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Neometric Spaces
, 1996
"... this paper is quite the opposite: we work within nonstandard analysis to formally prove, as promised in [9], that rich adapted spaces exist. Moreover, we explicitly show how nonstandard analysis provides the inspiration for the main notions and results presented in [9]. The paper [9] gives a large n ..."
Abstract

Cited by 12 (9 self)
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this paper is quite the opposite: we work within nonstandard analysis to formally prove, as promised in [9], that rich adapted spaces exist. Moreover, we explicitly show how nonstandard analysis provides the inspiration for the main notions and results presented in [9]. The paper [9] gives a large number of applications of rich adapted spaces to probability theory. Nonstandard analysis gives us tools to dig deeper into the structure of subsets of metric spaces. The purpose of this paper is to refine these tools. The contents of this paper are as follows. In Section 2 we review the basic notions concerning neometric spaces from [9]. In Sections 3 and 4 we study these notions within the nonstandard setting. We give an explicit definition of basic and neocompact sets that captures the way internal sets are used in nonstandard probability practice, and then present a huge neometric family which contains all neometric spaces studied so far. In Section 5 we prove that the huge neometric family contains rich adapted spaces, and hence that rich adapted spaces exist. Section 6 gives a detailed study of the function spaces related to the existence theorems in [9]. In Section 7 we consider neometric spaces whose elements are functions from a probability space into another neometric space. We finish up in Section 8 with a stronger theory of neometric spaces which requires a saturated nonstandard universe where is a cardinal greater than ! 1 . In other publications we will present other aspects of our work. In [10] we develop the logical aspects that are behind our results. In [11] we discuss another nonstandard approach to neometric families which uses long sequences. The paper [17] gives a general quantifier elimination result showing that for many neometric families, every n...
Foundations Of Nonstandard Analysis  A Gentle Introduction to Nonstandard Extemsions
 In Nonstandard analysis (Edinburgh
"... this paper is to describe the essential features of the resulting frameworks without getting bogged down in technicalities of formal logic and without becoming dependent on an explicit construction of a specific field ..."
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Cited by 10 (2 self)
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this paper is to describe the essential features of the resulting frameworks without getting bogged down in technicalities of formal logic and without becoming dependent on an explicit construction of a specific field