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We study the Bochner and Gel′fand integration of Banach space valued correspondences on a general Loeb space. Though it is well known that the Lyapunov type result on the compactness and convexity of the integral of a correspondence and the Fatou type result on the preservation of upper semicontinuity by integration are in general not valid in the setting(More)
This paper shows the existence of independent randommatching of a large (continuum) population in both static and dynamic systems, which has been popular in the economics and genetics literatures. We construct a joint agent-probability space, and randomized mutation, partial matching, and match-induced type-changing functions that satisfy appropriate(More)
Given a measurable mapping f from a nonatomic Loeb probability space (T; T ; P ) to the space of Borel probability measures on a compact metric space A, we show the existence of a measurable mapping g from (T; T ; P ) to A itself such that f and g yield the same values for the integrals associated with a countable class of functions on T A. A corollary(More)
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(More)
We present a particular class of measure spaces, hyperfinite Loeb spaces, as a model of situations where individual players are strategically negligible, as in large non-anonymous games, or where information is diffused, as in games with imperfect information. We present results on the existence of Nash equilibria in both kinds of games. Our results cover(More)
As is well known, a continuous parameter process with mutually independent random variables is not jointly measurable in the usual sense. This paper proposes an extension of the usual product measure-theoretic framework, using a natural “one-way Fubini” property. When the random variables are independent even in a very weak sense, this property guarantees(More)