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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
Inverse Problem For Upper Asymptotic Density II
"... Inverse problems study the structure of a set A when the "size" of A + A is small. In the article, the structure of an infinite set A of natural numbers is described when A + A has the least possible upper asymptotic density and A contains two consecutive numbers. For example, if the upper asymptot ..."
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Cited by 9 (6 self)
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Inverse problems study the structure of a set A when the "size" of A + A is small. In the article, the structure of an infinite set A of natural numbers is described when A + A has the least possible upper asymptotic density and A contains two consecutive numbers. For example, if the upper asymptotic density # of A is between 0 and , the upper asymptotic density of A + A is less than or equal to #, and A contains two consecutive numbers, then A is either a large subset of the union of two arithmetic sequences with same common di#erence k = or for any increasing sequence h n of positive integers such that the relative density of A in [0, h n ] approaches to #, the set A# [0, h n ] can be partitioned into two parts A [0, c n ] and A [b n , h n ] such that c n /h n approaches to 0, i.e. the cardinality of A [0, c n ] is relatively very small, and (h n b n )/h n approaches to #, i.e. the cardinality of A [b n , h n ] is relatively same as the cardinality of the interval [b n , h n ]. 1
Sumset phenomenon
 Proceedings of American Mathematical Society, 130, No.3
, 2002
"... Abstract. Answering a problem posed by Keisler and Leth, we prove a theorem in non–standard analysis to reveal a phenomenon about sumsets, which says that if two sets A and B are large in terms of “measure”, then the sum A+B is not small in terms of “order–topology”. The theorem has several corollar ..."
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Cited by 8 (3 self)
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Abstract. Answering a problem posed by Keisler and Leth, we prove a theorem in non–standard analysis to reveal a phenomenon about sumsets, which says that if two sets A and B are large in terms of “measure”, then the sum A+B is not small in terms of “order–topology”. The theorem has several corollaries about sumset phenomenon in the standard world; these are described in sections 2–4. One of these is a new result in additive number theory; it says that if two sets A and B of non–negative integers have positive upper or upper Banach density, then A + B is piecewise syndetic. 1. A theorem in non–standard analysis Let ∗V be a non–standard extension of a standard universe V, which contains all standard real numbers. The reader may consult [7] or [3] for basic knowledge of non–standard analysis. We denote by N the set of all standard non–negative integers, and denote by ∗N the set of all non–negative integers in ∗V. All integers in ∗N � N are called hyperfinite integers. For any two sets A and B, andabinary operator ◦ we write A ◦ B for the set {a ◦ b: a ∈ A and b ∈ B}. An infinite initial
Ultraproducts in Analysis
 IN ANALYSIS AND LOGIC, VOLUME 262 OF LONDON MATHEMATICAL SOCIETY LECTURE NOTES
, 2002
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Noncooperative games on hyperfinite Loeb spaces
, 1999
"... We present a particular class of measure spaces, hyperfinite Loeb spaces, as a model of situations where individual players are strategically negligible, as in large nonanonymous games, or where information is diffused, as in games with imperfect information. We present results on the existence of ..."
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Cited by 6 (4 self)
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We present a particular class of measure spaces, hyperfinite Loeb spaces, as a model of situations where individual players are strategically negligible, as in large nonanonymous games, or where information is diffused, as in games with imperfect information. We present results on the existence of Nash equilibria in both kinds of games. Our results cover the case when the action sets are taken to be the unit interval, results now known to be false when they are based on more familiar measure spaces such as the Lebesgue unit interval. We also emphasize three criteria for the modelling of such gametheoretic situations asymptotic implementability, homogeneity and measurabilityand argue for games on hyperfinite Loeb spaces on the basis of these criteria. In particular, we show through explicit examples that a sequence of finite games with an increasing number of players or sample points cannot always be represented by a limit game on a Lebesgue space, and even when it can be so rep...
The Complete Removal of Individual Uncertainty: Multiple Optimal Choices and Random Exchange Economies
, 1999
"... this paper is to develop some measuretheoretic methods for the study of large economic systems with individualspecific randomness and multiple optimal actions. In particular, for a suitably formulated continuum of correspondences, an exact version of the law of large numbers in distribution is cha ..."
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Cited by 4 (1 self)
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this paper is to develop some measuretheoretic methods for the study of large economic systems with individualspecific randomness and multiple optimal actions. In particular, for a suitably formulated continuum of correspondences, an exact version of the law of large numbers in distribution is characterized in terms of almost independence, which leads to several other versions of the law of large numbers in terms of integration of correspondences. Widespread correlation due to multiple optimal actions is also shown to be removable via a redistribution. These results allow the complete removal of individual risks or uncertainty in economic models where nonunique best choices are inevitable. Applications are illustrated through establishing stochastic consistency in general equilibrium models with idiosyncratic shocks in endowments and preferences. In particular, the existence of "global" solutions preserving microscopic independence structure is shown in terms of competitive equilibria for the cases of divisible and indivisible goods as well as in terms of core for a case with indivisible goods where a competitive equilibrium may not exist. An important feature of the idealized equilibrium models considered here is that standard results on measuretheoretic economies are now directly applicable to the case of random economies. Some asymptotic interpretation of the results are also discussed. It is also pointed out that the usual unit interval [0, 1] can be used as an index set in our setting, provided that it is endowed together with some sample space a suitable larger measure structure.
Natural and Formal Infinities
 Educational Studies in Mathematics 48 (2001
"... Concepts of infinity usually arise by reflecting on finite experiences and imagining them extended to the infinite. This paper will refer to such personal conceptions as natural infinities. Research has shown that individuals’ natural conceptions of infinity are ‘labile and selfcontradictory ’ (Fis ..."
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Cited by 4 (1 self)
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Concepts of infinity usually arise by reflecting on finite experiences and imagining them extended to the infinite. This paper will refer to such personal conceptions as natural infinities. Research has shown that individuals’ natural conceptions of infinity are ‘labile and selfcontradictory ’ (Fischbein et al., 1979, p. 31). The formal approach to mathematics in the twentieth century attempted to rationalize these inconsistencies by selecting a finite list of specific properties (or axioms) from which the conception of a formal infinity is built by formal deduction. By beginning with different properties of finite numbers, such as counting, ordering or arithmetic, different formal systems may be developed. Counting and ordering lead to cardinal and ordinal number theory and the properties of arithmetic lead to ordered fields that may contain infinite and infinitesimal quantities. Cardinal and ordinal numbers can be added and multiplied but not divided or subtracted. The operations of cardinals are commutative, but the operations of ordinals are not. Meanwhile an ordered field has a full system of arithmetic in which the reciprocals of
Kneser’s theorem for upper Banach density
 JOURNAL DE THEORIE DES NOMBRES DE BORDEAUX
, 2006
"... Suppose A is a set of nonnegative integers with upper Banach density α (see definition below) and the upper Banach density of A + A is less than 2α. We characterize the structure of A+A by showing the following: There is a positive integer g and a set W, which is the union of ⌈2αg − 1 ⌉ arithmetic ..."
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Cited by 3 (2 self)
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Suppose A is a set of nonnegative integers with upper Banach density α (see definition below) and the upper Banach density of A + A is less than 2α. We characterize the structure of A+A by showing the following: There is a positive integer g and a set W, which is the union of ⌈2αg − 1 ⌉ arithmetic sequences 1 with the same difference g such that A + A ⊆ W and if [an,bn] for each n is an interval of integers such that bn − an →∞and the relative density of A in [an,bn] approaches α, then there is an interval [cn,bn] ⊆ [an,bn] for each n such that (dn − cn)/(bn − an) → 1and(A + A) ∩ [2cn, 2dn] =W ∩ [2cn, 2dn].
Compactness of Loeb Spaces
, 1998
"... In this paper we show that the compactness of a Loeb space depends on its cardinality, the nonstandard universe it belongs to and the underlying model of set theory we live in. In x1 we prove that Loeb spaces are compact under various assumptions, and in x2 we prove that Loeb spaces are not compact ..."
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Cited by 3 (2 self)
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In this paper we show that the compactness of a Loeb space depends on its cardinality, the nonstandard universe it belongs to and the underlying model of set theory we live in. In x1 we prove that Loeb spaces are compact under various assumptions, and in x2 we prove that Loeb spaces are not compact under various other assumptions. The results in x1 and x2 give a quite complete answer to a question of D. Ross in #R1#, #R2# and #R3#. 0 Introduction In #R1# and #R2# D. Ross asked: Are #bounded# Loeb measure spaces compact? J. Aldaz then, in #A#, constructed a counterexample. But Aldaz's example is atomic, while most of Loeb measure spaces people are interested are atomless. So Ross reasked his question in #R3#: Are atomless Loeb measurespaces compact? In this paper we answer the question. Let's assume that all measure spaces mentioned throughout this paper are atomless probability spaces. Given a probability space## ; #;P#. A subfamily C # # is called compact if for any D #C, D has f.i....
Maharam Spectra of Loeb Spaces
 The Journal of Symbolic Logic
"... We characterize Maharam spectra of Loeb probability spaces and give some applications of the results. 0. Introduction In the nonstandard approach to probability theory, a central role is played by a family of very rich probability spaces, known as Loeb spaces. It is natural to ask for a description ..."
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We characterize Maharam spectra of Loeb probability spaces and give some applications of the results. 0. Introduction In the nonstandard approach to probability theory, a central role is played by a family of very rich probability spaces, known as Loeb spaces. It is natural to ask for a description of the Loeb spaces, or at least a description of their measure algebras, in standard terms. By Maharam's Theorem (see x1), the measure algebra of any atomless probability space(\Omega ; B; ) is determined up to isomorphism by a finite or countable set of "weighted" infinite cardinals, which we will call the Maharam spectrum of(\Omega ; B; ). In this paper we will study the Maharam spectra of Loeb probability spaces. We will concentrate on the two classes of Loeb spaces which are most frequently used in applications: the hyperfinite Loeb spaces and the Loeb spaces generated by standard probability spaces. In general, the possible Maharam spectra for these Loeb spaces will depend on the n...