multiple optimal choices and random exchange ... - Semantic Scholar

Economic Theory 14, 507–544 (1999)

Research Articles The complete removal of individual uncertainty: multiple optimal choices and random exchange economies? Yeneng Sun1,2 1 2

Department of Mathematics, National University of Singapore, Singapore 119260, SINGAPORE (e-mail: [email protected]) Cowles Foundation, Yale University, New Haven, CT 06520, USA

Received: September 14, 1998; revised version: January 6, 1999

Summary. The aim of this paper is to develop some measure-theoretic methods for the study of large economic systems with individual-specific 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 non-unique 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 measure-theoretic 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.

? Part of the results in Theorems 1 and 3 was included in an earlier unpublished draft “The law of large numbers for set-valued processes and stationary equilibria”, first written in 1994. The rest of the work was done while the author was visiting the Cowles Foundation for Research in Economics at Yale University in 1996-1997. The author is very grateful to Donald Brown, Ali Khan and Heracles Polemarchakis for useful conversations and help.

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Keywords and Phrases: Individual uncertainty, Removal of widespread correlations, General equilibrium models with idiosyncratic shocks. JEL Classification Numbers: C60, D51, D80. 1 Introduction In many economic models, an economic agent is required to choose optimal actions under certain constraints. It is often the case that best actions are not unique, and thus one is naturally led to work with sets of multiple optimal choices as functions of some underlying parameters. This has brought the theory of correspondences into an important component of modern economic theory. In fact, fixed point theorems based on correspondences have become the standard tools for showing the existence of equilibria in a variety of economic contexts since the path-breaking work of Nash [46], Arrow-Debreu [4] and McKenzie [43]. Though it is possible in some cases to impose strong conditions, such as strong convexity, to obtain a unique best choice,1 it often happens that either the underlying economic model itself does not allow a convex structure (for example, equilibrium models with nonconvex consumption sets or nonconvex technologies, and game-theoretic models with finitely many pure strategies) or a linearization procedure excludes the possibility of strong convexity (such as economic models with expected utility based on mixed strategies). Thus the consideration of multiple optimal actions is also inevitable in many theoretical situations. When uncertainty is explicitly introduced into an economic model, one usually distinguishes two different approaches. The first involves the so-called state contingent commodities or actions and all the results on a deterministic model continue to hold for the more general setting after a suitable reinterpretation.2 The second approach is based on the fact that there are some sources of uncertainty which are intrinsically unsuited to the first treatment in a realistic way. This often involves a large number of economic entities among which the correlations are weak, i.e., randomness appears on the individual level. In the literature, this kind of economic situation is described by various terminologies, such as individual risks/uncertainty, idiosyncratic uncertainty/risks/shocks, and information smallness. By appealing to the law of large numbers, one can somehow remove this type of uncertainty from macroscopic point of view. However, it is common sense that the classical law of large numbers in the sequential setting is only an asymptotic result, and it is usually difficult to build up further constructions based on an approximate structure. It is thus postulated in the economic literature that for a continuum of independent and identically distributed random variables (or stochastic processes), the means or distributions of sample functions (or the 1

See, for example, [16] and [31]. One only needs to note that all the relevant economic notions are now state contigent. See, for example, Debreu [16]. 2

The complete removal of individual uncertainty

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finite dimensional distributions of the relavant empirical processes) are almost constant. This complete cancellation of individual risks has been extensively used in various areas of economic theory.3 It is by now well understood that the exact law of large numbers, as formalized by the usual continuum model based on a Lebesgue interval, has no clear scientific meaning. As discussed by Doob [19], [20], Feldman-Gilles [23] and Judd [33], one has to face the measurability problem as well as the possible failure of the equality involving sample and theoretical means as required by the statement of such a law of large numbers. Even though one can produce a process satisfying both a microscopic independence assumption and an exact macroscopic stability condition,4 one can also construct an iid process based on the usual continuum for which the law of large numbers completely fails.5 In fact, the law of large numbers itself should be a result that gives sufficient (and possibly also necessary) conditions such that macroscopic stability is obtained whenever these conditions are satisfied by a general process. Such exact versions of law of large numbers have been systematically studied in [56], [59] and [60], where a number of conditions, including almost sure pairwise independence and almost mutual independence, are characterized to be necessary and sufficient for versions of the law to hold. These laws of large numbers are much more general than the postulated iid case. The ideal framework used in these papers involves a collection of random variables indexed by a special standard measure space with various useful properties not shared by other measure spaces. This measure space was introduced by Loeb in [40] and is called Loeb space in the literature. It is recognized in [56] and [59] that hyperfinite processes on Loeb product spaces exactly correspond to probabilistic situations with a large number of random variables,6 and thus especially amenable to the theoretical study of economic phenomena with a large number of random entities. The aim of this paper is to bring individual risks or uncertainty as formulated through the special measure spaces together with the case of multiple optimal choices and to provide a viable framework for the study of relevant economic problems. In particular, our Theorem 1 shows that versions of almost independence are necessary and sufficient for macroscopic stability of the distributions of the sample correspondences in a set-valued process. This theorem can be used to show that for an economic model in which the primitive data of the agents are almost independent, the set of distributions of certain equilibria are stable across different states of nature in a probabilistic sense. On the other hand, if several best actions are available to each economic agent, then some of those 3

see, for example, [3], [8], [23], [48], and their references. This can be difficult, as in Green [26], or by interpreting the classical law of large numbers in a nonstandard model, remarkably easy, as, for example, in [3], [34], [47] and [55]. 5 See [33] and [23]. Since the law of large numbers does hold in the discrete case, this also indicates that the usual concept of a continuum of random variables is a poor model for the ideal limit of a large finite of economic entities with low intercorrelation. 6 It is pointed out that exact properties of these processes correspond to asymptotic properties of triangular arrays of random variables. See [61] for a survey of the results on the characterizations of the exact law of large numbers. 4

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agents may face common optimal choice sets at certain non-negligible events. This allows the possibility of widespread correlation among different agents even though the primitive economic data of those agents could be independent. Our Theorem 2 essentially guarantees that different economic agents can still act independently in a “global” equilibrium from which equilibria with a common distribution are obtained for individual states, and moreover, every equilibrium distribution can, in principle, be achieved by such an independent equilibrium. In a dynamic model, when the initial data of the agents are independent, one might need some temporary equilibria to be part of the inputs for a next period. Then, Theorem 2 can hopefully be used to claim a sort of independence for the new set of data. It was argued by Block and Marschak [11] that “in interpreting human behavior there is a need to substitute stochastic consistency of choices for absolute consistency of choices. The latter is usually assumed in economic theory but is not well supported by experience. It is, in fact, not assumed in empirical econometrics and psychology.” To establish this type of stochastic consistency in a general equilibrium model, Hildenbrand [30] allowed independent shocks in individual agents’ endowments and preferences in a sequence of economies and then used the classical law of large numbers to show the existence of approximate deterministic equilibrium price systems for the random economies.7 Note that the use of the law of large numbers already requires that the economies considered must be “large”. On the other hand, due to the intrinsic difficulties associated with the usual continuum model as noted earlier, one cannot introduce individual uncertainty into economies with a continuum of agents in the usual sense so that results on measure-theoretic economies become relevant in the new setting. Thus, one is forced to take the sequential approach to have a rigorous theoretical model. In order to obtain a unique optimal consumption plan for an agent so that the classical law of large numbers could be applied, strictly convex and complete preferences were also used in [30]. In contrast to the literature on large economies as summarized in [31], this treatment is hardly satisfactory. Even though, for the model considered in [30], one could impose these type of strong conditions on preferences, these conditions may not make sense for other economic models. For example, the consumption sets for economies with indivisible commodities are already non-convex, which leads to the impossibility of introducing convex preferences. Based on the exact versions of the law of large numbers we have developed, we are able to propose a new model to establish the type of desired stochastic consistency as studied in [30] in a more exact and stronger sense. In particular, we show in Theorem 3 that in almost all states of nature, the random economies have exactly the same mean excess demand correspondence and the same nonempty set of deterministic equilibrium prices; their Walras allocations corresponding to a common derterministic price system are also exactly the same in terms of distributions, and more interestingly, each of these distributions can be achieved 7

For some related work and further developments, see [9], [45], [54], [63] and [64].


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by a “global” competitive equilibrium resulting from stochasticly independent actions of the individual agents, which also provides equilibria with a common distribution across different states of nature. Thus, from a macroscopic point of view, the random economies behave exactly the same as a derterministic economy, and uncertainty completely disappears.8 The preference relations considered are also very general in the usual framework of measure-theoretic economies, and in particular could be non-convex and non-complete. In fact, some results on deterministic measure-theoretic economies are directly used in the proof of Theorem 3. It was shown by Mas-Colell in [44] that with a suitable dispersedness assumption on a divisible commodity together with another special condition on the desirability of the divisible good, the main results on deterministic measuretheoretic economies with divisible commodities remain valid for economies with indivisible commodities. Our Theorem 4 shows that all the properties in Theorem 3 can be restated for the case of indivisible goods with some additional assumptions as used in [44]. In [37], Khan-Yamazaki pointed out that for a measure-theoretic economy, the notion of core could be a more attractive solution concept than the competitive equilibrium in the presence of indivisible commodities. In contrast to the divisible case, they showed that even if there is no competitive equilibrium, the core of the relevant economy is in general still nonempty. This means that the two notions are indeed different in their setting. To illustrate that the general approach developed here can also be used to study genuinely different notions other than competitive equilibrium as discussed in the two previous cases, we consider in Theorem 5 core allocations for random economies with indivisible commodities corresponding to the case studied by Khan-Yamazaki in [37]. It shows that the random economies have exactly the same set of core distributions in almost all states of nature and every core distribution can be realized by a “global” core allocation with stochastic independence. The paper is organized as follows. In Section 2, we state and discuss Theorems 1 and 2. Section 3 is devoted to the study of general equilibrium models with individual shocks on the endowments and preferences. The proofs of Theorems 3-5 in Section 3 rely on Theorems 1 and 2 as well as some relevant results on the corresponding deterministic measure-theoretic economies.9 Some asymptotic interpretations of our theorems are presented in Section 4. Such kind of interpretation was already used in Brown-Robinson [12] to obtain results on large finite economies from results on internal economies.10 Section 5 includes some concluding remarks. Section 6 is the Appendix which contains the detailed proofs of Theorems 1 and 2 as well as some versions of the law of large numbers for set-valued processes in terms of integrals in general infinite dimensional spaces. 8 9 10

Note that here no transfers are allowed among different individual states. These elegant results will not be available if one works with a discrete model as in [30]. See also [13] and [35] for such translations.

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To conclude this section, we emphasize that except for some technical lemmas in the appendix, all other results in this paper only involve special measuretheoretic properties of Loeb spaces. These properties can be understood and applied without prior knowledge about the construction of this measure space itself, much as Lebesgue spaces can be applied without detailed knowledge about their particular construction or the Dedikind set-theoretic basis for the set of real numbers. In fact, even if one prefers to use the unit interval I = [0, 1] to index random variables in a set-valued process, one can still work with a measure µ on I induced by a bijection between I and a hyperfinite set in an ultrapower construction based on N, since a collection of random variables indexed by hyperfinite Loeb spaces in this setting indeed has the cardinality of the continuum.11 All the results in Theorems 1-5 in this paper can be reproduced in terms of the new measure µ,12 provided that the unit interval I with measure µ is to be endowed together with some sample space a suitable larger product measure structure as above. The point is that the particular choice of a parameter space itself is not an issue; what is really relevant is the associated measure structure. Finally, we note that Theorems 3-5 are formulated to illustrate how the exact law of large numbers together with Theorem 2 can be used to obtain new results for economic models with individual uncertainty and non-unique optimal choices.13 The general idea can be applied to other situations as well.

2 Almost independence and the law of large numbers We shall first fix some notation. We work with a special type of probability spaces, which are called Loeb spaces in the literature. Let (T , L(T ), L(λ)) be such a special probability space, where L(T ) is a σ-algebra of subsets of T and L(λ) is a countably additive probability measure on the measurable space (T , L(T )). As a measure space, (T , L(T ), L(λ)) is also complete. We shall use (T , L(T ), L(λ)) as our index space, for example, as a parameter space of a collection of random variables or correspondences, or as the space of economic agents. Here, we note that the notation for the probability measure L(λ) might look rather redundant and confusing to those who are not familiar with Loeb measures. This is, however, the usual notation for such measures. We adopt it for consistency with the literature. One can certainly choose other simpler notations. Let (Ω, L(A), L(P )) be another Loeb probability space which is to be used as our sample space. We shall assume both Loeb probability spaces considered here to be atomless. With these two probability spaces in hand, one can certainly 11

See [1] and [32]. Hereafter, we may often omit the adjective “hyperfinite” for simplicity. Of course, this new measure µ on I cannot be the Lebesgue measure. In fact, it can be shown that for a nontrvial almost independent process, almost all the sample functions are µ-measurable but not Lebesgue measurable (for details, see [62]). 13 Theorem 3 shows that stronger results can be obtained in the new framework with much less restrictive conditions. Theorems 4 and 5 indicate that the approach taken here can still cover the case where a unique optimal choice is in general not possible within the economic models. 12


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take their usual product to obtain a product measure space (T × Ω, L(T ) ⊗ L(A), L(λ) ⊗ L(P )) and note that the Fubini theorem is satisfied by this usual product measure space.14 However, we shall not work with this usual product measure space. As we are working with a special type of measure spaces, we are also going to work with a special type of product spaces, called Loeb product spaces. Let (T × Ω, L(T ⊗ A), L(λ ⊗ P )) be a Loeb product space, which is a standard probability space itself. For our readers, what needs to be remembered is only that a Loeb product space is an extension of the usual product measure space and the Fubini theorem is still valid for this special product space.15 One point of crucial importance for this paper is that the Loeb product σ-algebra L(T ⊗ A) strictly contains the usual product σ-algebra L(T ) ⊗ L(A).16 This is due to the fact that the processes to be considered here are usually only measurable with respect to the bigger σ-algebra.17 In fact, the main technical strength of this work depends on this larger measure-theoretic framework. Hereafter, unless it is explicitly pointed out, any product space over T × Ω will refer to the Loeb product space. Let X be a complete separable metric space. A measurable function f from the product space (T × Ω, L(T ⊗ A), L(λ ⊗ P )) to X is called a process. For each t ∈ T , and ω ∈ Ω, ft denotes the function f (t, ·) on Ω and fω denotes the function f (·, ω) on T . Note that the measurability (in an almost sure sense) of ft and fω is guaranteed by the relevant Fubini Theorem. The functions ft are usually called the random variables of the process f , while the fω form the sample functions of the process. A correspondence is a mapping whose values are nonempty sets. A closed valued measurable correspondence from the product space (T ×Ω, L(T ⊗A), L(λ⊗ P )) to X will be called a set-valued process.18 Similarly, for each ω ∈ Ω, let Fω be the correspondence defined by Fω (t) = F (t, ω), which is called a sample correspondence in the set-valued process F ; for each t ∈ T , let Ft be the correspondence defined by Ft (ω) = F (t, ω), which is simply called a correspondence in F . The measurability of the correspondences Ft and Fω again follows from the Fubini property. For a correspondence G from a probability space (Λ, C , ν) to a complete separable metric space X , a measurable mapping g from Λ to X is said to be 14

See, for example, [31], p.47 and [14], p.156. General Loeb product spaces were already used in [40] to construct a sample space for hyperfinite coin-tossing. From its construction, it is easy to see that a Loeb product space contains its classical counterpart. This was first pointed out explicitly by Anderson in [2]. The Fubini property for a Loeb product space was first proven by Keisler in [34] (see also [1] and [41]). The general references for Loeb product spaces are [1], [32] and [55]. 16 See [1] for a special example of this type of proper inclusion. However, a complete understanding about the relationship between the two types of product spaces is also available in [56] and [59]. In particular, Proposition 6.6 in [59] characterizes that a Loeb product space is strictly bigger than its classical counterpart if and only if the individual Loeb probability spaces are non-purely atomic. 17 See the second paragraph in Section 5. 18 A closed valued correspondence from a probability space (Λ, C , ν) to a complete separable / metric space X is said to be measurable if for any open set O in X , F −1 (O) = {α ∈ Λ : F (α) ∩ O = ∅} is a measurable set in C . See [38], p.153. 15

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a selection of G if g(α) ∈ G(α) for ν-almost all α ∈ Λ. Let DG be the set of all the distributions on X induced by the selections of G, i.e., DG = {νg −1 : g is a selection of G}. For the case that X is the EuclideanR space Rn for R some n, the integral of the correspondence G is defined to be Λ Gd ν = { Λ gd ν : g is an integrable selection of G}.19 A process f is said to be a selection of a setvalued process F if f (t, ω) ∈ F (t, ω) for almost all (t, ω). When F is regarded as a correspondence on the product probability space, DF denotes the set of distributions of the selections of F , viewed as random variables on the product probability space. For ω ∈ Ω, the meaning of DFω is obvious. A set-valued process F is said to satisfy the law of large numbers if for almost all ω ∈ Ω, DFω = DF . Let G1 , G2 , . . . , Gn be n closed valued measurable correspondences from a probability space (Λ, C , ν) to a complete separable metric space X . They are said to be independent if for any open sets O1 , O2 , . . . , On in X , the events G1−1 (O1 ), G2−1 (O2 ), . . . , Gn−1 (On ) are independent. For a set-valued process F from (T × Ω, L(T ⊗ A), L(λ ⊗ P )) to X , if for L(λ ⊗ λ)-almost all (t1 , t2 ) ∈ T × T ,20 the correspondences Ft1 and Ft2 are independent, we say that the correspondences Ft ’s are almost surely pairwise independent; if for any n ≥ 2 and for almost all (t1 , t2 , . . . , tn ) ∈ T n , the n correspondences Ft1 , Ft2 , . . . , Ftn are independent, we say that the correspondences Ft ’s are almost mutually independent. Theorem 1 below characterizes those set-valued processes satisfying the consistent law of large numbers (or simply the consistency law). Some scalar versions of this law can be viewed as a formal version of the intuitive observation, as characterized with the aphorism “No betting system can beat the house” (see, for example, [49]), which simply means that a gambler cannot change the expectation of his return by betting at a particular subsequence. It also means that probabilistic stability occurs not only for a whole system but also for macroscopically visible subsystems. The necessity of such macroscopic stability for some economic models is also documented in [23], where it is pointed out that if the special case of a continuum of iid random variables is used to model individual risks, then one should require sample averages of any nonnegligible subcollection of the random variables to be constant, though this is shown in [23] to be impossible in the usual setting. In [56], [59] and [60], a number of conditions on the random variables in a point-valued process,21 including almost sure pairwise independence and almost mutual independence, are characterized to be necessary and sufficient for versions of the consistency law to hold. This means that this type of independence conditions provides an accurate description of risks or un19 For various properties of the integral of correspondences, see [31] and [38]. [57] contains a number of useful properties on the distributions of correspondences on Loeb spaces. 20 Again, when a measure on some product space is mentioned, the measure is the relavant Loeb product measure. Thus L(λ ⊗ λ) is the Loeb product measure in the Loeb product space (T × T , L(T ⊗ T ), L(λ ⊗ λ)). 21 Here a point-valued process simply means a process in the usual sense, i.e., it takes values in a metric space rather than in the power set of a metric space.


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certainty on the individual level but without randomness on macroscopic level. Theorem 1 is the set-valued analog of those results. The following is a formal definition of the consistency law for set-valued processes. Definition 1 Let F be a set-valued process from (T × Ω, L(T ⊗ A), L(λ ⊗ P )) to a complete separable metric space X . We say that F satisfies the consistent law of large numbers (or simply the consistency law) if for any measurable set A ∈ L(T ) with L(λ)(A) > 0, the set-valued process F A from (A × Ω, L(T ⊗ A)A , L(λ ⊗ P )A ) to X satisfies the law of large numbers, i.e., for L(P )-almost all ω ∈ Ω, DFωA = DF A , where F A and L(T ⊗ A)A are respectively the restrictions of F and L(T ⊗ A) to A × Ω, L(λ ⊗ P )A denotes L(λ ⊗ P )|A×Ω /L(λ)(A), a probability measure on (A × Ω, L(T ⊗ A)A ) rescaled from L(λ ⊗ P ). We are now ready to state Theorem 1. Its proof is given in Subsection 6.2 of the Appendix. The proof takes the view that a set-valued process F can also be regarded as a point-valued process F¯ in a hyperspace of sets. Such an approach goes back at least to Debreu [17]. Since the process being considered is a general closed valued measurable correspondence from the product space into a general complete separable metric space X , a suitable measurable structure on the hyperspace of nonempty closed sets in X , as used in [57], is needed.22 Once the relationships between the corresponding properties of F and F¯ on measurability, independence and distributions are established, the proof follows easily from the relevant results for case of the point valued processes shown in [56], [59] and [60]. Theorem 1 Let F be a set-valued process from the product space (T ×Ω, L(T ⊗ A), L(λ ⊗ P )) to a complete separable metric space X . Then the following are equivalent: (1) F satisfies the consistency law; (2) the correspondences Ft ’s are almost surely pairwise independent; (3) the correspondences Ft ’s are almost mutually independent. Note that pairwise independence is almost the weakest version of independence while mutual independence is almost the strongest.23 Since their almost versions are equivalent for random variables as shown in Theorem 3 in [60] and for correspondences as shown by Theorem 1 here, we will simply refer to the almost versions as almost independence. The following simple corollary of Theorem 1 presents a law of large numbers for set-valued processes in terms of integrals in an Euclidean space. It is easy to generalize this corollary to the case where the set-valued processes take values 22 For convenience, this measurable structure is also discussed in Subsection 6.1. In the literature, one often views compact valued correspondences or closed valued correspondences in a locally compact separable metric space as point valued mappings into some hyperspaces; see, for example, [17], [31] and [38]. Our case is more general. 23 It is common sense that for a finite collection of random variables, pairwise independence is strictly weaker than mutual independence; see, for example, [24], p.126.

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from the power set of an infinite dimensional space and the integrals are either the strong Bochner or the weak Gel0 fand integrals. The details are given in Corollaries 3 and 4 in the Appendix. Corollary 1 Let F be a set-valued process from the product space (T ×Ω, L(T ⊗ A), L(λ ⊗ P )) into Rn . If the correspondences Ft ’s are almost independent, then for L(P )-almost all ω ∈ Ω, ZZ Z Fω dL(λ) = FdL(λ ⊗ P ). T ×Ω

T

Proof. It is easy to check that Z Z Fω dL(λ) = { id d µ : µ ∈ DFω and id is µ-integrable on Rn }, ZZ

T

T ×Ω

Rn

FdL(λ ⊗ P ) = {

Z Rn

id d µ : µ ∈ DF and id is µ-integrable on Rn },

where id is the identity function on Rn . Then the result clearly follows from Theorem 1. As in the case of classical law of large numbers for a sequence of real valued random variables, there is a sizable liturature on the law of large numbers and its applications in the setting of a sequence of correspondences.24 Following the approach in Debreu [17], Artstein-Vitale [7] considered a sequence of iid correspondences as a sequence of iid random variables in a Banach space of compact convex sets and then appeal to a Banach space valued law of large numbers to obtain their result.25 In order to obtain a law for a sequence of iid correspondences with possibly unbounded sets, a new approach was needed in Artstein-Hart [6]. Applications of those laws of large numbers to the allocation model of Arrow-Radner [5] were given in [6] and [27]. We shall now compare Theorem 1 and its corollaries with those sequential laws of large numbers for correspondences in the liturature. First note that those sequential laws mostly focused on the iid case in terms of means,26 our Corollary 1 as well as Corollaries 3 and 4 in the Appendix cover the general case. Unlike in [6], the unbounded (even for iid) case has to be treated in a different way, our Theorem 1 in the distributional form allows a very simple and uniform treatment for all kinds of integrals with neither compactness nor boundedness assumptions nor distributional restrictions such as identical distributions. While the iid condition on correspondences, as commonly used in the previous liturature, is a very strong condition, the equivalence of macroscopic stability and almost independence in our Theorem 1 shows that this latter independence condition is essentially the weakest possible to derive the law of large numbers. 24

These includes [6], [7], [15], [27], [25], [29], [52]. This approach was also adopted in [15], [25], [29] and [52]. 26 Theorem 7 in [29] is one possible exception, where a law is proven for a sequence of independent random variables from some intricate Lp space together with several other conditions. 25


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The implications (1) =⇒ (2) and (2) =⇒ (3) in Theorem 1 have no sequential analog in the previous literature. In terms of applications to economic models which need some versions of the law of large numbers for correspondences, we note that with suitable reformulations of the relevant models, the exact laws of large numbers studied here are certainly applicable to those situations where the sequential laws are used. The main advantages of using these exact laws are that unnecessary estimations involving inequalities could be avoided and simple measure-theoretic methods and results are now directly available. In the discrete case, messy computations are not only required for new deduction procedures but also for the cases where some corresponding exact results are already known. These additional complications create artificial difficulties for discovering and proving new results. Next, we point out some weakness in the versions of law of large numbers for correspondences as presented in Theorem 1 and its corollaries. We illustrate this by discussing their possible applications in a general equilibrium model. These applications are realized later in Section 3. When a law of large numbers is applied to a large random economy E with agent space T , sample space Ω and an almost independence assumption on the agents’ primitive data, one may obtain, for almost every sample ω, some sort of equilibrium xω , as a measurable function on T , for the deterministic economy Eω associated with the sample realization ω. The problem is that the function x on T ×Ω obtained by putting all the xω together may not be measurable with respect to the product space. On the other hand, by viewing the random economy E as a deterministic economy with agents’ space T ×Ω, an equilibrium y on T ×Ω may not have the property that yω is an equilibrium for Eω for almost all ω. Furthermore, due to the possibility of multiple optimal choices, it is not clear at all that the microscopic independence assumption itself can still be preserved in equilibrium. In short, Theorem 1 is not strong enough to give any useful “global” equilibrium. To solve the problems raised in the previous paragraph, we formulate the following Theorem 2 which says that for any given measurable selection of a set-valued process with almost independence,27 there is another measurable selection of the set-valued process such that, as a process, this new selection has almost independent random variables and also induces the same distribution as the given selection when they are viewed as random variables on the product space.28 To see the main idea underlying the applications of Theorem 2, we still illustrate through the random economy E as above. By viewing the random economy E as a deterministic economy with agent space T × Ω, one may 27 It is easy to see that almost independence may not be preserved by the measurable selections of an almost independent set-valued process. Widespread correlations may exist in some selections. Consider a trivial example. Let F be a set-valued process to R such that Ft (ω) = R for all t and ω. The correspondences Ft are obviously independent of each other. Let α be a real-valued random variable on Ω which is not essentially constant and define ft (ω) = α(ω) for every t. Then f is a selection of F but its random variables are all perfectly correlated. 28 Note that it is very easy to obtain one measurable selection of F which has almost independent random variables. As in [6] for the discrete case, one can just take any selection from the hyperspace of closed sets and then compose it with F .

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obtain an equilibrium f . By Theorem 2, one can find an almost independent equilibrium g for the deterministic economy E with the same distribution as f .29 The almost independence then guarantees that gω is an equilibrium for Eω for almost all ω, i.e., g also induces an equilibrium essentially per state of nature and is thus a “global” equilibrium which, in addition, preserves the microstructure on independence. Moreover, every “equilibrium distribution” can be achieved by such an independent “global equilibrium”.30 Note that when one has a unique optimal choice under relevant parameters, the problems raised in the previous paragraph will not arise at all. In particular, if the set-valued process F below is, in fact, point valued in X , then Theorem 2 is completely trivial. Thus, Theorem 2 presents a distinctive new result for analyzing economic models with multiple optimal choices. It also has no known sequential analogs in the previous literature. Theorem 2 Let F be a set-valued process from the product space (T ×Ω, L(T ⊗ A), L(λ ⊗ P )) to a complete separable metric space X such that the correspondences Ft ’s are almost independent. Let µ be the distribution of a selection f of F as a random variable on T × Ω. Then there is a selection g of F such that the random variables gt ’s are almost independent and the distribution of g viewed as a random variable on T × Ω is µ. The proof of Theorem 2 involves delicate computations. The main difficulty comes from the atoms of the distribution of F . A closed set D in X is an atom for the distribution of F if there is a non-negligible set R in T × Ω such that F (t, ω) = D for all (t, ω) ∈ R. The events Rt ’s are almost independent. In some sense, one then has to choose a process with almost independence from R to D with a given distribution. The idea is to cut through all the events Rt ’s to obtain subevents with a given common proportion of the probability of the Rt ’s so that almost independence remains valid for those subevents. Eventually, one also has to paste all the constructions arising from the atoms as well as the atomless part together. The details of the proof is in Subsection 6.3 of the Appendix. The following corollary of Theorem 2 is obvious. Corollary 2 Let F be a set-valued process from the product space (T ×Ω, L(T ⊗ A), L(λ ⊗ PRR)) to Rn . If the correspondences Ft ’s are almost independent, then for any x ∈ T ×Ω FdL(λ ⊗ P ), there is a selection g of F such that the random RR variables gt ’s are almost independent and x = T ×Ω gdL(λ ⊗ P ). Before moving to the next section, we have to address the important question on the existence of nontrivial set-valued processes on T × Ω with almost independent correspondences. Note that with a suitable topology, the hyperspace FX of all nonempty closed sets in X becomes a complete separable metrizable space and the measurability of a closed valued correspondence in X reduces to the measurability of the relevant point valued mapping in FX .31 By the universality 29 30 31

The relevant set-valued process is the associated individual demand correspondence. For details, see the proof of Theorem 3 in Subsection 3.1. See Subsection 6.1 for details.


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result in Theorem 6.2 of [59], one can construct a process G from T × Ω to FX with almost independent random variables so that these random variables can take any given variety of distributions. By viewing G as a set-valued process, the Gt ’s are then almost independent correspondences which may be allowed any variety of distributions as well. 3 The complete removal of individual uncertainty in large random economies We shall first fix some notations for this section. Let ` be the number of distinguishable commodities in a market. We shall work with a common consumption set X which is a suitable closed subset of the positive orthant R`+ of R` .32 A preference relation is a transitive and irreflexive binary relation on X such that is a relatively open set in X × X . The set of all preference relations is denoted by P . The topology on P is the one induced by the topology of closed convergence on the closed sets X × X − . The space of preferences P with this topology is a compact and metrizable space.33 The atomless probability space (T , L(T ), L(λ)) represents the space of economic agents. An economy α = (, e) is a measurable mapping from the space of economic R agents (T , L(T ), L(λ)) to the space of agents’ characteristics P ×X such that T e(t)dL(λ) is finite, where t Rand e(t) are respectively the preference relation and endowment of agent t, and T e(t)dL(λ) is the mean endowment of the economy α. A price system p is a vector in R`+ such that the sum of its components is one. The set of all price systems is denoted by ∆. For an economic agent with preference and endowment e, the demand set ϕ(, e, p) under price system p is the set of maximal elements in the budget set ([31], p.92 and p.148). Thus the demand set of agent t is ϕ(t , et , p) or ϕ(α(t), p). When p is strictly positive, i.e., positive in every component, the demand set ϕ(, e, p) is always nonempty and compact. Note that the demand relation ϕ(, e, p) has a Borel measurable graph ([31], p.102), and hence under a stricly positive price system p, the individual demand relation ϕ(α(·), p) defines a compact valued, measurable correspondence from T to X , which will be called the individual demand correspondence. The mean demand correspondence Φ(α, p), R as a mapping of the price system p, is simply the integral T ϕ(α(t), p)dL(λ); and the mean excess demand Z (α, p) is the difference of the mean demand with R the mean endowment, i.e., Z (α, p) = Φ(α, p) − T e(t)dL(λ). Hereafter, we shall always regard various demand correspondences as set-valued mappings on the set of strictly positive prices unless otherwise noted. An allocation f for the economy α is simply an integrable function from T into the consumption set X . An allocation f and a price system p is called a competitive equilibrium for α if for L(λ)-almost all t ∈ T , f (t) ∈ ϕ(α(t), p), and It will be R`+ in Subsection 3.1 and N`−1 × [0, ∞] in Subsections 3.2 and 3.3. See [31], p.86 for the definition of a preference relation, p.19 and p.96 for the topology of closed convergence on preference relations. Note that no convexity is needed on X to show the closedness of P in our setting. 32 33

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R R also T fdL(λ) = T edL(λ). That is, almost all agents choose optimal consumptions within their budgets and the market is clear. We shall also call f a Walras allocation as in [31] (p.129).34 The atomless probability space (Ω, L(A), L(P )) formalizes the uncertainty for all the agents. A random economy E = (, e) is a measurable mapping or simply a process from the product space (T ×Ω, L(T ⊗A), RR L(λ⊗P )) to the space of agents’ characteristics P × X such that the vector T ×Ω e(t, ω)dL(λ ⊗ P ) is finite, where (t,ω) and e(t, ω) are respectivelyRRthe preference relation and endowment of agent t at sample realization ω, and T ×Ω e(t, ω)dL(λ ⊗ P ) is the expected mean endowment of the random economy. For almost all t ∈ T , Et is a measurable mapping from the sample space Ω to P × X ; et (·) represents the possible random shocks in agent t’s endowment, and t (ω) his random preferences. We can also view E as a deterministic economy with the measure space (T × Ω, L(T ⊗ A), L(λ ⊗ P )) as the space of economic agents. The relevant economic notions defined in the previous paragraphs can be trivially restated for this case. For almost all ω ∈ Ω, Eω is a measurable mapping from the space of economic agents (T , L(T ), L(λ)) toR the space of agents’ characteristics P × X such that the mean endowment T eω (t)dL(λ) is finite, and hence an exchange economy in the usual sense. The purpose of this section is to investigate the probabilistic behavior of the random economies Eω . In particular, we show that when the primitive economic data are almost independent for different economic agents, the set of equilibrium prices for Eω is in general nonempty and does not depend on particular sample realizations. As already illustrated in the previous section to motivate Theorem 2, this theorem allows us to find a “global” equilibrium which preserves the microscopic independence structure through a redistribution of an equilibrium for E , viewed as a deterministic economy;35 and moreover, the distribution of every Walras allocation can be achieved by such an independent “global” equilibrium. When there is no competitive equilibrium for an economy with indivisible commodities, similar results on distributions can be proven for core allocations.

3.1 Competitive equilibria with random preferences and endowments In this subsection, we work with ` perfectly divisible goods. The common consumption set X is the positive orthant R`+ . Let E = (, e) be a random economy, i.e., a process from (T × Ω, L(T ⊗ A), L(λ ⊗ P )) to P × X . The following assumptions on the economy E will be needed for Theorem 3 below. 34 For earlier work on general equilibrium models using nonstandard analysis, see the survey chapter [3] of Anderson and the book [53] of Rashid. In contrast to this earlier work, our approach here only uses special measure-theoretic properties of Loeb spaces, and thus no “nonstandard” methods are used to prove our results on general equilibrium models. 35 Note that when E is viewed as a deterministic economy, an arbitrary equilibrium may contain widespread correlation due to multiple optimal choices. Here we also point out that the terminology of “random economies” is used for both E and Eω ’s.


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RR (A1) The expected mean endowment T ×Ω e(t, ω)dL(λ ⊗ P ) is strictly positive. (A2) For L(λ ⊗ P ))-almost all (t, ω) ∈ T × Ω, (t,ω) is monotonic, i.e., for 0 ≤ x ≤ y, x = / y implies y (t,ω) x . In the following theorem, we illustrate how Theorems 1 and 2 can be used to derive results on probabilistic stability from known results on measure-theoretic economies. Here the random economies Eω are compared with a common reference economy E , which is viewed as a deterministic economy with the product space as the space of agents. Theorem 3 Let E be a random economy. Assume that (A1) and (A2) are satisfied. If the Et ’s are almost independent, then we have the following (1) there is a competitive equilibrium (p, g) for E , viewed as a deterministic economy, such that the gt ’s are almost independent, and for almost all ω ∈ Ω, (p, gω ) is a competitive equilibrium for the economy Eω . Moreover, the distribution of every Walras allocation of E can be attained by such an independent Walras allocation g; (2) for almost all ω ∈ Ω, the economies Eω and E have the same mean excess demand correspondence and the same nonempty set of strictly positive equilibrium price systems; (3) for any fixed equilibrium price p of E , essentially all the economies Eω have the same set of distributions for the relevant Walras allocations with equilibrium price p as that of the economy E , viewed as a deterministic economy on the product space. Proof. We first consider (2). By the law of large numbers in distribution as presented in Theorem A in the Appendix, the independence assumption on the economy E implies that for almost all ω ∈ Ω, the economy Eω has the same preference-endowment distribution as E . Since both economies are atomless, Proposition 4 on p.114 in [31] shows that the mean demand correspondences Φ(Eω , p) and Φ(E , p) are the same, and hence so are the relevant mean excess demand correspondences Z (Eω , p) and Z (E , p). Since p is an equilibrium price for E (or Eω ) if and only if 0 ∈ Z (E , p) (or 0 ∈ Z (Eω , p)), it is thus obvious that Eω and E have the same set of equilibrium prices for almost all ω ∈ Ω. By Condition (A1) above and Theorem 2 on p.151 in [31], E indeed has at least one strictly positive equilibrium price. Note that the monotonicity assumption also implies that any equilibrium price must be strictly positive. For (3), fix an equilibrium price p. Since ϕ(, e, p) has a measurable graph, the individual demand correspondence ϕ(E (t, ω), p) for the economy E , viewed as deterministic economy, is thus measurable. For simplicity, we denote ϕ(E (t, ω), p) by F (t, ω). Note that for any ω ∈ Ω, Fω is the individual demand correspondence for the economy Eω . By the almost independence assumption on the economy E , it is easy to see that the correspondences Ft ’s are almost independent. By Theorem 1, DFω = DF for almost all ω ∈ Ω. By the law of large numbers in Theorem A in the Appendix or simply

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Corollary 1 in the point value version, we obtain that for almost all ω ∈ Ω, R RR e(t, ω)dL(λ ⊗ P ) = T eω (t)dL(λ)(t). Then, it is obvious that the set T ×Ω ( Mω =

Z

µ ∈ DFω :

is the same as the set ( M=

x ∈R`+

Z

µ ∈ DF :

x ∈R`+

)

Z xd µ =

eω (t)dL(λ) T

)

ZZ xd µ =

T ×Ω

e(t, ω)dL(λ ⊗ P )

for almost all ω ∈ Ω. Note that Mω (or M) is set of distributions of the Walras allocations of the economy Eω (or E ) with equilibrium price p. Thus (3) is proven. Finally, we consider (1). Let (p, f ) be a competitive equilibrium of the economy E 36 and F the individual demand correspondence under price p defined in the previous paragraph. By Theorem 2, there is a measurable selection g of F such that g has the same distribution µ as f and the gt ’s are almost independent. It is clear that (p, g) is also a competitive equilibrium of the economy E . By the law of large numbers in Theorem A in theR Appendix, we RR obtain that for almost all ω ∈ Ω, T ×Ω e(t, ω)dL(λ ⊗ P ) = T eω (t)dL(λ)(t), µ as g. For almost all ω ∈ Ω, since and R RR gω has the same distribution e(t, ω)dL(λ ⊗ P ) = T eω (t)dL(λ)(t), we thus obtain T ×Ω ZZ

Z gω (t)dL(λ)(t) = T

T ×Ω

ZZ g dL(λ ⊗ P ) =

T ×Ω

Z e dL(λ ⊗ P ) =

eω (t)dL(λ)(t). T

This shows that (gω , p) is a competitive equilibrium for essentially every economy Eω , and thus g is a “global” equilibrium.

3.2 Competitive equilibria in random economies with indivisible goods As noted earlier, some economic model itself does not allow a structure with strict convexity so that a unique optimal choice can be obtained. Thus, to establish the type of stochastic consistency as in Theorem 3 for the indivisible case, the consideration of correspondences becomes inevitable. To illustrate, we consider a market with `−1 indivisible commodities and one perfectly divisible commodity. Let X = N`−1 × [0, ∞] be the common consumption set and b = (0, · · · , 0, 1). Let E = (, e) be a random economy, i.e., a process from (T × Ω, L(T ⊗ A), L(λ ⊗ P )) to P × X . The following assumptions on the economy E will be needed for Theorem 4 below. The statement of Theorem 4 is almost completely the same as that of Theorem 3.37 36 37

(2) already shows the existence of at least one competitive equilibrium. The set of conditions (B1)-(B4) is different from the set of conditions (A1) and (A2).


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RR (B1) The expected mean endowment T ×Ω e(t, ω)dL(λ ⊗ P ) is strictly positive. (B2) For L(λ ⊗ P ))-almost all (t, ω) ∈ T × Ω, (t,ω) is monotonic, i.e., for 0 ≤ x ≤ y, x = / y implies y (t,ω) x . (B3) For L(λ ⊗ P ))-almost all (t, ω) ∈ T × Ω, (t,ω) satisfies the following condition: for every x ∈ X , there is a positive real number β such that (βb) (t,ω) x. (B4) The distribution of the perfectly divisible endowment function e ` (t, ω) in the economy E is atomless.

Theorem 4 Let E be a random economy with indivisible goods. Assume that (B1)-(B4) are satisfied. If the Et ’s are almost independent, then we have the following (1) there is a competitive equilibrium (p, g) for E , viewed as a deterministic economy, such that the gt ’s are almost independent, and for almost all ω ∈ Ω, (p, gω ) is a competitive equilibrium for the economy Eω . Moreover, the distribution of every Walras allocation of E can be attained by such an independent Walras allocation g; (2) for almost all ω ∈ Ω, the economies Eω and E have the same mean excess demand correspondence and the same nonempty set of strictly positive equilibrium price systems; (3) for any fixed equilibrium price p of E , essentially all the economies Eω have the same set of distributions for the relevant Walras allocations with equilibrium price p as that of the economy E , viewed as a deterministic economy on the product space.

Proof. The proof of the existence of a competitive equilibrium for E follows that of Theorem 1 in Mas-Colell (1977). Note that the assumption of complete preference relations on p.444 of [44] is not needed. By the monotonicity of the preferences, strict positivity of the equilibrium prices and the fact that the `-th commodity is perfectly divisible, it is clear that at equilibrium the mean demand is exactly equal to the mean supply as opposed to the inequality in [44] (see p.445 and p.451). In addition, the atomless assumption on the distribution of the perfectly divisible endowment function is sufficient for the set I 0 defined on p.451 in [44] to have full measure. The rest of the proof of (1) is the same as its counterpart in Theorem 3. Note that Proposition 4 on p.114 of [31] is still valid in the setting with nonconvex consumption sets. Hence the same proof as in Theorem 3 (1) shows that essentially all Eω have the same mean excess demand correspondence as E , and hence the same set of equilibrium prices as well. The monotonicity assumption also implies that any equilibrium price must be strictly positive. Thus part (2) is proven. The proof of (3) here is the same as its corresponding part in Theorem 3.

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3.3 The core of random economies with indivisible goods Mas-Colell also presented a simple example in [44] showing that without the type of dispersedness assumption as (B4), there could be no equilibrium for an economy with indivisible goods. However, it is shown by Khan-Yamazaki that this situation can be salvaged by focusing on the core of an economy with indivisible goods. Among other results, they show that the set of core allocations is nonempty and equivalent to the set of some weakly competitive allocations.38 In this subsection, we shall consider an analog of Theorems 3 and 4 in the setting of core allocations. As in the previous subsection, we still work with a common consumption set X = N`−1 × [0, ∞]. For an economy α = (, e) with (T , L(T ), L(λ)) being the space of economic agents, a coalition S , i.e., a set in the σ-algebra L(T ) with positive measure, is said to improve upon an allocation f if there exists an allocation g for α such that (i) g(t) R t f (t), for Ralmost all t ∈ T , (ii) S g(t)dL(λ) ≤ S e(t)dL(λ). R R An allocation f is said to be in the core if T f (t)dL(λ) ≤ T e(t)dL(λ), and there is no coalition S which improves upon f . The distribution of a core allocation will simply be called a core distribution. Now let E = (, e) be a random economy, i.e., a process from (T ×Ω, L(T ⊗ A), L(λ ⊗ P )) to P × X . The following assumptions on the economy E will be needed for Theorem 5 below. RR (C1) The expected mean endowment T ×Ω e(t, ω)dL(λ ⊗ P ) is strictly positive. (C2) For L(λ ⊗ P ))-almost all (t, ω) ∈ T × Ω, (t,ω) is locally nonsatiated, i.e., for every x ∈ X and every neighborhood U of x , there exists x 0 ∈ U such that x 0 (t,ω) x . (C3) For L(λ ⊗ P ))-almost all (t, ω) ∈ T × Ω, (t,ω) satisfies the property of overriding desirability of the divisible commodity, i.e., for all x , y ∈ X , there exists z ∈ X such that z ` > y ` , z i ≤ y i for all i < `, and z (t,ω) x . (C4) For L(λ ⊗ P ))-almost all (t, ω) ∈ T × Ω, the holding of the divisible commodity is positive, i.e., e ` (t, ω) > 0. Theorem 5 Let E be a random economy with indivisible goods. Assume that (C1)-(C4) are satisfied. If the Et ’s are almost independent, then (1) essentially all the economies Eω have the same set of core distributions as that of the economy E , viewed as a deterministic economy on the product space; (2) there is a core allocation g for E , viewed as a deterministic economy, such that the gt ’s are almost independent, and for almost all ω ∈ Ω, gω is a core allocation for the economy Eω ; and every core distribution of E can be achieved by such an independent core allocation. 38

See Khan-Rashid [36] for some results on large finite economies with indivisible goods.


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Proof. As in Khan-Yamazaki [37], a set-valued mapping G from P × X × ∆ to X is defined by letting G(, e, p) = {x ∈ X : p · x ≤ p · e and z x ⇒ p · z ≥ p · e} for any ∈ P , e ∈ X , and p ∈ ∆. The usual argument can be used to show that G has a measurable graph. Let Ψ (t, ω, p) = G(E (t, ω), p). As usual, Ψω (t, p) denotes G(Eω (t), p). An allocation f for E is called a weakly competitive allocation, if there is a price systemRRp such that for L(λ ⊗ P ))-almost all (t, ω) ∈ T × Ω, f (t, ω) ∈ RR Ψ (t, ω, p) and T ×Ω f (t, ω)dL(λ ⊗ P ) ≤ T ×Ω e(t, ω)dL(λ ⊗ P ). We shall call (p, f ) a weakly competitive equilibrium and p a weak equilibrium price system. It is obvious to define weakly competitive allocations for economies Eω ’s. As in the proof of Theorem 3 (2), the law of large numbers in Theorem A in the Appendix together with the almost independence assumption on the economy E implies that for almost all ω ∈ Ω, the economy Eω has the same preference-endowment distribution as E , viewed as a deterministic economy on the product space. By the Fubini property, we can choose a subset B of Ω with full probability such that for every ω 0 ∈ B , Eω0 has the same preferenceendowment distribution as E , and the properties of local nonsatiation, overriding desirability and the positive holding of the divisible commodity are satisfied by the economy Eω0 . Since Eω0 and E have the same endowment distribution, their mean endowments must be the same strictly positive vector. Fix ω 0 ∈ B . For (1), we need to show that Eω0 and E have the same set of core distributions. By the overriding desirability of the divisible commodity, it is easy to see that any price system p associated with a weakly competitive allocation for Eω0 or E must have positive price p ` for the divisible commodity `.39 Thus, Theorem 4 in [37] also implies that the core for Eω0 or E coincides with its corresponding set of weakly competitive allocations. Now, since Eω0 and E have the same preference-endowment distribution, it is clear that for a fixed p ∈ ∆, G(Eω0 (t), p) is nonempty for almost all t ∈ T if and only if G(E (t, ω), p) is nonempty for almost all (t, ω) ∈ T × Ω, and if so, the correspondences Ψω0 (t, p) on T (written as Ψωp 0 ) and Ψ (t, ω, p) on T × Ω (written as Ψ p ) certainly satisfy part (1) of Lemma 3 in the Appendix. Then, by Lemma 3 (2), DΨ p 0 = DΨ p . If Ψ p (t, ω) is empty on a non-negligible set, so is ω Ψωp 0 (t); in this case, we can still write DΨ p 0 = DΨ p , since both are the empty ω

set.40 It is now easy to see the equivalence of the following statements. (a) µ is a core distribution for the economyREω0 ; R (b) there is p ∈ ∆ such that µ ∈ DΨ p 0 and x ∈X xd µ ≤ T eω0 dL(λ); ω R RR (c) there is p ∈ ∆ such that µ ∈ DΨ p and x ∈X xd µ ≤ T ×Ω e dL(λ ⊗ P ); (d) µ is a core distribution for the economy E , viewed as a deterministic economy 39

See the last paragraph on p.224 in [37]. There is a slight abuse of notation here. Since DF is usually used to denote the set of distributions of the slections of a correspondence F , which is assumed to be nonempty set valued. 40

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on the product space. Therefore (1) is shown. Next, Proposition 2 (b) in [37] shows that there is a core allocation f for the economy E . Let µ be the distribution of f . ItRRis clear that f is a measurable R selection of Ψ p for some p ∈ ∆ and x ∈X xd µ ≤ T ×Ω e dL(λ⊗P ). By Theorem 2, there is a measurable selection g of Ψ p such that g has the same distribution µ as f and the gt ’s are almost independent. Since the law of large numbers in Theorem A in the Appendix implies that for almost all ω ∈ Ω, the distribution of gω is µ. Thus, for almost all ω, (p, gω ) is a weakly competitive equilibrium of the economy Eω , and hence also a core allocation for the economy. This shows that g is a “global” core allocation. Since the core allocation f is chosen arbitrarily, (2) is thus proven. By the same ideas used in the proof of Theorem 5, one can choose a common null set of ω for different equilibrium price systems in part (3) of Theorems 3 and 4. Thus, it is also true that for almost all ω, Eω and E have the same set of distributions for Walras allocations.41

4 Some asymptotic interpretations Note that Loeb spaces are constructed within a model which is elementarily equivalent to the corresponding standard model.42 It is routine to interpret results from one model to the other. The general possibility of such a translation was already demonstrated by Brown and Robinson in [12], where large finite results on the cores of exchange economies were obtained from some internal counterparts. The procedure usually involves the so-called lifting, pushing-down and transfer, and by now is standard. To illustrate how Theorems 1-5 here can be rewritten in the asymptotic setting, we present in this section some large finite analogs of part of the results in Theorems 1 and 3. The proofs are omitted.43 41 As used in the proof of Theorems 3-5, the exact law of large numbers in Theorem A in the Appendix implies that the preference-endowment distribution of the economy Eω is essentially equal to that of E . Note that two atomless economies with the same preference-endowment distribution on Pmo × R`+ have the same closure with respect to weak convergence for their respective sets of distributions of Walras allocations (see Propostion 5 in [31], p.155). Since a basic feature of nonstandard methods is to capture an ideal limit in a systematic way, the closedness of those sets of distributions of Walras allocations can be shown by the usual argument, and thus the type of equality as in Theorem 3 (3) can be obtained. The first problem with such an approach is that it may still need considerable work to show the closedness rigorously; and the second problem is that it inherently depends on the known approximate result. When such an approximate result is not available as in the case considered in Theorems 4 and 5 , one has to show the approximate result first. The approach used here depends on those general results on correspondences, which are already shown in [57] and here. We have seen that based on those general results, simple and uniform proofs can be given for all the different cases. 42 See, for example, [1] and [32]. 43 A (set-valued) process in the ideal setting corresponds to a triangular array of (correspondences) random variables. [59] and [60] include some detailed proofs for translating idealized exact results on processes to large finite approximate results on triangular arrays of random variables. Since


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4.1 An asymptotic law of large numbers for correspondences Let X be a complete separable metrizable space with a totally bounded metric d , ρX the corresponding Prohorov metric on the space M(X ) of Borel probability distributions on X , and σ the Hausdorff metric on subsets of M(X ) induced by ρX .44 Let (Ω, A, P ) be a common atomless sample probability space. For each n ≥ 1, let {cin }ni=1 be a finite sequence of positive real numbers with sum 1 and probability space limn→∞ max{cin : 1 ≤ i ≤ n} = 0. Let (Tn , Tn , λn ) be a finite P with Tn = {1, 2, · · · , n}, Tn – the power set of Tn , and λn (A) = i ∈A cin for each A ⊆ Tn , and Fn a closed set-valued process from Tn × Ω to X . Since for each fixed level n and index t ∈ Tn , Fn (t, ·) defines a correspondence on Ω, we shall 45 call the collection {Fn }∞ n=1 a triangular array of correspondences. We also need a function to measure the “degree” of independence for correspondences on Ω. As noted in Lemma 1 in the Appendix,46 a closed valued measurable correspondence G from Ω to X can be viewed as a random variable G¯ from Ω to the space FX of closed subsets of X endowed with the Hausdorff distance derived from the totally bounded metric d . Let ρ(FX )2 be the Prohorov metric on the space of distributions on the product space FX × FX . For two closed valued measurable correspondences G1 , G2 from Ω to X , let −1 −1 −1 τ (G1 , G2 ) = ρ(FX )2 P G¯ 1 , G¯ 2 , , P G¯ 1 × P G¯ 2 −1 is the joint distribution of G¯ 1 and G¯ 2 . By Lemma 2 in where P G¯ 1 , G¯ 2 the Appendix, the correspondences G1 and G2 are independent if and only if τ (G1 , G2 ) is zero. Thus, we can say that the correspondences G1 and G2 are approximately independent, if τ (G1 , G2 ) is small, Assume that the following tightness condition holds: for any ε > 0, there is a compact set Kε in X such that for any n ≥ 1, (λn ⊗ P )({(tn , ω) : Fn (tn , ω) ⊆ Kε }) > 1 − ε. Let An be a nonempty subset of Tn , and FnAn the restriction of Fn to An × Ω, which is viewed as a set valued process on (An × Ω, Tn An ⊗ A, λAn n ⊗ P ). Here Tn An is the power set of An , and λAn n the probability measure on An An rescaled from λn . For a sample realization ω, Fnω is simply the relevant sample An , D An correspondence. Thus, σ DFnω measures the Hausdorff distance between Fn An , the respective sets of distributions of selections of the sample correspondenceFnω An as a correspondence on the product space. For (t , t ) ∈ T ×T and Fn , viewed 1 2 n n, τ Fnt1 , Fnt2 simply measures the degree of independence between Fnt1 and Fnt2 . Proposition 2 itself only involves a triangular array of random variables, it is clear that those proofs work for this case. In addition, the punchlines of those proofs can also be used to prove the result on a triangular array of correspondences in Proposition 1. 44 For the definitions of the Prohorov metric and the Hausdorff distance, see, respectively, [10], p.238, and [31], p.16. 45 If F is, in fact, point valued, the sequence is called a triangular array of random variables in n the probabilistic literature. 46 The result is proven in Proposition 2.3 in [57].

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The following proposition is a discrete analog of the equivalence of (1) and (2) in Theorem 1. Proposition 1 The following are equivalent: (1) the processes Fn ’s are asymptotically independent, which means that for any δ > 0, lim λn ⊗ λn (t1 , t2 ) ∈ Tn × Tn : τ Fnt1 , Fnt2 ≤ δ = 1; n→∞

(2) the Fn ’s satisfy the asymptotic law of large numbers, i.e., there is an α in the open interval (0, 1) such that for any An ⊆ Tn , if λn (An ) > α, then for any ε > 0, n o An , D An ≤ ε = 1. lim P ω ∈ Ω : σ DFnω Fn n→∞

In contrast to the law of large numbers for correspondences in the liturature,47 Proposition 1 covers the cases of non-iid, general weighted averages and weak dependence;48 and moreover, asymptotic independence is not only sufficient, as the independence condition in the liturature, but also necessary. 4.2 Asymptotic equilibria in random economies In this subsection, we translate part of Theorem 3 to an asymptotic setting. For simplicity, we take a compact subset P0 of P and a compact subset E0 of R`+ . We assume that every preference in P0 is monotonic and every vector in E0 is strictly positive. Let S = P0 ×E0 and c = (1, 1, . . . , 1). For a metric space Y , we shall use ρY to denote the Prohorov metric for distributions on Y . For each n ≥ 1, define an economy En to be a process (n , en ) from (Tn × Ω, Tn ⊗ A, λn ⊗ P ) to S , where Tn = {1, 2, . . . , n} with the counting probability measure λn on its power set Tn ,49 and (Ω, A, P ) is a common sample space. Thus the sequence {En }∞ n=1 is a triangular array of S -valued random variables. For (t1 , t2 ) ∈ Tn ×Tn , let µt1 t2 , µt1 and µt2 be respectively the joint and individual distributions of the random preference-endowments Ent1 and Ent2 of agents t1 and t2 in the n-th random economy En . Proposition 2 Assume that the random economies En ’s have asymptoticly independent agents, i.e., for any δ > 0, lim λn ⊗ λn {(t1 , t2 ) ∈ Tn × Tn : ρS 2 (µt1 t2 , µt1 × µt2 ) ≤ δ} = 1.

n→∞

Then for any given ε > 0, there is a positive integer N such that for any n > N , there exists a strictly positive deterministic price system pn , an allocation xn : Tn × Ω → R`+ , and a subset An of Ω with the following properties. 47

These known laws are, of course, all in the discrete form. Note that condition (1) is a version of weak dependence (see [60] for more discussion on this type of conditions). 49 This means that for A ⊆ T , λ (A) = |A|/n. To be consistent with the literature on large finite n n economies, we work in this subsection with arithmetic averages rather than the weighted averages in the Subsection 4.1. 48


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(a) The processes xn ’s are approximately independent. That is, λn ⊗ λn {(t1 , t2 ) ∈ Tn × Tn : ρX 2 (νt1 t2 , νt1 × νt2 ) ≤ ε} > 1 − ε, where νt1 t2 , νt1 and νt2 are respectively the joint and individual distributions of xnt1 and xnt2 . (b) The event An has large probability in the sense that P (An ) > 1 − ε. (c) For all ω ∈ An , the sample distribution τnω of xnω is close to the distribution τn of xn , viewed as a random variable on the product space, i.e., ρX (τnω , τn ) ≤ ε. (d) For all ω ∈ An , xnω is an approximate equilibrium of the economy Enω . That is, (1) the budget is exactly satisfied: for all t ∈ Tn , pn xnω (t) = pn enω (t), (2) the demand relation is approximately satisfied: |{t ∈ Tn : ∀y ∈ R`+ , pn y ≤ pn enω (t) ⇒ y − εc 6n(t,ω) xnω + εc}| ≥ 1 − ε. n (3) mean demand is approximately equal to mean supply: P P t∈Tn xnω (t) t∈Tn enω (t) − < ε, n n Note that Proposition 2 only contains asymptotic analogs of part of Theorem 3 (1). Comparing with the neat expression in the ideal case, the large finite result here is already too cumbersome. It will certainly be more complicated to prove this asymptotic result directly. On the other hand, in contrast to the discrete model considered in [30], the preference relations used here could be neither convex nor complete, and a rather weak condition of independence is allowed for the primitive data. 5 Concluding remarks The common sensical idea about individual uncertainty is that random events related to an individual economic entity essentially have no correlation with those of most others in an economic system and the overall effect of all such sources of randomness is small. Just as the usual continuum model having provided a usuful framework to express and analyze the idea of perfect competition where individual agents have small influences over a large market, the hyperfinite processes on Loeb spaces together with the associated law of large numbers also capture the idea of smallness of the overall effect (now in terms of randomness) as well as the meaning of “most others” in a right way so that this type of “smallness” becomes exactly negligible. This reduces considerably the analytic complexity of the relevant economic models as shown in the proofs of Theorems 3-5.50 50 It is certainly not easy to prove all the asymptotic analogs of Theorems 3-5 directly. On the other hand, even though the proof of Theorem 2 is also delicate, it is very easy to be applied to other problems as shown in the proof of Theorems 3-5; one certainly does not expect its asymptotic analog to be as useful.

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Let F be a set-valued process from the Loeb product space (T × Ω, L(T ⊗ A), L(λ ⊗ P )) to a complete separable metric space such that the Ft ’s are almost independent. If F is, in fact, measurable with respect to the usual product algebra L(T ) ⊗ L(A), then, Proposition 6.5 in [59] implies that for almost all t ∈ T , Ft is a constant correspondence, i.e., F is completely trivial. Thus, the larger measure-theoretic framework with Fubini property taken here is crucial for all the results presented in Theorems 1-5 of this paper. Such a larger framework does not exist when the unit Lebesgue interval is used as an index space.51 By rewriting the classical law of large numbers in terms of a Banach limit, one obtains, as in [23], an exact law of large numbers in terms of sample means over a purely finitely additive atomless measure µ on N. The problem with such a measure is that the usual convergence property fails. For example, let hn be the characteristic function of the set {0, 1, · · · , n}. Then the hn ’s are monotone R increasing with the constant R function 1 as the limit. However, limn→∞ N fn d µ = 0, which is not equal to N 1d µ = 1. Thus, common analytic techniques are not applicable to such measures. It is also important to note that since the underlying measure is only finitely additive, one cannot formulate sample distributions with useful properties.

6 Appendix In this Appendix, we give detailed proofs of Theorems 1 and 2. The proof of Theorem 2 is based on delicate computations and estimations involving both internal and external entities. Some basic knowledge of nonstandard analysis52 is needed to understand the proof. In Subsection 6.1, we collect some background results in [56], [57], [59] and [60]. The proof of Theorem 1 is given in Subsection 6.2, which is relatively easy. Subsection 6.3 presents the proof of Theorem 2. The idea of allocating almost independent events into several collections of almost independent subevents lies at the heart of this proof. Two more corollaries of Theorem 1 are also given in Subsection 6.4, which involves the Bochner and Gel0 fand integrals of correspondences in a separable Banach space.53

6.1 Some background results We shall first discuss briefly about Loeb spaces and Loeb product spaces. Let T be a hyperfinite set, T the internal algebra of all internal subsets of T , and λ an internal finitely additive probability measure on (T , T ). Define a real valued set function ◦ λ on (T , T ) such that for each A ∈ T , ◦ λ(A) is the standard part ◦ (λ(A)) of λ(A). It is observed in [40] that a decreasing sequence 51

See the Appendix of [59]. For example, a thorough reading of the survey paper [39] might already provide a good basis. 53 For earlier work on integration of Banach space valued correspondences, see the survey by Yannelis in [65]. [58] includes a number of special properties for the integrals of correspondences on a Loeb space. 52


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of internal sets in T with empty intersection must have empty set as its n-th term for large enough n ∈ N. This means that ◦ λ is, in fact, a countably additive measure on (T , T ). By the usual Caratheodory extension theorem, ◦ λ can be extended to a probability measure L(λ) on the σ-algebra generated by A. Let (T , L(T ), L(λ)) be the completion of the space (T , σ(T ), L(λ)). This completion is usually refered to as a Loeb space.54 This standard probability space has been used as the index space for processes or set-valued processes or as a space of economic agents in this paper. Starting with another internal probability space (Ω, A, P ), the corresponding standardization, the Loeb space (Ω, L(A), L(P )) is our sample probability space. Both Loeb probability spaces are assumed to be atomless. Note that the internal product space (T ×Ω, T ⊗A, λ⊗P ) is also an internal probability space. The corresponding Loeb space is denoted by (T × Ω, L(T ⊗ A), L(λ ⊗ P )), which is called the Loeb product space. All our analysis has been centered around this special product space.55 Similarly, for a positive integer n, let (T n , T n , λn ) be the n-fold internal product space of (T , T , λ); the associated n-fold Loeb product spaces are denoted by (T n , L(T n ), L(λn )). For the sake of completeness, we include a formal definition of the consistency law in distribution for a general process taking values in X .56 Definition A. Let f be a process from a Loeb product space (T × Ω, L(T ⊗ A), L(λ⊗P )) to a complete separable metric space X . We say that f satisfies the consistent law of large numbers in distribution (or simply the consistency law in distribution) if for any internal set A ∈ T with L(λ)(A) > 0, the process f A from (A × Ω, L(T A ⊗ A), L(λA ⊗ P )) to X satisfies the law in distribution. In other words, for L(P )-almost all ω ∈ Ω, the distribution µAω induced by the random variable fωA from (A, L(T A ), L(λA )) to X is equal to the distribution µA induced by the random variable f A from (A × Ω, L(T A ⊗ A), L(λA ⊗ P )) to X . Here f A is the restriction of f to A × Ω, T A is the collection of all internal subsets of A, and λA is the internal probability measure on (A, T A ) rescaled from λ. In the following theorem, we collect from [56], [59] and [60] a number of results which are used in this paper.57 The notion of a separating class is used in part (3) of the theorem. Let {ϕm }∞ m=1 be a sequence of bounded realvalued continuous functions on X . This sequence is called a separating class if it distinguishes different distributions on X . That is, any distributions µ and ν on R R X are equal if X ϕm d µ = X ϕm d ν for all m ≥ 1. 54

For details, see [40], [1] and [32]. As noted earlier, this product is strictly bigger than the usual product. 56 This concept is studied in detail in [56] and [59]. In contrast to the definition below, we did not require the set A to be internal in Definition 1 in Section 2. There is no essential difference between the two, since any Loeb measurable set can be approaximated by an internal set with a null symmetric difference (see [40]). 57 The equivalence of (1), (2) and (3) is part of Theorem 7.6 in [59] (see also Theorem 4 in [56]). The equivalence of (2) and (4) is Theorem 3 in [60]. The implication (5) =⇒ (2) is obvious, while (2) =⇒ (5) is part of Proposition 4.9 in [60]. 55

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Theorem A. Let f be a process from a Loeb product space (T × Ω, L(T ⊗ A), L(λ ⊗ P )) to a complete separable metric space X . Then the following are equivalent. (1) The process f satisfies the consistency law in distribution. (2) The random variables ft ’s are almost surely pairwise independent, i.e., for L(λ ⊗ λ)-almost all (t1 , t2 ) ∈ T × T , ft1 and ft2 are independent. (3) Let {ϕm }∞ m=1 be a sequence of bounded real-valued continuous functions on X , which is a separating class for all distributions on X . Then, for any m ≥ 1, the real-valued random variables ϕm (ft )’s are almost surely uncorrelated, i.e., for L(λ ⊗ λ)-almost all (t1 , t2 ) ∈ T × T , ϕm (ft1 ) and ϕm (ft2 ) are uncorrelated. (4) The random variables ft ’s are almost mutually independent, i.e., for any n ≥ 2, ft1 , ft2 , . . . , ftn are mutually independent for L(λn )-almost all (t1 , t2 , . . . , tn ) ∈ T n . (5) Let g be any process from (T × Ω, L(T ⊗ A), L(λ ⊗ P )) to X . Then, the processes f and g are independent in the sense that for L(λ ⊗ λ)-almost all (t1 , t2 ) ∈ T × T , ft1 and gt2 are independent. Unless otherwise noted, we shall use (X , d ) to denote a Polish space with a totally bounded metric d on X .58 Then the space FX of nonempty closed subsets of X endowed with the Hausdorff distance ρ derived from d is still a Polish / ∅}. For a space.59 For each open set O in X , let EO = {A ∈ FX : A ∩ O = closed valued correspondence F from some measure space to X , let F¯ denote the mapping from the measure space to the hyperspace FX of nonempty closed sets such that the value of F¯ at each point is the nonempty closed set obtained by evaluating the correspondence at that point. The following lemma, which is Proposition 2.3 in [57], characterizes measurable correspondences as well as the Borel σ-algebra B (FX ) of the Polish space FX . Note that the characterization of compact valued measureable correspondences in terms of measurable functions in a hyperspace of compact sets was established by Debreu in [D]. Here a general case is considered. Lemma 1 The Borel σ-algebra B (FX ) is generated by the collection {EO : O is an open set in X }. Let F be a closed valued correspondence from a probability space (Λ, C , ν) to (X , d ), then F is a measurable correspondence if and only if F¯ is a measurable mapping from (Λ, C , ν) to (FX , B (FX )). Let X be a locally compact seprable metric space. Then the collection F (X ) of nonempty closed sets in X , joined by the empty set, is compact and metrizable when the enlarged collection is endowed with the topology of closed convergence ([31], p.19). As noted in Remark 2.4 (2) in [57], The σ-algebra generated by this topology is the same as B (FX ). Thus, as long as measurability, distributions and independence are concerned on FX , it is irrelevent to know whether the underlying topology on FX is the topology of closed convergence or the topology induced by the Hausdorff distance defined from an equivalent totally bounded metric on X . 58 59

A Polish space is a complete separable metrizable topological space. See, for example, [22]. For the definition of Hausdorff distance, see [31], p.16.


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6.2 Proof of Theorem 1 In this subsection, we adopt the same notation as the above subsection. The following lemma shows that the notions of independence for correspondences and for their associated point valued mappings into the hyperspace FX are the same. Lemma 2 Let F and G be closed valued measurable correspondences from a probability space (Λ, C , ν) to a Polish space X with a totally bounded metric d . Then F and G are independent correspondences if and only if the random variables F¯ and G¯ (Λ, C , ν) to FX are independent. Proof. If the random variables F¯ and G¯ are independent, then for any open sets O1 , O2 in X , the events F¯ −1 (EO1 ), G¯ −1 (EO2 ) are independent, so are the events F −1 (O1 ) and G −1 (O2 ). This means that the correspondences F and G are independent. On the other hand, if F and G are independent as correspondences, then for any open sets O1 , O2 in X , F¯ −1 (EO1 ) and G¯ −1 (EO2 ) are independent events, and so are the complements F¯ −1 (FX − EO1 ) and G¯ −1 (FX − EO2 ). By Lemma 1, {FX − EO : O is an open set in X } also generates the Borel σ-algebra B (FX ). Since this collection is closed under finite intersections,60 the independence of F¯ and G¯ follows from the usual extension theorem ([42], p.237). The following lemma establishes some equivalence results for various notions relevant to distributions of correspondences on Loeb spaces. For Rn -valued correspondences on a general probability space, [28] showed that versions of (1) or (3) imply the sets of distributions of selections of the relevant correspondences having the same closure; a converse is also shown in Proposition 2.7 of [6].61 The Loeb measure framework allows an exact equality as in (2), which is very useful for applications. Lemma 3 Let F and G be closed valued measurable correspondences from atomless Loeb probability spaces (Ω 1 , L(A1 ), L(P1 )) and (Ω 2 , L(A2 ), L(P2 )) to a Polish space X respectively. Then the following are equivalent: (1) for every open set O in X , L(P1 )(F −1 (O)) = L(P2 )(G −1 (O)); (2) DF = DG ; (3) F¯ and G¯ have the same distributions on the hyperspace FX . Proof. (1) =⇒ (2) is already shown in Proposition 3.5 in [57]. To show (2) =⇒ (1), we need the following claim: for any given open set O in X , L(P1 )(F −1 (O)) = sup{µ(O) : µ ∈ DF }. Denote the Loeb measurable set F −1 (O) by A, and let FO be the correspondence from the measurable space Tn 60 It is easy to check that

i =1

F X − EO i

= FX − E∪ n

O i =1 i

. This also means that {FX − EO :

O is an open set in X } is a π-system. 61 Note that our proof of (2) =⇒ (1) establishes the type of equality L(P )(F −1 (O)) = sup{µ(O) : 1 µ ∈ DF }, which makes the implication very transparent. The equality is not used in [6].

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(A, L(A1 ) ∩ A) into O defined by FO (ω1 ) = F (ω1 ) ∩ O for ω1 ∈ A. Then the of F with A×O, which is measurable graph of FO is the intersection of the graph in the product σ-algebra L(A1 ) ∩ A ⊗ B (O). Since O is still a Polish space, it follows from Theorem 14.3.2 in [38] (on p.166) that there exists a measurable selection f1 of FO . Take any measurable selection f2 of F . Let f (ω1 ) = f1 (ω1 ) if / A. Then f is a measurable selection of F ω1 ∈ A, and f (ω1 ) = f2 (ω1 ) if ω1 ∈ with L(P1 )(f −1 (O)) ≥ L(P1 )(A). Hence L(P1 )(F −1 (O)) ≤ sup{µ(O) : µ ∈ DF }. Since the other direction of the inequality is obvious, the claim is thus proven. Similarly, L(P2 )(G −1 (O)) = sup{µ(O) : µ ∈ DG }. Therefore, (2) =⇒ (1) is now obvious. By evaluating the distributions of F¯ and G¯ on sets of the form EO in FX , we can obtain (3) =⇒ (1). If (1) holds, then the distributions of F¯ and G¯ agree on sets of the form EO in FX , and hence also on sets of the form FX − EO . As noted in the proof of Lemma 2, this latter collection is a π-system. By the uniqness of measures ([14], p.45), the two distributions must be the same. Thus (3) holds. We are now ready to give a Proof of Theorem 1. By Lemma 3, it is clear that the set-valued process F satisfies the consistency law if and only if the process F¯ taking values in FX satisfies the consistency law in distribution. By Theorem A, the latter assertion is equivalent to the fact that for L(λ ⊗ λ)-almost all (t1 , t2 ) ∈ T × T , F¯ t1 and F¯ t2 are independent. Lemma 2 says that the independence of the correspondences Ft1 and Ft2 is equivalent to that of the random variables F¯ t1 and F¯ t2 .62 Hence (1) and (2) are equivalent. Next, (3) =⇒ (2) is obvious. If (2) holds, then the above paragraph shows that for L(λ ⊗ λ)-almost all (t1 , t2 ) ∈ T × T , F¯ t1 and F¯ t2 are independent. By Theorem A, we know that for all n ≥ 2, F¯ t1 , F¯ t2 , . . . , F¯ tn are independent for L(λn )-almost all (t1 , t2 , . . . , tn ) ∈ T n , which certainly implies (3) by taking the inverse images of these point mappings on the sets of the form EO in FX . 6.3 Proof of Theorem 2 To prove Theorem 2, we need a number of preliminary lemmas. The first of such lemmas is a simple corollary of Theorem A. Lemma 4 Let C be a Loeb product measurable set in L(T ⊗ A). Then the following are equivalent: (1) for L(λ⊗λ)-almost all (t1 , t2 ) ∈ T ×T , the events Ct1 and Ct2 are independent; (2) let D be any given Loeb product measurable set in L(T ⊗ A); then, for L(λ ⊗ λ)-almost all t1 , t2 ∈ T × T , the events Ct1 and Dt2 are independent. 62 By using the notion of semi-independence and Corollary 7.12 in [59], the almost sure pairwise independence of the F¯ t ’s is equivalent to the fact that the events F¯ t−1 (O)’s are independent for all open sets O. Hence, the correspondences Ft ’s are independent if and only if for any open set O in X the events Ft−1 (O)’s are almost surely pairwise independent.


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Proof. Define real-valued processes f and g on the Loeb product space by letting f (t, ω) = χC (t, ω) and g(t, ω) = χD (t, ω), the indicator functions of C and D. Then the equivalence of (1) and (2) in the lemma follows from the equivalence of (2) and (5) in Theorem A (or the equivalence of (2) and (4) in Theorem 2 of [56]). The following lemma shows that almost sure pairwise independence is preserved under disjoint unions and proper differences. Lemma 5 Let A and B be Loeb product measurable sets in L(T ⊗ A). Assume that the events At ’s as well as the events Bt ’s are almost surely pairwise independent. Then (1) if A and B are disjoint, and C = A ∪ B , then the events Ct ’s are almost surely pairwise independent; (2) if B ⊆ A and C = A − B , then the events Ct ’s are almost surely pairwise independent. Proof. We first consider (1). By the assumptions of disjointness and almost sure pairwise independence, we can obtain for L(λ ⊗ λ)-almost all (t1 , t2 ) ∈ T × T , = P (At1 ∪ Bt1 ) ∩ Ct2 = P At1 ∩ Ct2 + P Bt1 ∩ Ct2 P Ct1 ∩ Ct2 ' P (At1 )P (Ct2 ) + P (Bt1 )P (Ct2 ) = P (Ct1 )P (Ct2 ), where P At1 ∩ Ct2 ' P (At1 )P (Ct2 ) and P Bt1 ∩ Ct2 ' P (Bt1 )P (Ct2 ) follow from Lemma 4. Hence the events Ct ’s are almost surely pairwise independent. For (2), we note that Ct = At −Bt . By applying Lemma 4 (2) to the pairs A, C and B , C , we know that for L(λ ⊗ λ)-almost all (t1 , t2 ) ∈ T × T , P (At1 ∩ Ct2 ) ' P (At1 )P (Ct2 ) and P (Bt1 ∩ Ct2 ) ' P (Bt1 )P (Ct2 ), and hence P (Ct1 ∩ Ct2 )

=

P (At1 ∩ Ct2 ) − P (Bt1 ∩ Ct2 )

' P (At1 )P (Ct2 ) − P (Bt1 )P (Ct2 ) = P (Ct1 )P (Ct2 ). Therefore the events Ct ’s are also almost surely pairwise independent. In the following two lemmas, we show that almost independent events can be uniformly decomposed into two parts with common proportion of the probability of the relevant original events such that almost independence remains valid for those subevents. We consider the case of “fair-cutting” first, and then the general case. Note that the atomlessness of both L(λ) and L(P ) is used in an essential way. Lemma 6 Let A be an internal set in T ⊗ A. Assume that the events At ’s are almost surely pairwise independent. Then, A can be partitioned into two disjoint internal parts C and D such that for L(λ)-almost all t ∈ T , L(P )(Ct ) = L(P )(Dt ) = 1/2L(P )(At ) and the events Ct ’s as well as the events Dt ’s are almost surely pairwise independent.

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Proof. For any n ∈ N, the atomlessness of both L(λ) and L(P ) implies that there is an internal partition T1 , · · · , Tn of T such that L(λ)(Ti ) = 1/n; and n independent internal events B1 , · · · , Bn in Ω with probability 1/2.63 By applying Lemma 4 (2) to the pair, A and T × Bi , we know that Bi is independent of At for L(λ)-almost all t ∈ T . Similarly, each Bi ∩ Bj is independent of At for L(λ)almost all t ∈ T . Thus, for L(λ)-almost all t ∈ T , P (At ∩ Bi ) ' P (At )P (Bi ) and P (At ∩Bi ∩Bj ) ' P (At )P (Bi ∩Bj ) for all i , j , and hence |P (At ∩Bi )−P (At )/2| < 1/n for all i and |P (At ∩ Bi ∩ Bj ) − P (At )/4| < 1/n / j . By applying for all i = Lemma 4 (2) to the sets A and A ∩ T × (Bi ∩ Bj ) , we obtain that for L(λ ⊗ λ)almost all (t1 , t2 ) ∈ T × T , P (At1 ∩ At2 ∩ Bi ∩ Bj ) ' P (At1 )P (At2 ∩ Bi ∩ Bj ) for all i , j , and hence |P (At1 ∩ At2 ∩ Bi ∩ Bj ) − P (At1 )P (At2 )/4| < 1/n for all i = / j . By the ℵ1 -saturation of the nonstandard model, the above three inequalities with 1/n on the right side hold for some n ∈ ∗ N∞ . Define an internal set C = {(t, ω) : for t ∈ Ti , ω ∈ At ∩ Bi }. Then Ct = At ∩ Bi if t ∈ Ti . Since for L(λ)-almost all t ∈ T , |P (At ∩ Bi ) − P (At )/2| < 1/n for all i , we obtain that P (Ct ) ' P (At )/2. For L(λ ⊗ λ)-almost all (t1 , t2 ) ∈ T × T , we have P (Ct1 ∩ Ct2 ) = P (At1 ∩ At2 ∩ Bi ∩ Bj ) ' P (At1 )P (At2 )/4 ' P (Ct1 )P (Ct2 ), and hence the events Ct ’s are almost surely pairwise independent. Next define D = A − C . Then Dt = At − Ct , and hence L(P )(Dt ) = 1/2L(P )(At ). By Lemma 5, the events Dt ’s are also almost surely pairwise independent. Lemma 7 Let α ∈ [0, 1] and A be an internal set in T ⊗ A. Assume that the events At ’s are almost surely pairwise independent. Then, A can be partitioned into two disjoint internal parts C and D such that for L(λ)-almost all t ∈ T , L(P )(Ct ) = αL(P )(At ), L(P )(Dt ) = (1 − α)L(P )(At ), and the events Ct ’s as well as the events Dt ’s are almost surely pairwise independent. Proof. First assume that α = m/2n for some n ≥ 1 and 1 ≤ m < 2n . By applying Lemma 6 repeatedly, we can partition A into 2n parts Ai , i = 1, 2, . . . , 2n such that the events Ait ’s are almost surely pairwise independent, and for L(λ)-almost all t ∈ T , L(P )(Ait ) = L(P )(At )/2n for all i . Take C to be union of m such Ai ’s and D to be the union of the rest. Then the result follows from Lemma 5 (1). Next, we work with an arbitrary α ∈ [0, 1]. Choose a sequence αn , n = 1, 2, . . . , of numbers in the form specified in the previous paragraph such that 63 This independence is in terms of the relevant Loeb probability space, which is very different from ∗-independence, the transferred version of independence. Starting from Ω, one can use the Lyapunov theorem to keep cutting events into two parts with equal Loeb probability to obtain an n internal S partition {Ci1 ...in : i1 , . . . , in = 0, 1} with L(P )(Ci1 ...in ) = 1/2 . For j = 1, . . . , n, take Bj = i =1 Ci1 ...in . This procedure is also used in the proof of Theorem 6.2 in [59]. j


537

its limit is α. For each n, we can find an internal subset C n of A such that for L(λ)-almost all t ∈ T , P (Ctn ) ' αn P (At ), and the events Ctn ’s are almost surely pairwise independent. By the extention principle, we can obtain an internal sequence C n , n ∈ ∗ N which extends the original sequence of C n ’s. Since for all n ∈ N, C n ⊆ A, λ {t ∈ T : |P (Ctn ) − αn P (At )| < 1/n} > 1 − 1/n and λ ⊗ λ {(t1 , t2 ) ∈ T × T : |P (Ctn1 ∩ Ctn2 ) − P (Ctn1 )P (Ctn2 )| < 1/n} > 1 − 1/n, the spillover principle implies that these relations hold for some n = h ∈ ∗ N. Let C = C h . Since the numbers αn ’s converge to α, we have αh ' α. Thus for L(λ)-almost all t ∈ T , L(P )(Ct ) = αL(P )(At ), and the events Ct ’s are almost surely pairwise independent. Define D = A − C . By Lemma 5 (2), the events Dt ’s are also almost surely pairwise independent. The following lemma is very much in the spirit of Proposition 3.3 in [57]. Instead of working with a sequence of correspondences on the simple Loeb space (Ω, L(A), L(P )) there, we work with a fixed correspondence and a sequence of its selections on the Loeb product space. The proofs are essentially the same except we have to ensure an almost sure pairwise condition to be satisfied by a limit process. Lemma 8 Let µ be a Borel probability measure on a Polish space X , F a closed valued measurable correspondence from the Loeb product space (T × Ω, L(T ⊗ A), L(λ ⊗ P )) to X , and f n , n = 1, 2, . . . , a sequence of measurable selections of F . Assume that for each n ≥ 1, the random variables ftn ’s are almost surely pairwise independent. If the sequence of distributions induced by f n on the Loeb product space converges weakly to µ, then there is a selection f of F with distribution µ which also has almost surely pairwise independent random variables. Proof. We adopt the proof of Proposition 3.3 in [57] here. The situation here is simpler in the sense that we are working with a constant sequence of correspondences. Take a sequence {ϕm }∞ m=1 of bounded continuous functions as in that proof. By Theorem 6.1 on p.40 of [50], this sequence is a separating class for all distributions on X .64 For each n ∈ N, we can take an internal T ⊗A-measurable lifting g n : Ω → ∗ X for f n . Since our nonstandard model is ℵ1 -saturated, we can n extend the sequence {g n }∞ n=1 to an internal sequence {g }n∈∗ N of measurable ∗ ∗ functions from (T × Ω, T ⊗ A) to ( X , B (X )). By the proof of Proposition 3.3 in [57], we know that there is H ∈ ∗ N∞ such that for any r ∈ ∗ N∞ , r ≤ H , there is a measurable selection f r of F such that the distribution of f r is µ and ◦ (g r (t, ω)) = f r (t, ω) for L(λ ⊗ P )-almost all (t, ω) ∈ T × Ω. 64

For a definition, see the paragraph above Theorem A.

538

Y. Sun

Since each f n has almost surely pairwise independent random variables, the fact that for L(λ)-almost all t ∈ T , gtn is still an internal lifiting of ftn 65 implies that Z Z Z ∗ ∗ ∗ n ∗ n n ϕm (gt1 ) ϕm (gt2 )dP ' ϕm (gt1 )dP ϕm (gtn2 )dP Ω

Ω

Ω

holds for each m ∈ N and for L(λ ⊗ λ)-almost all (t1 , t2 ) ∈ T × T . Hence, for each m ∈ N, the following inquality λ ⊗ λ {(t1 , t2 ) : | E ∗ ϕm (gtn1 ) ∗ ϕm (gtn2 ) −E ∗ ϕm (gtn1 ) E ∗ ϕm (gtn2 ) | < 1/n} > 1 − 1/n holds for all n ∈ N. By the Permanence Principle, we can find H 0 ∈ ∗ N∞ such that the previous inequality holds for all m ∈ N and for all n with 1 ≤ n ≤ H 0 . Take r = min{H , H 0 } and let f = f r . Then, for each m ∈ N, it is clear that the process ϕm (f ) has almost surely uncorrelated random variables. Since the ϕm ’s form a separating class for all distributions on X , Theorem A implies that the process f has almost surely pairwise independent random variables. The other required properties of f are already shown, and hence we are done. We are now ready to give a Proof of Theorem 2. We endow X with a totally bounded metric d . For any two finite Borel measures ν1 and ν2 on X , define ρ(ν1 , ν2 ) = inf{δ > 0 : for all Borel set C in X , ν1 (C ) ≤ ν2 (B (C , δ)) + δ}, where B (C , δ) = {x ∈ X : ∃y ∈ C , d (x , y) < δ}. Note that when ν1 and ν2 are probability distributions, ρ is the same as the usual Prohorov metric (see [51], p.75). Let F¯ denote the mapping from T × Ω to the hyperspace FX of nonempty closed sets such that the value of F¯ (t, ω) is the closed set F (t, ω), and τ the distribution of F¯ on FX with atomless part τc and atoms {D1 }, {D2 }, . . .. Let R i = F¯ −1 ({Di }) for i ≥ 1. One can choose internal sets H i ’s such that L(λ ⊗ P )(R i ∆H i ) = 0 for all i ≥ 1 and the H i ’s are disjoint. Let H0 be the complement of all the Hi for i ≥ 1. For i ≥ 0, denote the restriction of the measure L(λ ⊗ P ) to H i by σi , and σi (fH i )−1 by µi . It is clear that −1 τc = σ0 F¯ |H 0 . We shall now fix any > 0. Let G be the correspondence from (FX , τc ) to X defined by G(C ) = C for all closed set C in X . Then, it is clear that the correspondences F |H 0 and G are equally distributed.66 It is known that for two equally distributed correspondences, each on an atomless measure space, the closure of the sets of distributions of measurable selections are the same.67 Since f |H 0 is a measurable selection of F |H 0 with distribution µ0 , one can thus find a 65

See, for example, the proof of the Fubini theorem for Loeb measures in [1]. See [28], [31], p.74, and Lemma 3 in this subsection for this notion. 67 See [28] and [31], p.74, where the result is stated for Euclidean spaces, but the proof works for Polish spaces with minor modifications. 66


539

measurable selection ψ of G such that ρ(τc ψ −1 , µ0 ) < .68 Let φ0 = ψ ◦ F¯ |H 0 −1 −1 0 −1 ψ = τc ψ −1 . Then, φ0 is a measurable and ν0 = σ0 (φ ) = σ0 F¯ |H 0 selection of F |H 0 and ρ(ν0 , µ0 ) < . Next, for each i ≥ 1, we can find a partition Eji , j = 1, . . . , m, of Di such that the Eji ’s are of diameter less than . Pick xji ∈ Eji . By Lemma 7, we can partition H i into m parts Hji such that for each pair i , j , Hji has measure µi (Eji ) and also the events (Hji )t ’s are almost surely pairwise independent. Define a mapping φi from H i to Di by letting φi (t, ω) = xji if (t, ω) ∈ Hji , and let νi = σi (φi )−1 . For / ∅}. Then we have any given Borel set C in X , let Ji = {j : 1 ≤ j ≤ m, C ∩ Eji =   [ X C ∩ Eji  ≤ νi (Eji ) νi (C ) = νi  j ∈Ji

=

X

σi (Hji ) =

j ∈Ji

j ∈Ji

X



µi (Eji ) = µi 

j ∈Ji

[

 Eji  ≤ µi (B (C , )) .

j ∈Ji

Define φ by pasting all the φi ’s together, i.e., for (t, ω) ∈ H i , let φ(t, ω) = φ (t, ω). Then, φ is a measurable selection of F , whose distribution on X is denoted by ν. For any Borel set C in X , i

ν(C ) = ν0 (C ) +

∞ X

νi (C ) ≤ µ0 (B (C , )) + +

i =1

∞ X

µi (B (C , )) = µ(B (C , )) + ,

i =1

and hence ρ(ν, µ) ≤ . We shall now show that the random variables φt ’s are almost surely pairwise independent. Take O to be a countable open base of X which is also closed under finite intersections. For any given O ∈ O , denote Q i = (φi )−1 (O) for i ≥ 0. Then −1 −1 −1 (O) = F¯ |H 0 ψ (O) . Q 0 = (φ0 )−1 (O) = ψ ◦ F¯ |H 0 Since the distribution of F¯ |H 0 is atomless, the symmetric difference of Q 0 with the set −1 −1 ψ (O) − {D1 , D2 , . . . , Di , . . .} F¯ |H 0 / Di , then is thus zero. Note that if for some i ≥ 1, (t, ω) ∈ H i and F¯ (t, ω) = (t, ω) ∈ H i − R i , which is a null set. This means that −1 ψ −1 (O) − {D1 , D2 , . . . , Di , . . .} F¯ |S H i i ≥1

is a null set. Hence, 68 One can also give a direct proof of this fact by using a continuous version of the marriage lemma (see, for example, [31], p.74). It is clear that for any open set O in X , µ0 (O) ≤ τc (G −1 (O). Then part of the detailed argument used in the proof of (iv) =⇒ (i) in Proposition 3.5 in [57] can be used to prove the existence of such a selection ψ.

540

Y. Sun

L(λ ⊗ P ) Q 0 ∆F −1 ψ −1 (O) − {D1 , D2 , . . . , Di , . . .}

=0

which implies that the events Qt0 ’s are almost surely pairwiseSindependent. Now, for i ≥ 1, let Ki = {j : xji ∈ O}. Then Q i = (φi )−1 (O) = j ∈Ki Hji . Since all the events Hji ’s are disjoint, and the events Hji t ’s are almost surely pairwise independent for each j , Lemma 5 (1) implies that the events Qti ’s are almost surely pairwise independent. S∞ Sn Next, it is clear that φ−1 (O) = i =0 Q i . Denote i =0 Q i by W n . Since the Q i ’s are also disjoint, by Lemma 5 (1) again, we know that the events Wtn ’s are almost surely pairwise independent. We can group countably many L(λ ⊗ λ)null sets together to obtain an L(λ ⊗ λ)-null set N0 in T × T such that for all / N0 , and for all n ≥ 1 (t1 , t2 ) ∈ L(P )(Wtn1 ∩ Wtn2 ) = L(P )(Wtn1 ) L(P )(Wtn2 ); by taking n → ∞, we obtain −1 −1 −1 L(P )(φ−1 t1 (O) ∩ φt2 (O)) = L(P )(φt1 (O))L(P )(φt2 (O)).

For any O 0 ∈ O , by applying Lemma 4 (2) to the pair φ−1 (O), φ−1 (O 0 ), we −1 0 obtain that for L(λ⊗λ)-almost all (t1 , t2 ) ∈ T ×T , the events φ−1 t1 (O) and φt2 (O ) 0 are independent. Since one can only obtain countably many pairs (O, O ) from O , by grouping countably many L(λ ⊗ λ)-null sets together, we know that for −1 0 L(λ ⊗ λ)-almost all (t1 , t2 ) ∈ T × T , φ−1 t1 (O) and φt2 (O ) are independent for all 0 O, O ∈ O , which implies that φt1 and φt2 are independent by the usual extension theorem ([42], p.237). Finally, note that the process φ defined above depends on . For = 1/n, we denote the corresponding φ by g n . Then, we obtain a sequence of measurable selections g n , n = 1, 2, . . . , of F . For each n ≥ 1, the random variablesgtn ’s are almost surely pairwise independent, and moreover ρ L(λ ⊗ P )(g n )−1 , µ ≤ 1/n. Since ρ defines the Prohorov metric on the space of probability distributions on X , we know that the sequence of distributions induced by g n on the Loeb product space converges weakly to µ. By Lemma 8, there is a selection g of F with distribution µ which also has almost surely pairwise independent random variables.

6.4 The law of large numbers for Banach space valued correspondences In this subsection, we generalize Corollaries 1 and 2 to the setting of a separable Banach space or the dual of a separable Banach space. Corollary 3 considers the case for the Bochner integral.69 69 For the definition of Bochner integrable functions, see [18], p.44. A survey of some earlier work on the Bochner integrals of correspondences and its applications to economics is precented in [65].


541

Corollary 3 Let X be a separable Banach space and F a set-valued process from the Loeb product space (T × Ω, L(T ⊗ A), L(λ ⊗ P )) to X . Assume that for L(λ ⊗ λ)-almost all (t1 , t2 ) ∈ T × T , Ft1 and Ft2 are independent. Then (1) for L(P )-almost all ω ∈ Ω, ZZ Z Fω dL(λ) = FdL(λ ⊗ P ). T ×Ω

T

where the integral RR is the Bochner integral of correspondences; (2) for any x ∈ T ×Ω FdL(λ⊗P ), there is a selection g of F such that the random RR variables gt ’s are almost surely pairwise independent and x = T ×Ω gdL(λ ⊗ P ). Proof. It is easy to check that Z Z Fω dL(λ) = { idX d µ : µ ∈ DFω and idX is Bochner integrable on (X , µ}, T

X

ZZ T ×Ω

FdL(λ ⊗ P ) Z = { idX d µ : µ ∈ DF and idX is Bochner integrable on (X , µ)}, X

where idX is the identity function on the separable Banach space X . Thus, (1) follows from Theorem RR 1 easily. For (2), take x ∈ T ×Ω FdL(λ ⊗ P ). Then, there is a Bochner integrable RR selection f of F such that x = T ×Ω fdL(λ ⊗ P ). By Theorem 2, there is a selection g of F such that the random variables gt ’s are almost surely pairwise independent RR and the distribution of g is the same as that of f . Hence, we also have x = T ×Ω gdL(λ ⊗ P ). The next corollary considers the Gel0 fand integrals of correspondences taking values in the dual of a separable Banach space.70 Note that the measurability and independence used below are based on σ-algebra generated by the weak∗ topology on X . Corollary 4 Let X be the dual Y ∗ of a separable Banach space Y and F a correspondence from the Loeb product space (T × Ω, L(T ⊗ A), L(λ ⊗ P )) to X . Assume that F is weak∗ measurable and weak∗ compact valued. If for L(λ⊗λ)almost all (t1 , t2 ) ∈ T × T , Ft1 and Ft2 are independent, then (1) for L(P )-almost all ω ∈ Ω, ZZ Z Fω dL(λ) = FdL(λ ⊗ P ). T ×Ω

T 0

where the integral RR is the Gel fand integral of correspondences; (2) for any x ∈ T ×Ω FdL(λ⊗P ), there is a selection g of F such that the random RR variables gt ’s are almost surely pairwise independent and x = T ×Ω gdL(λ ⊗ P ). 70 See [18], p.53 for the definition of Gel0 fand integrable functions. [58] also includes several useful properties on the Gel0 fand integrals of correspondences on Loeb spaces.

542

Y. Sun

Proof. Since Y is separable, we can choose a sequence {ym }∞ m=1 from the unit ball of Y such that the linear space spanned by the sequence is dense in Y . Define a metric dw on the dual Y ∗ = X by letting ∞ X 1 |x ∗ (ym ) − y ∗ (ym )| dw (x , y ) = 2m ∗

∗

m=1

∗

∗

for each pair of x , y in X (see [21], p.426). As used in [58], p.149, it is easy to verify that this topology is weaker than the weak∗ topology on X . Thus the two topologies agree on any weak∗ compact subset of X and hence on any norm bounded set in X by the Alaoglu theorem. Let (Z , dw ) be the completion of the metric space (X , dw ). Since X is the union of all the balls with radius n ≥ 1 and thus a union of countably many weak∗ compact sets, we know that X is a σcompact set in (Z , dw ). It is clear that the Borel σ-algebras on X generated by the weak∗ topology and the topology induced by the metric dw are the same. Thus weak∗ measurability and measurability with respect to (X , dw ) are the same. Now F is a closed valued measurable correspondence from (T × Ω, L(T ⊗ A), L(λ ⊗ P )) to the Polish space (Z , dw ). By Theorem 1, DFω = DF for almost all ω ∈ Ω. Note that any distribution µ in DF (or in DFω ) on Z concentrates on X , and we shall use the same µ denote its restriction to X . It can also be checked that Z Fω dL(λ) = T Z { idX d µ : µ ∈ DFω and idX is Gel0 fand integrable with respect to µ} X

and ZZ T ×Ω

FdL(λ ⊗ P ) = Z { idX d µ : µ ∈ DF and idX is Gel0 fand integrable with respect to µ}. X

Thus (1) is proven. As in the proof Corollary 3, (2) is also a simple consequence of Theorem 2. References 1. Albeverio, S., Fenstad, J.E., Hoegh-Krohn, R., Lindstrom, T.L.: Nonstandard methods in stochastic analysis and mathematical physics. Orlando, FL: Academic Press 1986 2. Anderson, R.M.: A nonstandard representation for Brownian motion and Ito integration. Israel Journal of Mathematics 25, 15–46 (1976) 3. Anderson, R.M.: Non-standard analysis with applications to economics. In: Hildenbrand, W., Sonnenschein, H. (eds.) Handbook of mathematical economics, vol. IV. New York: NorthHolland 1991 4. Arrow, K.J., Debreu, G.: Existence of equilibrium for a competitive economy. Econometrica 22, 265–290 (1954) 5. Arrow, K.J., Radner, R.: Allocation of resources in large teams. Econometrica 47, 361–385 (1979)


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6. Artstein, Z., Hart, S.: Law of large numbers for random sets and allocation processes. Mathematics of Operations Research 6, 485–492 (1981) 7. Artstein, Z., Vitale, R.A.: A strong law of large numbers for random compact sets. Annals of Probability 3, 879–882 (1975) 8. Bewley, T.F.: Stationary monetary equilibrium with a continuum of independently fluctuating consumers. In: Hildenbrand, W., Mas-Colell, A. (eds.) Contributions to mathematical economics: in Honor of Gerard Debreu. New York: North-Holland 1986 9. Bhattacharya, R.N., Majumdar, M.: Random exchange economies. Journal of Economic Theory 6, 37–67 (1973) 10. Billingsley, P.: Converegnce of probability measures. New York: Wiley 1968 11. Block, H.D., Marschak, J.: Random orderings and stochastic theories of response. In: Olkin, I. (ed.) Contribution to probability and statistics. Stanford, CA: Stanford University Press 1960 12. Brown, D.J., Robinson, A.: A limit theorem on the cores of large standard exchange economies. Proceedings of the National Academy of Sciences USA 69, 1258–1260 (1972) 13. Brown, D.J., Robinson, A.: Nonstandard exchange economies. Econometrica 43, 41–55 (1975) 14. Cohn, D.J.: Measure theory. Boston: Birkh¨auser 1980 15. Cressie, N.: A strong limit theorem on random sets. Supplement to Advances in Applied Probability 10, 36–46 (1978) 16. Debreu, G.: Theory of value. New York: Wiley 1959 17. Debreu, G.: Integration of correspondences. In: Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability 2, part 1 (1967) 18. Diestel, J., Uhl, J.J.: Vector measures. Mathematical Surveys, Vol. 15. Providence, RI: American Mathematical Society 1977 19. Doob, J.L.: Stochastic processes depending on a continuous parameter. Transactions of the American Mathematical Society 42, 107–140 (1937) 20. Doob, J.L.: Stochastic processes. New York: Wiley 1953 21. Dunford, N., Schwartz, J.: Linear operators, Part I. New York: Interscience 1958 22. Effros, E.: Convergence of closed subsets in a topological space. Proceedings of the American Mathematical Society 16, 929–931 (1965) 23. Feldman, M., Gilles, C.: An expository note on individual risk without aggregate uncertainty. Journal of Economic Theory 35, 26–32 (1985) 24. Feller, W.: An introduction to probability theory and its applications, 3rd edn. New York: Wiley 1968 25. Gine, E., Hahn, M.G., Zinn, J.: Limit theorems for random sets: an application of probability in Banach space results. In: Beck, A., Jacobs, K. (ed.) Probability in Banach spaces, vol. IV. Lecture Notes in Mathematics, No. 990. Berlin Heidelberg New York: Springer 1983 26. Green, E.J.: Individual-level randomness in a nonatomic population Mimeo, University of Minnesota (1994) 27. Groves, T., Hart, S.: Efficiency of resource allocation by uninformed demand. Econometrica 50, 1453–1481 (1982) 28. Hart, S., Kohlberg, E.: Equally distributed correspondences. Journal of Mathematical Economics 1, 167–174 (1974) 29. Hiai, F.: Strong laws of large numbers for multivalued random variables. In: G. Salinetti (ed.) Multifunctions and integrands. Lecture Notes in Mathematics 1091, pp. 160–172. New York: Springer 1984 30. Hildenbrand, W.: Random preferences and equilibrium analysis. Journal of Economic Theory 3, 414–429 (1971) 31. Hildenbrand, W.: Core and equilibria of a large economy. Priceton, NJ: Princeton University Press 1974 32. Hurd, A.E., Loeb, P.A.: An introduction to nonstandard real analysis. Orlando, FL: Academic Press 1985 33. Judd, K.L.: The law of large numbers with a continuum of iid random variables. Journal of Economic Theory 35, 19–25 (1985) 34. Keisler, H.J.: Hyperfinite model theory. In: Gandy, R.O., Hyland, J.M.E. (eds.) Logic Colloquium 76. Amsterdam: North-Holland 1977 35. Khan, M.A.: Oligopoly in markets with a continuum of traders: an asymptotic interpretation. Journal of Economic Theory 12, 273–297 (1976)

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