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Strictly positive measures on Boolean algebras
 MR 2467227 (2010b:03077
"... We investigate strictly positive finitely additive measures on Boolean algebras and strictly positive Radon measures on compact zerodimensional spaces. The motivation is to find a combinatorial characterisation of Boolean algebras which carry a strictly positive finitely additive finite measure with ..."
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We investigate strictly positive finitely additive measures on Boolean algebras and strictly positive Radon measures on compact zerodimensional spaces. The motivation is to find a combinatorial characterisation of Boolean algebras which carry a strictly positive finitely additive finite measure with some additional properties, such as separability or nonatomicity. A possible consistent characterisation for an algebra to carry a separable strictly positive measure was suggested by Talagrand in 1980, which is that the Stone space K of the algebra satisfies that its space M(K) of measures is weakly separable, equivalently that C(K) embeds into l∞. We show that there is a ZFC example of a Boolean algebra (so of a compact space) which satisfies this condition and does not support a separable strictly positive measure. However, we use this property as a tool in a proof which shows that under MA + ¬CH every atomless ccc Boolean algebra of size < c carries a nonatomic strictly positive measure. Examples are given to show that this result does not hold in ZFC. Finally, we obtain a characterisation of Boolean algebras
Randomizing a Model
"... 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 randomizati ..."
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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 wellknown examples will be given in this introduction, and many others will be given in Section 4. The aim of thi...
Why saturated probability spaces are necessary
, 2009
"... 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 ..."
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Cited by 3 (1 self)
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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 nonsaturated 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 outperforms the Lebesgue unit interval in almost any way at all is already saturated.
Complete embeddings of the Cohen algebra into three families of c.c.c., nonmeasurable Boolean algebras
 Pacific J. Math
, 2004
"... The Cohen algebra embeds as a complete subalgebra into three classic families of complete, atomless, c.c.c., nonmeasurable Boolean algebras; namely, the families of Argyros algebras andGalvinHajnal algebras, andthe atomless part of each Gaifman algebra. It immediately follows that the weak (ω, ω) ..."
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The Cohen algebra embeds as a complete subalgebra into three classic families of complete, atomless, c.c.c., nonmeasurable Boolean algebras; namely, the families of Argyros algebras andGalvinHajnal algebras, andthe atomless part of each Gaifman algebra. It immediately follows that the weak (ω, ω)distributive law fails everywhere in each of these Boolean algebras. 1. Introduction. Von Neumann conjectured that the countable chain condition and the weak (ω, ω)distributive law characterize measurable algebras among Boolean σalgebras [Mau]. Consistent counterexamples have been obtained by Maharam
On Large Games with a BioSocial Typology
, 2009
"... We present a comprehensive theory of large nonanonymous games in which agents have a name and a determinate socialtype and/or biological trait to resolve the dissonance of a (matchingpennies type) game with an exact purestrategy Nash equilibrium with finite agents, but without one when modeled ..."
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We present a comprehensive theory of large nonanonymous games in which agents have a name and a determinate socialtype and/or biological trait to resolve the dissonance of a (matchingpennies type) game with an exact purestrategy 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.
ALGEBRAIC CHARACTERIZATIONS OF MEASURE ALGEBRAS
, 2007
"... We present necessary and sufficient conditions for the existence of a countably additive measure on a Boolean σalgebra. For instance, a Boolean σalgebra 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 a ..."
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We present necessary and sufficient conditions for the existence of a countably additive measure on a Boolean σalgebra. For instance, a Boolean σalgebra 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.
AMS JOURNAL STYLE Volume 000, 0000 PROPERTIES OF IDEALS ON THE GENERALIZED CANTOR SPACES
"... Abstract. We define a class of productive σ−ideals of subsets of the Cantor space 2 ω and observe that both σ−ideals of meagre sets and of null sets are in this class. From every productive σ−ideal J we produce a σ−ideal Jκ of subsets of the generalized Cantor space 2 κ. In particular, starting from ..."
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Abstract. We define a class of productive σ−ideals of subsets of the Cantor space 2 ω and observe that both σ−ideals of meagre sets and of null sets are in this class. From every productive σ−ideal J we produce a σ−ideal Jκ of subsets of the generalized Cantor space 2 κ. In particular, starting from meagre sets and null sets in 2 ω we obtain meagre sets and null sets in 2 κ, respectively. Then we investigate additivity, covering number, uniformity and cofinality of Jκ. For example, we show that Our results generalizes those from [5]. non(J) = non(J ω1) = non(J ω2). In this paper we shall discuss the properties of canonical σ−ideals of subsets of generalized Cantor spaces 2 κ, for example the σ−ideal of null sets and of meagre sets. In the 80’s several people investigated relations between the σ−ideal N of null