Abstract. The aim of the paper is to describe the necessary and sufficient conditions for a Boolean algebra to admit the largest possible sequential convergence structure. We present examples of complete algebras, known from construction of various generic extensions of set theory, carrying such converegence structures. 1. Introduction. In this section we review some basic notions and facts concerning sequential convergence structures and continuity of submeasures on a Boolean algbera B. The motivation for the research described in this paper comes from [Ja] and [Ja1], where it is shown that the maximal possible convergence structure is attained for (ω, 2)distributive

A randomization of a first order structure M is a new structure with certain closure properties whose universe is a set K of "random elements" of M. Randomizations assign probabilities to sentences of the language of M with new constants from K. Our main theorem shows that all randomizations of M are models of the same first order theory T , which has a nice set of axioms and admits elimination of quantifiers. Moreover, the class of substructures of models of T is characterized by a natural set V of universal axioms of T , so that T is the model completion of V . 1 Introduction A common theme in mathematics is to start with a first order structure M, and introduce a new structure K which has a set K of "random elements" of M as a universe and which assigns probabilities to sentences of the language of M with new constants from K. There are several ways to do this; three well-known examples will be given in this introduction, and many others will be given in Section 4. The aim of thi...

An atomless probability space (Ω, A,P) is said to have the saturation property for a probability measure μ on a product of Polish spaces X × Y if for every random element f of X whose law is margX(μ), there is a random element g of Y such that the law of (f, g) is μ. (Ω, A,P) is said to be saturated if it has the saturation property for every such μ. We show each of a number of desirable properties holds for every saturated probability space and fails for every non-saturated probability space. These include distributional properties of correspondences, such as convexity, closedness, compactness and preservation of upper semicontinuity, and the existence of pure strategy equilibria in games with many players. We also show that any probability space which has the saturation property for just one “good enough” measure, or which satisfies just one “good enough ” instance of the desirable properties, must already be saturated. Our underlying themes are: (1) There are many desirable properties that hold for all saturated probability spaces but fail everywhere else; (2) Any probability space that out-performs the Lebesgue unit interval in almost any way at all is already saturated.

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We present necessary and sufficient conditions for the existence of a countably additive measure on a Boolean σ-algebra. For instance, a Boolean σ-algebra B is a measure algebra if and only if B −{0} is the union of a chain of sets C1 ⊂ C2 ⊂... such that for every n, (i) every antichain in Cn has at most K(n) elements (for some integer K(n)), (ii) if {an}n is a sequence with an / ∈ Cn for each n, then limn an = 0,and (iii) for every k,if{an}n is a sequence with limn an = 0, then for eventually all n, an / ∈ Ck. The chain {Cn} is essentially unique.

Abstract: We present a comprehensive theory of large non-anonymous games in which agents have a name and a determinate social-type and/or biological trait to resolve the dissonance of a (matching-pennies type) game with an exact pure-strategy Nash equilibrium with finite agents, but without one when modeled on the Lebesgue unit interval. We (i) establish saturated player spaces as both necessary and sufficient for an existence result for Nash equilibrium in pure strategies, (ii) clarify the relationship between pure, mixed and behavioral strategies via the exact law of large numbers in a framework of Fubini extension, (iii) illustrate corresponding asymptotic results. (99 words)