Weak measurability and characterizations of riskw - CiteSeerX

Economic Theory 13, 541±560 (1999)

Weak measurability and characterizations of riskw M. Ali Khan1 and Yeneng Sun2 1 2

Department of Economics, The Johns Hopkins University, Baltimore, MD 21218, USA (e-mail: [email protected]) Department of Mathematics, National University of Singapore, Singapore 119260, SINGAPORE (e-mail: [email protected])

Received: April 23, 1998; revised version: April 28, 1998

Summary. In the context of a continuum of random variables, arising, for example, as rates of return in ®nancial markets with a continuum of assets, or as individual responses in games with a continuum of players, an important economic issue is to show how idiosyncratic risk can be removed through some device of aggregation or diversi®cation when such risk is explicitly introduced into the model. In this paper, we use recent work of AlNajjar [1] as a general backdrop to provide a review of the basic issues involved when the continuum is formulated as the Lebesgue interval. We present two examples to argue that the fundamental problem of the nonmeasurability of sample functions, originally identi®ed by Doob, and further elaborated by Feldman, Gilles and Judd in the economic literature, simply cannot be bypassed by reinterpretations of standard results. We also provide an equivalence result in the spirit of Al-Najjar's e€orts; but argue that this elementary result does not go beyond the standard law of large numbers for a sequence of real-valued iid random variables, and as such, is incapable of yielding anything of substantive economic interest beyond this law. Keywords and Phrases: Weak measurability, Pettis integrability, Bochner integrability, Decomposition, Law of large numbers, Large games. JEL Classi®cation Numbers: C60, D80. w

Part of this work was done while Khan was visiting CORE, and Sun was visiting the Cowles Foundation; both authors thank the Directors and the Sta€ of the two institutions for their support. The authors would also like to thank Roko Aliprantis, Parkash Chander, Chiaki Hara, Chris Harris, Preston MacA€ee, Lou Maccini, Bernard de Mayer, Tapan Mitra, Heracles Polemarchakis, Debraj Ray, Max Stinchcombe, Michel Talagrand, FrancËois Velde, Rajiv Vohra and an anonymous referee for comments and discussion. Previous versions of this work were circulated as CORE Discussion Paper No. 9738 and Johns Hopkins Working Paper No. 383. Correspondence to: M. Ali Khan

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1 Introduction It is now well-understood in the economic literature that uncertainty involving a single economic entity is modelled by a random variable, and that an economic system with uncertainty and many agents, in or out of an intertemporal setting, is modelled by a collection of random variables. The mathematical idealization is then a process, a joint function f …t; x†; with t an element in a parameter space T and x an element in a sample space X: If time is continuous or if the number of agents is a continuum, the parameter space is also a continuum and typically taken to be the unit interval ‰0; 1Š or the real line R endowed with the Lebesgue measure. A process is also commonly viewed as a mapping from the parameter space or from the sample space into a function space. Thus, for example, if the p-th moment of the random variables ft exists, f induces a mapping from T into an Lp space; or if the sample functions fx are continuous, f induces a mapping from X into the space of continuous functions on T . When f is jointly measurable, it is called a measurable process. Doob [19] is a standard reference for the details of a theory of measurable processes. When idiosyncratic uncertainty is injected into the usual continuum model, one encounters the well-known measurability problem, and the consequent diculty of making any sense of the desired cancellation of this type of uncertainty via aggregation. This is the basic economic problem. To be more precise and speci®c, in the context of a continuum of random variables with low intercorrelation, say a process consisting of a continuum of iid random variables, joint measurability of the process is in general lost. Since the Fubini type property associated with joint measurability is vital for the mathematical manipulation of the process, this lack of joint measurability is a basic diculty. Indeed, for even a typical sample x 2 X; fx as a function on T may not be measurable, and this failure of measurability of the sample functions precludes at the very outset a meaningful exact law of large numbers since the statement of the law itself requires the sample functions to be measurable.1 Of course, the classical law of large numbers o€ers approximate cancellation for a large but ®nite number of sources of risk, but it is the exact cancellation of individual risk that plays an important role in applications in macroeconomics and ®nance, as well as in general equilibrium and game theory.2 This exact cancellation requires the law of large numbers for a continuum of random variables or for a continuum of stochastic processes.3 The fundamental problem, then, is to give meaning to the statement that aggregation removes idiosyncratic uncertainty.

1

See the discussion in [18], [19], [21] and [29]. See the detailed documentation in [21] and [29]. A larger sample from the literature consists of [7], [9], [12], [23], [35], [39], and [41]. 2

3

For the distinctive implications of approximate versus exact, see [44, especially last two paragraphs].

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Though a continuum of random variables with low intercorrelation does not give a jointly measurable process, a standard example due to Birkho€ illustrates a process which is a weakly measurable vector valued function when viewed as a mapping into an L2 space. There is of course a considerable literature on weakly measurable functions,4 and one can ask if this literature can be repackaged in such a way that economic issues can be meaningfully addressed without making any progress on obtaining a law of large numbers for a continuum of random variables. In a recent paper, Al-Najjar [1] presents a theorem on the equivalence of four conditions on a process which is bounded in L2 …X†, and then uses this theorem to claim the complete characterization of risks generated by a continuum of random variables fft : t 2 ‰0; 1Šg. In particular, he writes that the ``force of the theorem lies in establishing the equivalence of four seemingly unrelated aspects of the process f (p. 1205)'', and, more signi®cantly, that the diculties arising from the non-measurability of sample realizations can be completely bypassed in applications to ``parts of Game Theory, Finance, and the Economics of Information (pp. 1197-98, 1209-10). The paper's Main Theorem identi®es weakly measurable processes as the class for which typical large samples can be used to obtain consistent estimates of total risk of the underlying continuum model, [and] points to the remarkably simple structure and tractability of weakly measurable processes (p. 1205). The link with ®nite approximations is an essential ingredient of an economically motivated interpretation of the assumptions and implications of the continuum model (p. 1197).''5 Al-Najjar's equivalence theorem raises two questions at the outset. First, given the basic fact o€ered by the standard Birkho€ example that a continuum of mutually orthogonal random variables Pettis-integrate to zero, how can a bounded weakly measurable vector-valued function yield, what has been termed in the literature, the strong law of large numbers for a sequence of vector-valued iid random variables?6 Second, given the intuition that asymptotic results in general equilibrium theory cannot be derived simply as a consequence of the approximation of measurable functions by simple functions,7 how can this approximation by simple functions in the vector case, by itself, be used to justify an idealized continuum through a large but ®nite model? Put more succinctly, how can the equivalence theorem be true? and second, even if true, how can it be used in applications to obtain results for large ®nite models and to bypass the need for an exact law of large numbers? The facts are that part of the characterization of weak measurability in Al-Najjar's equivalence theorem is invalid, and that the remainder, though 4

For the Birkho€ example, see [13, Example 5, p. 43] or [49, Example 3.2.1, p. 33]. For weakly measurable functions, see [13], [49] and their references. 5 In the sequel, all unascribed page numbers refer to reference [1]. 6 For the law, see [6], [14], [15], [16], [17], [28], [50], and [51]. 7 See the references in footnote 23 below.

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already available in a general form in the literature, cannot, by itself, be used to ``bypass'' the diculties arising in applications from the non-measurability of sample realizations. In this paper, we substantiate these claims. However, it is important to emphasize that the point of this paper is not simply to show that there is a substantial error in the main (equivalence) theorem and Proposition 4(i) of Al-Najjar [1]; but rather to expose for the more general reader what are the main issues involved. Towards this end, we provide a self-contained treatment with a fairly detailed discussion and elaboration of counterexamples and relevant references. We hope that the sequel will provide an introductory review for a reader who is interested in seeing how far one can use the usual continuum model based on the unit interval to address relevant questions of topical economic interest without solving the central measurability problem. The outline of the paper is as follows. First, after a presentation of the notation, terminology and the alleged result in Sect. 2, we elaborate an explicit counterexample in Sect. 3. This is important because it allows us to gauge how complete and robust is the failure of the equivalence theorem. Next, in Sect. 4, we consider a possible correction based on a strengthening of the measurability hypothesis, and show, again through a counterexample, that it cannot restore even part of the equivalence theorem. In Sect. 5, we revisit the equivalence theorem, but show how it does not go beyond the standard law of large numbers for a sequence of real-valued iid random variables. This allows us a clearer perspective on the hypothesis of weak measurability of a process. In Sect. 6, we return to the ®rst three conditions of the equivalence theorem, and show how they are special cases of general results available for some time in the standard literature on Banach spaces; we do this not only for bibliographic completeness, but also for the insight that one obtains by looking at a proposition in a more unencumbered general setting. In Sect. 7, we turn to economic interpretation; since methods for dealing with collective risk are by now standard,8 we focus on the implications of the results for the handling of idiosyncratic uncertainty in the context of the usual continuum setting.9 In particular, we consider ``a new model of large games'' proposed10 in [1; Abstract], and reiterate that the applications require the exact cancellation of idiosyncratic risk, and that the well-known law of large numbers for a sequence of Banach-space valued iid random variables simply does not address this requirement. Section 8 concludes the paper.

8

See [4], [5], [11], [33] and the synthetical treatments in [9] and [36]. For the discrete setting, one simply applies the classical law of large numbers, as, for example, in [25], [37], and [38]. 10 A more general version of this model is available in [10], [31] and [32]. 9

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2 An equivalence result? We work with vector-valued functions de®ned on …T ; T; k†, the unit interval endowed with Lebesgue measure, and taking values in a Banach space X with a dual space X  : We now copy some relevant de®nitions from [13], a standard reference on vector integration; also see [6] and [54] and their references for applications in probability theory and economics respectively. A sequence fxn g chosen from X converges strongly if there exists x 2 X such that jjxn ÿ xjj converges to zero. The sequence converges weakly if there exists x 2 X such that for each x 2 X  ; the sequence x …xn † of real numbers converges to the real number x …x†: A function f : T ÿ! X is called Psimple if there exist x1 ; x2 ; . . . ; xn 2 X ; and E1 ; E2 ; . . . ; En 2 T such that f ˆ niˆ1 xi vEi ; where vEi is the indicator function of the set Ei : A function f : T ÿ! X is called bounded if there exists a real number M such that kf …t†k  M for all t 2 T . A function f : T ÿ! X is called strongly measurable11 if there exists a sequence of simple functions ffn g such that for k-almost all t 2 T ; fn …t† converges R strongly to f …t†, and it is said to be Bochner integrable if limn!1 T kfn …t† ÿ f …t†kdk…t† ˆ 0: A function f : T ÿ! X is called weakly measurable if for each x 2 X  ; the real-valued function x …f † is measurable. A weakly measurable function f : T ÿ! X is said Rto be Pettis integrable, with Pettis integral x 2 X ; if for each x 2 X  ; x …x† ˆ T x …f †…t†dk…t†. A function f : T ÿ! X is said to be scalarly equivalent to a function g : T ÿ! X if for each x 2 X  ; x …f †…t† ˆ x …g†…t† for k-almost all t 2 T .12 In this case, one can also say that f is decomposed as the sum of g and h, where h ˆ f ÿ g is scalarly equivalent to the zero function. Talagrand [49, Sections 3.3 and 3.4, p. 35] provides several characterizations of the concept, and points out that the problem of recognizing when a function is scalarly equivalent to a ``better'' function lies at the ``very core of the theory of Pettis integration.'' By de®nition, weak measurability and Pettis integrability do not depend on the particular choice from an equivalence class of scalarly equivalent functions. It is clear that these de®nitions specialize to the case where X is the Hilbert space L2 …X† modeled on some probability space …X; A; P †: Note that L2 …X† is self-dual with the usual inner products …
;
† formalizing the actions of continuous linear functionals; see, for example [43, Theorem 4.12]. This has the straightforward implication that any bounded weakly measurable function f : T ÿ! L2 …X† is Pettis integrable, since the self-duality indicates 2 that there is R y 2 L …X† such that2 the bounded linear functional w de®ned by w…x† ˆ T …x; f …t††dk…t†; x 2 L …X† can also be represented as w…x† ˆ …x; y†; x 2 L2 …X†. Thus, for L2 …X†-valued bounded functions, weak measurability and Pettis integrability are simply the same. 11 Strongly measurable functions are referred to as k-measurable in [13; p. 41], and also simply as measurable [13, p. 88]. This latter terminology is used in [1, Section 4]. 12

Such functions are referred to simply as equivalent in [13, p. 88], and as weakly equivalent in [46]. We adopt the terminology of [49, p. 35].

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We now state the main result of [1, p. 1205] in a terminology that is conventional in the relevant literature. Let …T 1 ; T1 ; k1 † be the product measure space generated by a countably in®nite number of Lebesgue unit intervals; see [20, Theorem 8.2.2, p. 202]. Theorem 1. For any bounded function f : ‰0; 1Š ÿ! L2 …X†, the following statements are equivalent: (i) f is weakly measurable. (ii) f is scalarly equivalent to an essentially unique strongly measurable function. (iii) There exists a sequence of simple functions fn : ‰0; 1Š ÿ! L2 …X† such that for any bounded linear functional x on L2 …X†, x …fn † converges almost surely to x …f †: P (iv) There exists x 2 L2 …X† such that the sequence …1=n† niˆ1 f …ti † con1 verges strongly to x for k -almost all …ti † 2 T 1 . If (iv) holds, and P f is scalarly equivalent to a strongly measurable function R1 g, then limn!1 …1=n† niˆ1 f …ti † ˆ 0 g…t†dk…t†, the Bochner integral of g and is independent of the residual f ÿ g. Al-Najjar writes that the ``force of the theorem lies in establishing the equivalence of four seemingly unrelated aspects of the process f (p. 1205).'' Unfortunately, (i) to (iv) are not equivalent, and we turn to an explicit counterexample. 3 The counterexample Dobric [14] presents several counterexamples to the assertion …i† ˆ) …iv† for functions taking values in a variety of classical Banach spaces; also see [16]. We reproduce his ideas in the probabilistically more standard context of a continuum of random variables indexed by elements of ‰0; 1Š; and in a Hilbert space L2 …X† modeled on a probability space …X; A; P †. Dobric [14, Lemma 1.1] presents the following lemma in the context of a di€used Polish space. We simply take the special case ± the unit interval endowed with Lebesgue measure. Lemma 1. There exists an index set S of cardinality of the continuum and a class F ˆ fFs : s 2 Sg of subsets of ‰0; 1Š such that (i) for all s 2 S, the car0 0 0 dinality S of Fs is less than or equal to @0 ,1(ii) F1s \ Fs ˆ ; for all s 6ˆ s ; s; s 2 S, (iii) s2S Fs ˆ ‰0; 1Š, (iv) for all B 2 T ; k …B† > 0, there exists s 2 S such that B \ Fs1 6ˆ ;. By Kolmogorov's existence theorem ([20, Theorem 12.1.2], for example), there exists a probability space …X; A; P † and identically distributed and independent random variables fxs gs2S with a common uniform distribution r on fÿ1; 1g. These random variables are thus mutually orthogonal with zero mean and unit variance, and constitute an orthonormal set in the space L2 …X†. Let f : ‰0; 1Š ÿ! L2 …X† be given by f …t† ˆ xs ; if t 2 Fs : By conditions (ii) and (iii) of Lemma 1 above, f is well-de®ned. Certainly f is a bounded

Weak measurability and characterizations of risk

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function. We shall show that it is weakly measurable but that it does not satisfy condition (iv) of Theorem 1. For any x in the dual of L2 …X†; Bessel's inequalityP(see, for example, [43; Corollary to Theorem 4.16]), allows us to assert that s2S j…x ; xs †j2  kx k2 ; and that therefore the set S0 ˆ fs 2 S : …x ; xs † 6ˆ 0g has at most S a countable number of elements. Hence M ˆ ft 2 ‰0; 1Š : …x ; ft † 6ˆ 0g  s2S0 Fs . Condition (i) of Lemma 1 and the countability of S0 allows us to Rconclude that M is 1 countable. Hence …x ; ft † ˆ 0 for k-almost all t 2 T , and 0 …x ; ft †dk…t† ˆ 0. 2  Since L …X† is self-dual and x is arbitrary, this shows that f is weakly measurable, that it is scalarly equivalent to the zero function, and that its Pettis integral is zero. Thus f satis®es condition (i) in Theorem 1. Now consider the set n X f …ti †k ˆ 1; 8n 2 Ng ; L ˆ f…ti † 2 T 1 : k…1=n† iˆ1

S and note that it contains the set s2S Fs1 , and by an appeal to condition (iv) of Lemma 1, that k1 …L† ˆ 1, where k1 is the outer measure generated by (iv) in Theorem 1 is satis®ed, then the strong limit k1 . If condition P limn!1 …1=n† niˆ1 f …ti † should be almost surely equal to the Pettis integral of f which is zero in this case. This contradicts the fact that L has outer measure 1. Thus f does not P satisfy condition (iv) in Theorem 1. In fact, …1=n† niˆ1 f …ti † does not even converge weakly to the Pettis integral of f on any set of positive measure. Suppose it does. PThen there is a set B 2 T1 with k1 …B† > 0 such that for all …ti † 2 B, …1=n† niˆ1 f …ti † converges weakly to zero. By condition (iv) of Lemma 1, there exists s 2 S such that B \ Fs1 P6ˆ ;. Hence, there is a sequence …ti † chosen from Fs such that …1=n† niˆ1 f …ti † converges weakly to zero, which means that the constant sequence …xs † with norm 1 converges weakly to zero. This is a contradiction. Note also that P f does not satisfy part (i) in Proposition 4; the distribution l…tn † of …1=n† niˆ1 f …ti † does not converge on any set of positive measure in T 1 to the Dirac measure df0g at 0; the distribution l…f † of the Pettis integral of f . Suppose it does. Then there is a set B 2 T1 with k1 …B† > 0 such that for all …ti † 2 B, l…tn † ! l…f †, where tn ˆ …t1 ; . . . ; tn †. By condition (iv) of Lemma 1, there exists s 2 S such that B \ Fs1 6ˆ ;. Hence there is a sequence …ti † chosen from Fs such that l…tn † ! l…f †. Since all the f …ti † equal xs ; this means that the constant sequence consisting of the uniform distribution on fÿ1; 1g converges to df0g in distribution. This is a contradiction. The counterexample exhibits a process constituted, in a well-speci®ed sense, of countable replicas of a process with a continuum of iid random variables. The strength of the example lies in the fact that any set of positive measure of sequential draws contains a draw in which a particular random variable is in®nitely repeated. It is this that leads to the negation of Theorem 1 and Proposition 4(i), and shows how totally the approximation claim can fail when a process has a purely idiosyncratic part. Furthermore, one ought not to loose sight of the fact that (iv) involves sequential sampling from the index set T , rather than from the sample space X, and it is

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the latter that is relevant for the removal of idiosyncratic uncertainty through aggregation. The counterexample makes transparent the error in the proof of the claim …ii† ˆ) …iv† and that of Proposition 4(i). Al-Najjar (p. 1221) uses the fact that ``the vectors ht are all orthogonal o€ a set A  T of s-measure zero,'' i.e., for all t1 ; t2 2j A with t1 6ˆ t2 , ht1 and ht2 are orthogonal. In the counterexample, for all t1 ; t2 in a particular Fs , ft1 and ft2 are identical, and hence certainly not orthogonal. Thus individual risk concerning entity t; as represented by the random variable ft , is correlated with at most countably many others, but this countable collection depends on the particular index t. On the other hand, Proposition 4(i) is deduced from the false implication …ii† ˆ) …iv†. 4 A possible correction? For any process f , Dobric [14] investigates the autocorrelation function of a process taking values in the classical Hilbert space `2 …S†.13 In the light of his work, it is then natural to ask whether the equivalence theorem can be restored by substituting the requirement of the joint measurability of the autocorrelation function for that of weak measurability in condition (i) of Theorem 1. Since conditions (i), (ii) and (iii) of Theorem 1 are equivalent and strictly weaker than (iv), the answer to this question is clearly negative. Put another way, the strengthening of condition (i) to restore an invalid implication …i† ˆ) …iv† obviously destroys the equivalence of that condition with the other conditions such as (ii) and (iii). However, one can bypass full equivalence and the original problem as discussed in Section 1 above, and focus solely on the equivalence of the ®rst and the fourth conditions of Theorem 1. On transposing and specializing Dobric [14, Theorem 1.3] to the context of this paper, one obtains Theorem 2. For any bounded function f : ‰0; 1Š ÿ! L2 …X†, the joint Lebesgue measurability of the autocorrelation function gf implies condition (iv) of Theorem 1. Note that the autocorrelation function gf of a process f is a joint Lebesgue measurable function on the Lebesgue square …T 2 ; T2 ; k2 † given R by …t1 ; t2 † ÿ! gf …t1 ; t2 † ˆ X f …t1 †…x†f …t2 †…x†dP …x† for …t1 ; t2 † 2 T 2 . We shall refer to the requirement of joint Lebesgue measurability of the autocorrelation function as condition …i†0 . The question then reduces to the validity of the assertion …iv† ˆ) …i†0 . Unfortunately, this assertion is false, and we show this through a modi®cation of the counterexample presented above ± this is again a reproduction of the ideas in [14].

13

We note here the standard result that every nontrivial Hilbert space is isomorphic to some `2 …S† for some S; see, for example, [43, Chapter 4].

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Dobric [14, Lemma 1.2] presents the following lemma in the context of a di€used Polish space. As in the case of Lemma 1 above, we simply take the special case ± the unit interval endowed with Lebesgue measure. Lemma 2. For any integer n 2 N, there exists an index set S of cardinality of the continuum and a class F ˆ fFs : s 2 Sg of subsets of ‰0; 1Š such that (i) for all s 2 S, the cardinality of FS s is less than or equal to …n ‡ 1†; (ii) Fs \ Fs0 ˆ ; 0 0 for all s 6ˆ s ; s; s 2 S, (iii) s2S Fs ˆ ‰0; 1Š, (iv) for all B 2 Tn ; kn …B† > 0, there exists s 2 S such that B \ Fsn 6ˆ ;. We now choose n equal to 2 in Lemma 2, and construct, just as in Section 3, a process f from ‰0; 1Š to L2 …X†. We shall show that f satis®es condition (iv) of Theorem 1 but does not satisfy condition …i†0 of Theorem 2. Since k is the Lebesgue measure, k1 …f…ti † 2 T 1 : tj 6ˆ tk ; for all j 6ˆ kg† ˆ 1 : For any …ti † 2 T 1 with tj 6ˆ tk ; for all j 6ˆ k, one can use the fact that the cardinality of Fs is less than or equal to 3 for any s 2 S to deduce that

2

n n X n n X X X

f …ti † ˆ …1=n2 † vF …ti ; tj †  …1=n2 † 3 ˆ 3=n ;

…1=n†

iˆ1 iˆ1 jˆ1 iˆ1 where F ˆ [s2S Fs  Fs ; and vF is the indicator function of the set F in T 2 . Thus f satis®es condition (iv) of Theorem 1. Since gf …t1 ; t2 † ˆ vF …t1 ; t2 † for any …t1 ; t2 † 2 T 2 ; the non-measurability of gf follows as a direct consequence of the non-measurability of F . Pick any B 2 T2 ; k2 …B† > 0, and appeal to condition (iv) of Lemma 2 to obtain B \ F ˆ B \ … [ Fs  Fs † ˆ [ …B \ …Fs  Fs †† 6ˆ ; : s2S

2

s2S

2

Hence k …F † ˆ 1, where k is the outer measure generated by k2 . Now for any t 2 ‰0; 1Š; by conditions (ii) and (iii) of Lemma 2, there is a unique s0 2 S such that t 2 Fs0 , and hence the section [ F …t† ˆ ft 2 ‰0; 1Š : …t; t† 2 F g ˆ ft 2 ‰0; 1Š : …t; t† 2 Fs  Fs g ˆ Fs0 ; s2S

which then implies that k…F …t†† ˆ k…Fs0 † ˆ 0 by an appeal to condition (i) of t 2 ‰0; 1Š; k…A…t†† Lemma 2. If A  F is a k2 measurable set, then for every R1  k…F …t†† ˆ 0 for every section A…t†: Hence k2 …A† ˆ 0 k…A…t††k…t† ˆ 0; and consequently k2 …F † ˆ 0, where k2 is the inner measure generated by k2 . Therefore F is not product Lebesgue measurable, and thus f does not satisfy condition …i†0 in Theorem 2. Note that every random variable in the process f is uncorrelated with all the rest except possibly two others, and yet condition …i†0 is not satis®ed. In other words, this condition allows ``very little'' correlation between the residual random variables. Thus the joint Lebesgue measurability of the autocorrelation function of the process is far too strong a condition, and does not ``characterize the class of processes whose representations of risk can be

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estimated from the corresponding representations of large ®nite samples in a statistically consistent manner (p. 1204).'' Even though joint Lebesgue measurability of gf may appear to be a ``minor technical'' matter of going from the measurability of the sections of a function, as required by the criterion of weak measurability, to the measurability of the function itself, it constitutes the key substantive diculty in handling non-trivial idiosyncratic uncertainty, and is clearly not a matter of mathematical cosmetics. As our example brings out, the assumption rules out of consideration important processes, and therefore imposing it does make a big di€erence in the content of the result. One can also view this di€erence of joint measurability and sectional measurability in a slightly di€erent setting. Let x…t; x† be the specially constructed iid process in [29]. Then, for a ®xed t, x…t;
† is always measurable on the continuum product space X. By extending the product measure on X; albeit in an ad hoc way, one can also make sure that x…
; x† is measurable on T almost surely. This means that sectional measurability can somehow be obtained for this special process; but as shown in Doob [19, p. 67], x will never be jointly measurable and even has no (jointly) measurable standard modi®cation. It is of course already evident from Talagrand [50, Theorem 8] that it is the subtler notion of ``proper measurability'' rather than ``weak measurability'' or ``joint Lebesgue measurability of autocorrelation functions'' that o€ers a necessary and sucient condition for condition (iv) in Theorem 1; the only problem is that proper measurability is nowhere as ``simple and tractable'' as is the criterion of weak measurability (but see [51]).

5 Equivalence revisited Since an equivalence theorem cannot be obtained by strengthening condition (i), the obvious course of action is to weaken condition (iv). This is accomplished simply by trivializing it to the real case to obtain the following elementary and obvious result. Theorem 3. Conditions (i), (ii) and (iii) in Theorem 1 are equivalent to a condition to be called …iv†0 where: R …iv†0 For any x 2 L2 …X†; and w : T ÿ! R with w…t† ˆ X x …x†f …t†…x†dP …x†, there is a real number a such that lim …1=n†

n!1

n X ÿ
w ti ˆ a for k1 -almost all …ti † 2 T 1 : iˆ1

Proof. For any ®xed x 2 L2 …X†, weak measurability of f implies that w is measurable. By the very construction of the product measure, the coordinate functions are independent, and since independence is preserved under com1 position, the sequence /i …ftn g1 nˆ1 † ˆ w…ti † on T , i ˆ 1; 2; . . . of real-valued random variables R classical law of large numbers thus says that P is iid. The limn!1 …1=n† niˆ1 w…ti † ˆ T w…t†dk almost surely. One can simply take

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R a ˆ T w…t†dk. For the converse, note that (iv) implies (i) is true when L2 …X† is trivialized to the real line (for a proof of this result of Talagrand, see [28, pp. 310±312]). We can now use this fact to claim the measurability of w. Hence weak measurability of a process is equivalent to the satis®ability of the law of large numbers for some sequences of real-valued iid random variables as formalized in condition (iv)0 . ( The equivalence of conditions (i), (ii) and (iii) with that of …iv†0 then clearly brings out the status of the result ± it is simply a restatement of the classical law of large numbers for a sequence of iid random variables. Note that (iv) is a statement of the law of large numbers for iid random variables in the vector case, while …iv†0 is the law essentially for the real case. While the basic intuition and the meaning of the law of large numbers is well-understood and readily amenable to common sense, it may be worth bringing out the di€erences between the vector and the scalar cases in the context of this paper.14 (iv) states that one canR use a typical sequential draw …ti † to approximate the random variable T g…t†…x†dk…t†, while …iv†0 states that approximation under the premise of a given linear functional x , and thus the null set in which the approximation fails may depend on the particular choice of the linear functional. It is precisely this fact that is illustrated by the counterexample in Sect. 3. It shows that for the general case of weakly measurable functions, where there is non-trivial idiosyncratic risk, there is no way that one can use a typical Rsequential draw t1 ; t2 ; . . . ; tn ; . . . from T to approximate the collective risk T g…t†…x†dk…t†, even weakly. Convergence can fail very badly in the sense Pn that the collection of sequential draws with the property that …1=n† iˆ1 f …ti †…x† is not weakly convergent to R g…t†…x†dk…t† has outer measure 1. Thus, one cannot applyR …iv†0 to claim T that a typical sequential draw …ti † can be used to approximate T g…t†…x†dk…t† in a setting with non-trivial idiosyncratic residual part f ÿ g. Put another way, even though …iv†0 shares a certain similarity in form with (iv), it has very di€erent, and inadequate, content in terms of solving the problem at hand. As brought out in the prescient remarks of Beck [6, Sects. 6 and 7], the problem arises from the inadequacy of the Pettis integral. There is also a clear statement in [1, Introduction, last paragraph] about the diculty in justifying the claim that Pettis integration somehow delivers a ``law of large numbers for the continuum;'' and Al-Najjar is explicit that ``I neither solve the measurability problem nor provide a law of large numbers of continuum models (p. 1197).'' But that is precisely what is needed. To illustrate, one can compare the meanings of the classical law of large numbers with a cosmetically-similar version based on the weak topology on the L2 space of random variables. Let an ; n ˆ 1; 2; . . . be an iid sequence of real-valued random variables on a sample space …X; A; P † with common Pmean 0 and variance 1. The classical law of large numbers says that …1=n† niˆ1 ai converges almost surely to 0. On the other hand, a natural corresponding version using the 14

It is some interest that (iv) is nowhere used in the ®nance application presented in [2].

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weak P topology on L2 …X† asserts that for eachP x 2 L2 …X†, n R 1   0 …1=n† iˆ1 X ai …x†x …x†dP converges to 0. Express Pn Rx as  nˆ1 bn an ‡ x 0 with x orthogonal iˆ1 X ai …x†x …x†dP is the P to all the an . Then …1=n† P Note that limn!1 …1=n† niˆ1 bi ˆ 0 simply follows CesaÁro sum …1=n† niˆ1 bi .P 1 2  2 from Bessel's inequality nˆ1 bn  kx k < 1. Hence the second type of convergence has nothing to do with the law of large numbers, but is rather a weaker statement deriving from the square integrability of relevant random variables. As alluded to in the introduction, the fact that a continuum of mutually orthogonal variables has a zero Pettis integral is a restatement of the standard Birkho€ example, and by itself, it does not provide enough of a basis for a meaningful probabilistic interpretation, or for further analyses of structures constructed from the original process such as the relevant sample functions or empirical processes in a setting with time. In the usual economic applications, one has to aggregate by considering integrals or distributions of the sample functions, or the ®nite dimensional distributions of the empirical processes.15 It does not make sense, as required under Pettis integration, to ®rst formulate the joint moment (or an inner product in the Hilbert space terminology) of a random variable in an underlying process with another random variable exogenously chosen outside the model, and then to aggregate this joint moment. Thus, Uhlig [53, p. 47] notes, ``Using the weak topology view, there is no distinction between a continuum economy with idiosyncratic risk and a continuum economy without any risk: if a given sequence of ®nite economies converges to one, it will also converge to the other.'' 6 Relationship with the literature The importance of Dobric [14] for the substantive issues at hand is by now clear. In terms of the characterization question, the only implication not available in the literature is …i† ˆ) …iv†. Before we take the other implications in turn, it is important to point out that the literature shows them to be valid for the more general context of functions that are not necessarily bounded and which take values in Banach spaces. Thus, it lays the groundwork for a reader who may possibly be interested in the L1 …X† context of shocks which have a mean but do not necessarily have a variance. From another perspective, it singles out the implications that are valid for a context where the operation of Pettis integral is well-de®ned but where the concept of orthogonality makes no sense.16 All this enables one to keep distinct probabilistic aspects of the problem from functional-analytical ones, and underscores the points already made at the end of the Section 5 above.

15 16

See the papers cited in footnote 1 for this kind of aggregation.

In [53, Theorem 4], Pettis integration is approached through Riemann integration, a line of investigation that is ascribed in [19, p. 625] to a 1928 paper of Slutsky's.

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The valid implications in Theorem 1 above simply involve the rewriting of available results in their bounded L2 …X† context. …i† () …ii†: The implication that …ii† ˆ) …i† is a triviality given that all strongly measurable functions are weakly measurable. …i† ˆ) …ii† follows directly from the facts that a Pettis integrable function naturally de®nes a vector measure by integrating it on subsets, and that any Hilbert space has the Radon-Nikodym property. The latter property of Hilbert spaces goes back to von Neumann in the late forties; see [13, p. 100, Corollary 4]. Indeed, Diestel-Uhl [13, p. 89] write ``If X is a dual space with the Radon-Nikodym property, then the result collapses to a triviality.''17 Note that the uniqueness of such a scalarly equivalent strongly measurable function is already pointed out in [13, p. 48]; this is Proposition 2 in [1]. …i† () …iii†: The implication …iii† ˆ) …i† is the standard result that the pointwise limit of a sequence of measurable functions is measurable; see [43, p. 15]. As regards the implication …i† ˆ) …iii†, note that (ii) ˆ) (iii) follows trivially from that fact that strongly measurable functions can be approximated by simple functions and the type of convergence in (iii) is invariant under scalar equivalence. In any case, since the Lebesgue interval [0, 1] is a perfect measure space, the desired equivalence is directly available in Geitz [22, Theorem 7(a)]. It is also available in [47, Corollary 3.3] and in [49, Theorem 5-3-2, p. 62] without the perfectness assumption but with an additional separability condition that is trivially satis®ed here as a consequence of (ii); see the comment in [49, Chapter 5, p. 210]. What makes the results of these authors nontrivial is that they are set in general Banach spaces where a bounded weakly measurable function may not be scalarly equivalent to a strongly measurable function (see [13, p. 43] and [46, p. 433] for the wellknown example of Hagler). However, Sentilles [46, Theorem 2.5] shows that a bounded weakly measurable function can always be essentially uniquely decomposed as the sum of a strongly measurable function and what he terms a purely weakly measurable function. For Hilbert spaces, this decomposition reduces to the one considered in (ii). …iv† ˆ) …i†: This assertion is a special case of that available in Ho€mannJorgensen [28, Theorem 2.4 and the remark following it; p. 310] for functions that take values in a Banach space and are not necessarily bounded. The proof is also contained in the proof of Theorem 22 in [50, p. 861]; since f is assumed to be bounded, we need only the third paragraph. 7 Interpretation and discussion We now turn from this somewhat bewildering array of mathematical concepts to the ®rmer ground of economic interpretation. Since the notion 17

For more general spaces, the result is a theorem due to Lewis and reported in [13, p. 88]; also see [15], [16], [17], [46], [47], Section 5] and [49, pp. 37±41] for a variety of generalizations. Note that for any Hilbert space, and therefore for the L2 …X† setting used here, weak and weak measurability are equivalent conditions, as are weak -Radon, weak-Radon and norm-Radon.

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of an uncertain prospect or a risky outcome or of a random shock is universally modelled by random variables in the social scienti®c literature, including that of economics, it is clear that any statement on processes or random variables can be straightforwardly translated, if one so prefers, as a property of uncertainty or of risk or of randomness. Accordingly, in Fig. 1, we summarize the relationships of the special classes of processes discussed in this paper, and restate their de®nitions in terms of economic terminology. A process f is simply a mathematical formalization of an ensemble of risk in the form of a collection of random variables or shocks. It is purely weakly measurable or purely idiosyncratic if it is scalarly equivalent to zero, or all, or almost all, of the shocks are uncorrelated with any given shock. It is simple or ®nitely generated if all the shocks are a linear combination of (are generated by) the same ®nite set of shocks. It is clear that a ®nitely generated process precludes the existence of nontrivial idiosyncratic risk. It is also clear from elementary linear algebra that, without any loss of generality, a ®nitely generated process, may be de®ned in terms of a ®nite

Figure 1

Weak measurability and characterizations of risk

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orthonormal set of shocks. The next stage in this development of ideas is to consider a bounded aggregate process; a process such that all, or almost all, of the shocks are a linear combination (are generated by) a countable set of orthonormal shocks. Again, it is clear that a bounded aggregate process precludes the existence of nontrivial idiosyncratic risk, and thereby may be described by some terminology emphasizing the opposite property of idiosyncrasy such as, say, systematic or collective risk. Fortunately, the terminology of an aggregate process is not needed because the celebrated Pettis measurability theorem guarantees that such a process is identical to a strongly measurable process. This theorem asserts that every weakly measurable process is strongly measurable if all, or almost all, of its shocks lie in a separable space; and moreover, this requirement, along with weak measurability is sucient for strong measurability; see [13, Chapter 2], [49, Chapter 3] and [52]. Now, by virtue of the fact that every separable Hilbert space has a countable basis, it is easy to see that every bounded aggregate process is the same as a bounded strongly measurable process which, given boundedness, is again the same as a Bochner integrable process.18 If we turn to the primitive de®nition of Bochner integrability o€ered in Section 2 above, then the de®nition itself already says that any bounded aggregate process g can be approximated by a ®nitely generated process.19 This is simply to say that, for a given  > 0, a Bochner integrable g can be approximated by a simple vector function g with Rvector function  T kg…t† ÿ g kdk < . Thus, in our consideration of any ensemble of risk, the two relevant conceptual markers are those of strong and weak measurability, and the criterion of the scalar equivalence of a weakly measurable process to a strongly measurable one is simply a way of saying that an ensemble of risk can be decomposed into its idiosyncratic and collective components. If we want to include non-trivial idiosyncratic risk, we will be thrown back to the diculty with which we began this paper; namely, the possible non-measurability of sample realizations. If ft is to be interpreted as the random actionRof an entity t, then the aggregate action at the state of nature x is given by T f …t; x†dk, and it is precisely the diculties concerning the measurability of sample realizations that render the de®nition of this integral problematic. To put the matter another way, when the distribution of aggregate actions is formalized simply as the distribution l…f † of the Bochner integral of the collective part of the risk, g, one is either assuming the aggregate of the idiosyncratic part to be essentially zero, or simply dropping it without any explicit emphasis. Note R g…t†…x†dk…t† means ®nding the that taking the distribution function of T R probability of all x such that T g…t†…x†dk…t† < x for all real numbers x. One cannot conclude that this distribution function is somehow the distribution function of the aggregate actionRbased on the individual actions ft simply R  f …t; x†x …x†dP ˆ g…t; x†x …x†dP for from the known identity X X 18 19

See [13, Chapter 2]. These matters are restated in [1, Proposition A1]. This is Proposition 1 in [1].

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x 2 L2 …X†, and for almost all t. Observe that the earlier integral is on T while R the latter integrals are on X, and T f …t; x†dk may not exist at all because of the possible non-measurability of sample realizations. This manner of ``bypassing the problem'' solves a problem simply by assuming it away. It may also be noted that the distribution of the aggregate l…f † is motivated in [1] through a decision maker who does not care about sample realizations, and it is explicitly argued that the measurability problem remains important in models where this assumption is not justi®ed (p. 1210). On the other hand, l…f ; x† is related to a decision maker who does care about the state of nature x, and yet the measurability problem is ``bypassed'' by assuming it away in the de®nition (pp. 1210±1211). Once the focus is narrowed down by necessity to the non-idiosyncratic part of risk, we are on the familiar ground of the existing literature on the convergence of real-valued random variables and strongly measurable functions. In particular, the so-called G-convergence of the processes, as well as the L2 -convergence of the random variables de®ned as the relevant Bochner integrals on subsets, as proposed in [1], is not needed for the validity of the convergence results in Propositions 4(ii), 5 and 6. It is well-known, for example, that a sequence of random variables converges in distribution if it converges in the L2 norm, and that a sequence of regular conditional distributions converges in distribution almost surely if the original sequence of random variables converges almost surely.20 Note also that conditional expectations can be written as integrals in terms of a regular conditional distribution for each x, and almost sure convergence of a sequence of random variables is preserved under the operation of conditional expectation; see, for example, [20, pp. 266 and 272]. There are also standard procedures for obtaining results on distributions from results on random variables. For example, by picking a sequence of bounded continuous functions which determines weak convergence of measures through integrals in the usual way (see, for example, [40, Theorems 6.2 and 6.6, pp. 43 and 47]), almost sure convergence of the random variables guarantees that outside a null set of X, the sequence of integrals of every chosen continuous function in terms of the sequence of regular conditional distributions converges for each x, which means that for these x, the relevant sequence of conditional distributions converges in distribution.21 From a more general point of view, the approximation of measurable functions by simple functions is a basic intuition underlying the standard theory of Lebesgue integration as well as its in®nite dimensional Bochner counterpart,22 and while it is only reasonable to expect that it will obtain in 20

See, for example, [34, pp. 166 and 170]. Proposition 4 (ii) and Proposition 6 in [1] are simply restatements of these well-known facts with extraneous conditions. 21 Propositions 3 and 5 in [1] then follow trivially from the relevant de®nitions, and from the fact that strongly measurable functions can be approximated by simple functions in a variety of strong ways. 22 See, for example, [43] for the former and [13] for the latter.

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some form or other in a more general setting,23 it is hard to see how, by itself, it can be used to justify an idealized continuum by a large ®nite model. Even in the case of purely deterministic economies with a continuum of agents, one cannot obtain corresponding results for large ®nite economies simply by an appeal to the fact that measurable functions on the Lebesgue unit interval can be approximated by simple functions. For such practical procedures for translating back and forth from an idealized economic model to one that is based on a large but ®nite setting, one can rely on the theory of weak convergence of measures, or on methods of nonstandard analysis.24 Chakrabarti-Khan [10] propose a model of a ``large'' game in which every player is endowed with his/her own private information formalized as a subr-algebra, and the player's strategy is a measurable function with respect to this sub-r-algebra; also see [31, 32]. In [1], a special case of this model is considered whereby uncertainty does not a€ect the fundamentals of the game, namely, players' action sets and payo€s, and the question is posed as to conditions under which statistics of ``societal responses'' are independent of the state of nature. This question can be alternatively phrased as asking for conditions under which the standard Schmeidler's 1973 model of ``large'' games adequately captures a setting with uncertainty generated by ``sunspots.'' This is the so-called ``validity'' condition for the distributional representation ([1, De®nition on p. 1215]). By now, it ought to be clear that the answer to this question must again run afoul of the problem of the nonmeasurability of sample functions. When a strategy pro®le s is considered for games, say with two actions …1; 0† and …0; 1†, for a ®xed x, the distribution of s…
; x† gives the fractions of the population taking the two actions, one simply cannot claim that the integral of the relevant aggregate part (denoted by v) of s delivers such a distribution (see p. 1213). Leaving aside the fact that the existence of the distribution for s…
; x† is problematic, this claim cannot be justi®ed on account of the fact that v usually takes values other than the two actions …1; 0† and …0; 1†, and its own game-theoretic meaning is far from clear. In any case, the so-called ``validity'' condition for the distributional representation [1, De®nition on p. 1215] is a more cumbersome statement of the fact that the Bochner integrals of the aggregate part of every strategy pro®le is non-random. To see this, one can simply take the i-th coordinate function on D as the utility function, and observe that the ``validity'' condition implies that the conditional expectation E…hi jft † is essentially the constant Ehi , where hi is the i-th component of the Bochner integral of the relevant aggregate part of the strategy pro®le s. Since E…sit jft † ˆ sit , E…hi sit † ˆR E…E…hi jft †sit † ˆ Ehi Esit . Since hi is the Pettis integral of sit , Ehi hi ˆ T Ehi sit dk…t† ˆ …Ehi †2 , which implies that hi is essentially constant. Now assume away the contribution of an idiosyncratic part from the de®nition of a state-dependent notion lt …s; x†. In this case, it is easy to make 23 24

See the discussion of the implication (i) ˆ) (iii) in Section 6 above. For the former, see [25], [26]; and for the latter, see [3], [8], [42], and their references.

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some statements on the ``validity'' condition. Since the independence assumption already guarantees that the Bochner integral of the relevant aggregate part is essentially a constant function ([53, Theorem 2]), its regular conditional distribution with respect to any sub-r-algebra is also essentially constant. On the other hand, since a strategy pro®le is an arbitrary process, taking values in the action set, and measurable according to the information structure, one can certainly ®nd one strategy pro®le to have nontrivial correlation when an informational structure allows nontrivial correlation.25 Thus, we again come to the conclusion that we reached earlier. Either the ensemble of risk facing a decision maker consists solely of collective risk, in which case l…f † has meaning and can be an argument in the payo€ function; or this assumption is not justi®ed, in which case the measurability problem remains important and the distribution of the sample function l…f ; x† cannot be given meaning, much less shown to ®lter out the particular state of nature x. 8 Conclusion To conclude, we note that the properties of weak measurability, scalar equivalence, approximation by simple functions, and the sequential version of the Banach space valued law of large numbers, already well-studied in the literature for the usual continuum model, fail to provide the right tools to investigate economic phenomena with idiosyncratic uncertainty. It is not enough to show the existence of a process satisfying both a microscopic independence assumption and an exact macroscopic stability condition. The fact that this can be technically dicult, or through an interpretation of the classical law of large numbers in a nonstandard model, remarkably easy, is not to the point;26 instead of an existence statement, what one needs for the applications is a genuine law of large numbers ± one that gives sucient (and possibly also necessary) conditions such that macroscopic stability is obtained whenever these conditions are satis®ed by a general process. Due to some intrinsic diculties associated with the usual continuum model, one needs to go beyond the existing constructions based on it, and look for a better ideal framework to model mass phenomena in economics. References 1. Al-Najjar, N. I.: Decomposition and characterization of risk with a continuum of random variables. Econometrica 63, 1195±1224 (1995) 2. Al-Najjar, N. I.: Factor analysis and arbitrage pricing in large asset economies. Journal of Economic Theory 78, 231±262 (1998) 3. Anderson, R. M.: Non-standard analysis with applications to economics. In: Hildenbrand, W., Sonnenschein H. (eds.) Handbook of mathematical economics, Vol. 4. New York: North-Holland 1991

25 26

These two results respectively constitute Propositions 7 and 8 in [1]. For the former, see [24], and for the latter, see, for example, [3] and [30].

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4. Arrow, K. J.: La role des valeurs boursieres pour la repartition la meilleure des risques. Econometrie, pp. 41±47 (1953). English translation: The role of securities in the optimal allocation of risk-bearing. Review of Economic Studies 31, 91±96 (1964) 5. Arrow, K. J., Lind, R.: Uncertainty and the evaluation of public investments. American Economic Review 60, 364±378 (1970) 6. Beck, A.: On the strong law of large numbers. In: Wright, F.B. (ed.) Ergodic theory. New York: Academic Press 1963 7. Bewley, T. F.: Stationary monetary equilibrium with a continuum of independently ¯uctuating consumers. In: Hildenbrand, W., Mas-Colell, A. (eds.) Contributions to mathematical economics: In Honor of Gerard Debreu. New York: North-Holland 1986 8. 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) 9. Cass, D., Chichilnisky, G., Wu, H.: Individual risk and mutual insurance. Econometrica 64, 333±341 (1996) 10. Chakrabarti, S. K., Ali Khan, M.: Equilibria of large games with imperfect observability. In: Aliprantis, C. D., Border, K. C., Luxemburg, W. A. J. (eds.) Positive operators, Reisz spaces and economics, pp. 49±68. New York: Springer 1991 11. Debreu, G.: Theory of value. New York: Wiley 1959 12. Diamond, D. W., Dybvig, P. H.: Bank runs, deposit insurance and liquidity. Journal of Political Economy 91, 401±419 (1983) 13. Diestel, J., Uhl, Jr., J. J.: Vector measures. Rhode Island: American Mathematical Society 1977 14. Dobric, V.: The law of large numbers, examples and counterexamples. Mathematica Scandinavica 60, 273±291 (1987) 15. Dobric, V.: The decomposition theorem for functions satisfying the law of large numbers. Journal of Theoretical Probability 3, 489±496 (1990) 16. Dobric, V.: The Glivenko-Cantelli Theorem in a Banach space setting. In: Dudley, R. M., Hahn, M. G., Kuelbs, J. (eds.) Probability in Banach spaces, pp. 267±272. Basel: BirkhaÈuser 1992 17. Dobric, V.: Weakly compactly generated Banach spaces and the strong law of large numbers. Journal of Theoretical Probability 7, 129±134 (1994) 18. Doob, J. L.: Stochastic processes depending on a continuous parameter. Transactions of the American Mathematical Society 42, 107±140 (1937) 19. Doob, J. L.: Stochastic processes. New York: Wiley 1953 20. Dudley, R. M.: Real analysis and probability. Paci®c Grove, California: Wadsworth &Brooks/Cole 1989 21. Feldman, M., Gilles, C.: An expository note on individual risk without aggregate uncertainty. Journal of Economic Theory 35, 26±32 (1985) 22. Geitz, R. F.: Pettis integration. Proceedings of the American Mathematical Society 82, 81± 86 (1981) 23. Green, E. J.: Lending and the smoothing of uninsurable income. In: Prescott, E. C., Wallace, N. (eds.) Contractual arrangements for international trade. Minneapolis: University of Minnesota Press 1987 24. Green, E. J.: Individual-level randomness in a nonatomic population. University of Minnesota, mimeo, (1994). First draft circulated in 1988 25. Hildenbrand, W.: Random preferences and equilibrium analysis. Journal of Economic Theory 4, 414±429 (1971) 26. Hildenbrand, W.: Core of an economy. In: Arrow, K. J., Intrilligator, M. D. (eds.) Handbook of mathematical economics, Vol. 2. Berlin Heidelberg New York: North-Holland 1982 27. Hildenbrand, W.: Cores. In: Eatwell, J., Newman, P. K., Milgate, M. (eds.) The new palgrave. London: MacMillan 1987 28. Ho€mann-Jorgenson, J.: The law of large numbers for non-measurable and non-separable random elements. Asterisque 131, 299±356 (1985)

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29. Judd, K. L.: The law of large numbers with a continuum of iid random variables. Journal of Economic Theory 35, 19±25 (1985) 30. Keisler, H. J.: Hyper®nite model theory. In: Gandy, R. O., Hyland, J. M. E. (eds.) Logic Colloquium 76. Amsterdam: North-Holland 1977 31. Khan, M. Ali, Rustichini, A.: Di€erential information and a reformulation of CournotNash equilibria. In: TheÂra, M. A., Baillon, J. B. (eds.) Fixed point theory and applications. Pitman Research Notes in Mathematics No. 252. London: Longman 1991 32. Khan, M. Ali, Rustichini, A.: On Cournot-Nash equilibrium distributions for games with uncertainty and imperfect information. Journal of Mathematical Economics 22, 35±59 (1993) 33. La€ont, J. J.: Economics of uncertainty and information. Cambridge: MIT Press 1989 34. LoeÁve, M., Probability theory I &II. Berlin Heidelberg New York: Springer 1977 35. Lucas, R. E., Prescott, E. C.: Equilibrium search and unemployment. Journal of Economic Theory 7, 188±209 (1974) 36. Magill, M., Shafer, W.: Allocation of aggregate and individual risks through futures and insurance markets. In: Majumdar, M. (eds.) Equilibrium and dynamics: Essays in Honour of David Gale. London: MacMillan 1992 37. Malinvaud, E.: The allocation of individual risks in large markets. Journal of Economic Theory 4, 312±328 (1972) 38. Malinvaud, E.: Markets for an exchange economy with individual risks. Econometrica 41, 383±410 (1973) 39. Mas-Colell, A., Vives, X.: Implementation in economies with a continuum of agents. Review of Economic Studies 60, 613±630 (1993) 40. Parthasarathy, K. R.: Probability measures on metric spaces. New York: Academic Press 1972 41. Prescott, E. C., Townsend, R. M.: Pareto optima and competitive equilibria with adverse selection and moral hazard. Econometrica 52, 21±45 (1984) 42. Rashid, S.: Economies with many agents. Baltimore: The Johns Hopkins University Press 1987 43. Rudin, W.: Real and complex analysis. New York: McGraw Hill 1974 44. Samuelson, P. A.: Risk and uncertainty: a fallacy of large numbers. Scientia 57, 1±6 (1963) 45. Schmeidler, D.: Equilibrium points of non-atomic games. Journal of Statistical Physics 7, 295±300 (1973) 46. Sentilles, F. D.: Decomposition of weakly measurable functions. Indiana University Mathematics Journal 32, 425±437 (1983) 47. Sentilles, F. D., Wheeler, R. F.: Pettis integration via the stonian transform. Paci®c Journal of Mathematics 107, 473±496 (1983) 48. Stokey, N. L., Lucas, R. E.: Recursive methods in dynamic economics. Cambridge: Harvard University Press 1989 49. Talagrand, M.: Pettis integral and measure theory, No. 307. Providence: Memoirs of the American Mathematical Society, 1984 50. Talagrand, M.: The Glivenko-Cantelli Problem. Annals of Probability 15, 837±870 (1987) 51. Talagrand, M.: The Glivenko-Cantelli Problem, ten years later. Journal of Theoretical Probability 9, 371±384 (1996) 52. Uhl, J. J. Jr.: Pettis's Measurability Theorem. Contemporary Mathematics 2, 135±144 (1980) 53. Uhlig, H.: A law of large numbers for large economies. Economic Theory 8, 41±50 (1996) 54. Yannelis, N.: Integration of Banach-valued correspondences. In: Khan M. Ali, Yannelis, N. C. (eds.) Equilibrium theory in in®nite dimensional spaces. Berlin Heidelberg New York: Springer 1991

Note added in proof After the galley proofs of this paper were reviewed in June 1998, the authors were noti®ed by Professor AI-Najjar in September 1998 that a Corrigendum to his paper (Reference 1) was formally accepted for publication in Econometrica.