Long Sequences and Neocompact Sets

Users of non-standard methods in mathematics have always been interested in the following question: what does non-standard analysis offer the mathematical community? The issues raised by this question are neverending and there is a whole spectrum of possible answers. The paper [HK] offered an explanation from the point of view of logic, explaining how the principles used in the superstructure approach to nonstandard analysis are related to standard mathematical practice. One difficulty is that the mathematical community is not agreed on what “standard mathematical practice” is. [HK] used mathematical logic to provide a formal framework where these issues can be discussed. The question posed above was approached from a different point of view in the series of papers beginning with [K1] and continuing with [FK1], [CK], [FK2], [K2] and [K3]. This series of papers develops the notion of a neometric space, and the whole program is explained in the survey paper [K6] in this volume. The approach may be intuitively described as follows. Start from a part of mathematics, probability theory, where nonstandard methods have clearly offered new insights and enriched the field with new and interesting results. Then isolate and present in “standard terms” those features of nonstandard practice that have made this success possible. The results appeared in [FK1] and [FK2] where the notions of neocompact sets and neometric families were presented, and the basic mathematical theory around these new concepts was developed. A few words about these two papers will help to explain our reason for writing the present paper. In [FK1], entitled “Existence Theorems in Probability Theory”, we developed a standard theory which captured the key elements from nonstandard analysis that made it possible to prove new existence theorems in stochastic analysis (see [AFHL], [K4] and [K5]). Using neocompact sets and neometric spaces we introduced a new class of probability spaces called “Rich Probability Spaces” and then proceeded to show that in those spaces the results obtained using nonstandard