the Riesz Representation Theorem. These theorems have played an important role in the development of nonstandard measure theory. The Caratheodory theorem was used in a measure theoretic approach (Loeb (1975)) to extend internal measures to standard ones.

INTRODUCTION

In (1977) Edward Nelson gave a new formulation of Abraham Robinson's Theory of Infinitesimals known as Internal Set. Theory. In (1980) Nelson refined this to give a solution to Robinson's (1973) Metamathematical Problem 11.

Nelson's approach to infinitesimal analysis has been taken up by a large number of workers in various fields. Lawler (1980) has obtained interesting results on a kind of self-avoiding random walk. The article by the Dieners below describes some of the many applications of Internal Set Theory to the study of differential equations. The article by Stroyan describes some extensions of Nelson's methods which are useful in topology and functional analysis. This article is an introduction to the ones by the Dieners and Stroyan.

This article is also a description of the common ground shared by the two approaches to Robinson's theory. We hope that our short presentation of 1ST and its interpretation in a superstructure will help those familiar with superstructures and those familiar with 1ST to understand each other.

In section 2 we give a brief introduction to 1ST (see references at the end for more complete introductions). In section 3 we restrict Nelson's methods to a superstructure as described above by Lindstrøm. This means that we only consider predicates from Nelson's formal language whose quantifiers are bounded by a standard entity. With this restriction, we shall prove in section 4 that Nelson's axiom schemes (I), (S) and (T) hold in a superstructure.

We shall work in Lindstrøm's polysaturated nonstandard model V(*S). We also need some of the syntactical ideas from the article of Diener & Stroyan in this volume. We shall use the notation and Syntactical Theorem on Monads from that article.

MONADS

If U is an entity of V(S), the set of standard elements of U is denoted σU = {U ∈ *U : st(U)} = {*V : V ∈ U}.

Definition

A nonempty set µ ∈ V(*S) is called a monad if and only if there is a standard family U of subsets of a standard set W such that

The monad µ ⊇ *W is nonempty if and only if the corresponding standard family tU is a filter base on W.

INTRODUCTION

The methodological background of this paper is the theory of rewrite rules, although it is never mentioned explicitly. Finite complete sets of rewrite rules for a finitely presented algebra solve a number of decision problems for this algebra such as the word problem, the finiteness problem or the existence of a nilpotent subgroup of finite index. These problems remain solvable also for many infinite complete systems; what really matters is that the irreducible words form a regular set in the sense of automata theory. The regularity of the set of irreducible words can be regarded as an asymptotic aspect of the so-called Knuth-Bendix completion procedure. This is connected with an ordering on the words and some of these orderings are the central subject of our investigation. For more information on these topics we refer, for example to Benninghofen, Kemmmerich & Richter (1987) or to Richter (1987). In Richter (1987) one finds also a proof of the main result of this paper in a much more restricted case.

The main result can be found in section 4 and shows the non-regularity of the minimal words of the free nilpotent group of class 2 for a certain set of widely used orderings. We study a nonstandard model of this group (which was introduced in van den Dries & Wilkie (1984) in the context of Gromov's theorem) and its nonstandard hull and use a connection with the growth function of the group and with (nonstandard) automata theory.

INTRODUCTION

Consider the Boltzmann equation, that famous kinetic model of rarefied gases driven by binary collisons. Since a linearization would remove some of the interesting behaviour, we shall here retain the truly non-linear collision mechanism.

A fairly common characteristic of non-linear evolution equations is that the behaviour close to equilibrium is smooth and regular with nice asymptotic convergence, whereas initial values further away may result in wilder, possibly non-smooth behaviour. For space-dependent Boltzmann gases in full space or bounded containers, various contraction mapping estimates can be used to prove the existence of unique, smooth solutions converging to an equilibrium (Maxwellian) value with time, if the initial values are close enough to equilibrium. Such methods break down when the initial values are further away from equilibrium, and so do natural compactness arguments.

For another approach, let us first recall the physical background. From the point of view of physics, what happens at distances or within volumes below, say, the scale of elementary particle phenomena, is an artefact of the model with little experimental relevance. From this perspective, the question whether the model starts from an underlying set of rationally, really, or infinitesimally spaced points, should be decided purely on mathematical grounds.

The aim of this article is to provide, when combined with the survey paper Henson & Moore (1983), a fairly complete description of the nonstandard hull construction and of the most important ways in which nonstandard methods have been used to solve problems in functional analysis. Most of the material concerning Banach spaces is already covered in that earlier survey; there are a few important recent developments, which we have included in the last two sections of this paper. Here we will, however, concentrate on the general nonstandard hull construction for topological vector spaces and for operators on such spaces. We have also tried to include here some more elementary variations on arguments which appeared there, and it may well be that this paper can in part serve as an introduction to the Banach space survey. (But also the reverse may be true for some readers.)

The nonstandard hull construction applied to topological vector spaces and to continuous operators on them plays very much the same role in functional analysis that the Loeb measure construction does in probability theory. It provides a systematic (functorial) procedure for obtaining a topological vector space or continuous operator (in the usual mathematical sense) from internal spaces and operators. Moreover, in the setting of functional analysis there is an elaborate and important structure of infinitesimals and finite points, which provide an elegant framework for the expression of complicated topological concepts, as well as for the study of the nonstandard hulls themselves.