Atomless Measure Space Research Articles

On the optimality conditions of vector-valued n-set functions

In a finite atomless measure space ( X, Γ, μ), the optimization problem of vector-valued n-set functions defined on a convex subfamily S of Γ n = Γ × … × Γ is investigated. The necessary and sufficient conditions of Pareto optimal solution or proper R + p -solution of optimization problem with differentiable vector valued n-set functions are given.

Some geometrical properties and fixed point theorems in Orlicz spaces

Let ( G, ∑, μ) be a finite, atomless measure space and let L φ be an Orlicz space of measurable functions on G. We consider some geometrical properties of the functional ρ(ƒ) = ∝ Gφ(ƒ(t)) dμ(t) , called the Orlicz modular. These properties, like strict convexity, uniform convexity or uniform convexity in every direction, can be equivalently expressed in terms of the properties of the corresponding Orlicz function φ. We use these properties in order to prove some fixed point results for mappings T: B → B, B ⊂ L φ , that are nonexpansive with respect to the Orlicz modular ρ, i.e., ρ(Tƒ − Tg) ⩽ ρ(ƒ − g) for all ƒ and g in B. We prove also existence and uniqueness in L φ of the best approximant with respect to ρ and some convex subsets of L φ . Our results are valid also in the case when the Orlicz function φ does not satisfy the Δ 2-condition. This demonstrates the advantage of our method because, in the latter case, both Luxemburg's and Orlicz's norms cannot possess suitable convexity properties.

Edgeworth's conjecture in economies with a continuum of agents and commodities

This paper contains the following results for economies with infinite dimensional commodity spaces. (i) We establish a core-Walras equivalence theorem for economies with an atomless measure space of agents and with an ordered separable Banach commodity space whose positive cone has a non-empty norm interior. This result includes as a special case the Aumann (1964) and Schmeidler-Hildenbrand [Hildenbrand (1974, p. 33)] finite dimensional theorems. (ii) We provide a counterexample which shows that the above result fails in ordered Banach spaces whose positive cone has an empty interior even if preferences are strictly convex, monotone weakly continuous and initial endowments are strictly positive. (iii) Using the assumption of an ‘extremely desirable commodity’ (which is automatically satisfied whenever preferences are monotone and the positive cone of the commodity space has a non-empty interior), we establish core-Walras equivalence in any arbitrary separable Banach lattice whose positive cone may have an empty (norm) interior.

Optimality for set functions with values in ordered vector spaces

Let (X, Г, μ) be a finite atomless measure space,L a convex subfamily of Φ, andY andZ locally convex Hausdorff topological vector spaces which are ordered by the conesC andD, respectively. LetF:L→Y beC-convex andG:L→Z beD-convex set functions. Consider the following optimization problem (P): minimizeF(ω), subject to ω∈L andG(ω)≤ D θ. The paper generalizes the Moreau-Rockafellar theorem with set functions. By applying this theorem, a Kuhn-Tucker type optimality condition and a Fritz John type optimality condition for problem (P) are established. The duality theorem for problem (P) is also studied.

Generalized holomorphic processes and differentiability

Holomorphic processes are studied when the parameter set is a general finite atomless separable measure space, using the differentiation operator and Malliavin calculus. Generalized martingales are connected with differentiable processes. In the two-parameter case, these methods give a new simpler proof of the characterization of holomorphic processes by power series expansion.

On Cournot-Nash Equilibria in Generalized Qualitative Games with an Atomless Measure Space of Agents

We present a result on the existence of Cournot-Nash equilibria in games with an atomless measure space of players each with nonordered preferences and with strategy sets in a separable Banach space. Our result dispenses with any convexity assumption on the preference correspondence.

On Cournot-Nash equilibria in generalized qualitative games with an atomless measure space of agents

We present a result on the existence of Cournot-Nash equilibria in games with an atomless measure space of players each with nonordered preferences and with strategy sets in a separable Banach space. Our result dispenses with any convexity assumption on the preference correspondence.

A continuous generalization of the transversal property

A transversal for a set-system is a one-to-one choice function. A necessary and sufficient condition for the existence of a transversal in the case of finite sets was given by P. Hall (see [4, 3]). The corresponding condition for the case when countably many countable sets are given was conjectured by Nash-Williams and later proved by Damerell and Milner [2]. B. Bollobás and N. Varopoulos stated and proved the following measure theoretic counterpart of Hall's theorem: if (X, μ) is an atomless measure space, ℋ = {Hi: i∈I} is a family of measurable sets with finite measure, λi (i∈I) are non-negative real numbers, then we can choose a subset Ti ⊆ Hi with μ(Ti) = λi and μ(Ti ∩ Ti′) = 0 (i ≠ i′) if and only if μ({U Hi: iεJ}) ≥ Σ{λi: iεJ}: for every finite subset J of I. In this note we generalize this result giving a necessary and sufficient condition for the case when I is countable and X is the union of countably many sets of finite measure.

The Multiobjective Optimization Problem of Set Function

Abstract The multiobjective optimization problem of a set function defined on a σ-algebra If in a finite atomless measure space (X, II, m) is investigated. A constrained local minimum is formulated and its necessary conditions are derived. Also, the sufficient conditions of a constrained local lower efficient point are given. Further, a necessary condition of a constrained minimum is shown in a form of a saddle point of a Lagrangian set function and a theorem for Lagrangian duality is given.

Diversified Consumption Characteristics and Conditionally Dispersed Endowment Distribution: Regularizing Effect and Existence of Equilibria

The paper introduces the concept of a conditionally endowment distribution (given the consumption set) in the study of regularizing effect of aggregation and existence of equilibria in an exchange economy with an atomless measure space of consumers. It is shown that there is a regularizing effect of aggregation whenever the endowment distribution of an economy is conditionally dispersed. On the basis of this regularizing effect, the existence of a competitive equilibrium is proved in a very general setting where some of the commodities are indivisible and where different consumers may have different consumption sets. GENERAL EQUILIBRIUM THEORY saw, in recent years, explicit formulation of models with indivisible commodities in which existence of competitive equilibria was discussed. (References are Henry [12], E. Dierker [10], Broome [7], and Mas-Colell [19].) A pathbreaking paper by Mas-Colell [19] clearly reconfirmed the power of a continuum of traders model introduced by Aumann [1], and it emphasized, for the first time in the literature of the equilibrium existence problem, the importance of a condition relating to the form of distribution of agents' characteristics. More specifically, a competitive equilibrium was shown to exist in an economy with indivisible commodities where a single perfectly divisible commodity exists and the distribution of its initial holdings among agents is dispersed in the sense that it is absolutely continuous with respect to the Lebesgue measure. (See [19, Theorem 1, p. 446].) This development in the existence literature is of much interest to equilibrium theory and coincides with the way in which the analysis of a certain aggregation problem is being pursued in equilibrium theory. (See, e.g., Sonderman [33], Mas-Colell [20], and Hildenbrand [18].) In both of these problems, elaboration of the concept of dispersedness in the distribution of agents' characteristics is required.3 The work of Mas-Colell [19] was followed by [35] in which the concept of a endowment distribution was introduced in the study of existence of equilibria in an exchange economy with an atomless measure space of consumers where no convexity assumptions on preferences and consumption sets were

Efficiency in an Atomless Economy with Fiat Money

L. Shapley in collaboration with M. Shubik has presented several variants of a pure exchange economy where money (fiat or commodity) serves as the medium of exchange. More specifically, each commodity is exchanged in a market where buyers bid money and sellers offer quantities of the commodity. Each buyer receives the proportion of the aggregate amount of the commodity equal to the proportion of his bid to the aggregate bids, and vice versa for sellers. This mechanism of exchange has several advantages over the Walrasian market mechanism. In a thin market, the mechaniism does not presume price taking behavior amonig traders. On the other hand, the optimizing behavior of a trader in a market with many small traders is essentially that of a price taker. Hence, the difference between oligopolistic and competitive behavior is determined endogenously. Another advantage of this model is that it distinguishes between feasible and optimal actions of economic agents. In the Walrasian analysis, there is no room for non-optimal behavior. Finally, the role of money and credit is naturally incorporated in the model. For more details, see the papers by Shapley and Shubik in the references. The purpose of this paper is to analyze the efficiency properties of equilibria resulting from the Shapley-Sliubik mechanism in a perfectly competitive economy with fiat money, i.e., money without any intrinsic value. The purely competitive feature of the economy is captured by representing economic agents by an atomless measure space. Even then, Nash equilibria of such an economy are not always efficient. The main results of this paper consist of a characterization of those Nash equilibria which are Pareto efficient and their relations to the Walras equilibria in Aumann's market with a continuum of traders. Since most of the variants of Shapley's model [2] are meaningless with fiat money (as opposed to commodity money), a different variant is discussed in this paper. The description of an economy in this paper is that of Aumainn's with the addition of an initial allocation of money over agents. In this respect, we have a straightforward extension of the Shapley-Shubik model to a market with a continuum of traders. We interpret the initial allocation of money as a credit from a bank which must be returned after trade ends. In this sense, the model is different from those in [2] where traders accept money for constimption. The usual interpretation of Nash equilibrium (which is relevant here) is that each player realizes his expecta-

The Limit of the Core of an Economy with Production

THE BEHAVIOR OF THE CORE of large economies has received considerable attention during the past decade during which two distinct lines of research have been followed. The first originates with Debreu and Scarf [7] who supplied a rigorous generalization and proof for a conjecture made already by Edgeworth [9], which since has entered economic textbooks as the statement that the core .shrinks to the set of competitive equilibria if the number of economic agents becomes infinitely large. The other line of research was started by Aumann [2] who used, as a representation of a large economy, an atomless measure space. Both approaches represent an attempt to describe the intuitive phenomenon of competition, that the power of any individual agent to influence the outcome of trading diminishes if there are sufficiently many other agents who also participate in the market. Recently, Hildenbrand [12] indicated that an atomless economy may be considered as the limit of a sequence of certain increasing but finite economies which provides a link between the two approaches. Most of the available results, however, deal only with pure exchange economies, except for Arrow and Hahn [1], Debreu and Scarf [7], Champsaur [6], and Hildenbrand [10] where production is treated in a very special way. Recently, the general case of the core with production has been formalized and existence proofs were given for the finite case in [3] and for the approach in a measure space by Sondermann in [14]. In [4] and [14] similar market equilibrium concepts were proposed which allow a comparison of the set of equilibrium allocations with the core. Since then a conjecture has been formulated which asserts that the same results may be obtained as in the case of pure exchange. This paper intends to show that this is, in principal, the case for the result of Debreu and Scarf with very general assumptions about the technology distribution. For the case of an atomless economy, however, the example in [5] indicates that the identity of the core and the set of equilibrium allocations may not be expected if the production correspondence is not strictly additive.

Fatou's Lemma in Several Dimensions

In this note the following generalization of Fatou's lemma is proved: Lemma. Let {fn)n_l be a sequence of integrable functions on a mea- sure space S with values in R+, the nonnegative orthant of a d-dimen- sional Euclidean space, for which ffn—*aGiR+. Then there exists an in- tegrable function f, from S to R+, such that a.e. f(s) is a limit point of VnisV^and ff^a. 1. Introduction. When d = l, the result is a form of Fatou's lemma. The assertion above is applied in mathematical economics (4).2 It is also strongly connected with the theory of set valued functions (2) or correspondences (3). The nontrivial part of the arguments is lim- ited to the case where 5 is an atomless measure space. In the purely atomic case the lemma is reduced to a simple exercise in series. In any case, the lemma cannot be proved by a successive application of Fatou's lemma d times. A few corollaries of the lemma are proved in §3. 2. Preliminary results and the proof of the lemma. Let (A)°=1 be a sequence of (nonempty) subsets of Rd. We denote by Lim Sup^4 the set of all the limit points of the sequences (a),T-i with anEAn, » = 1, 2, • • ■ . Denote by x-y the inner product, Yt=i xlyl, in Rd.