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Dans ces notes il sera expliqué que la propriété est vérifiée par le groupe de Heisenberg muni de la distance de Carnot-Carathéodory et de la mesure de Lebesgue. Cette propriété correspond pour les espaces métriques mesurés à une courbure de Ricci positive. Comme application, les mesures interpolées par transport de mesure sont absolument continues. En revanche, la courbure-dimension , une autre courbure de Ricci synthétique adaptée aux espaces métriques mesurés est fausse pour .
Let X and Y be two compact spaces endowed with
respective measures μ and ν satisfying the condition µ(X) = v(Y). Let c be a continuous function on the product space X x Y. The mass transfer problem consists in determining a measure ξ on
X x Y whose marginals coincide with μ and ν, and such that
the total cost ∫ ∫ c(x,y)dξ(x,y) be minimized. We first
show that if the cost function c is decomposable, i.e., can be
represented as the sum of two continuous functions defined on X and
Y, respectively,...
A properly measurable set (where are Polish spaces and is the space of Borel probability measures on ) is considered. Given a probability distribution the paper treats the problem of the existence of -valued random vector for which and -almost surely that possesses moreover some other properties such as “ has the maximal possible support” or “’s are extremal...
We prove an extension theorem for modular functions on arbitrary lattices and an extension theorem for measures on orthomodular lattices. The first is used to obtain a representation of modular vector-valued functions defined on complemented lattices by measures on Boolean algebras. With the aid of this representation theorem we transfer control measure theorems, Vitali-Hahn-Saks and Nikodým theorems and the Liapunoff theorem about the range of measures to the setting of modular functions on complemented...
Let be the set of upper strongly porous at subsets of and let be the intersection of maximal ideals . Some characteristic properties of sets are obtained. We also find a characteristic property of the intersection of all maximal ideals contained in a given set which is closed under subsets. It is shown that the ideal generated by the so-called completely strongly porous at subsets of is a proper subideal of Earlier, completely strongly porous sets and some of their properties were...
A subset of the plane is called a two point set if it intersects any line in exactly two points. We give constructions of two point sets possessing some additional properties. Among these properties we consider: being a Hamel base, belonging to some -ideal, being (completely) nonmeasurable with respect to different -ideals, being a -covering. We also give examples of properties that are not satisfied by any two point set: being Luzin, Sierpiński and Bernstein set. We also consider natural generalizations...
Doubling measures appear in relation to quasiconformal mappings of the unit disk of the complex plane onto itself. Each such map determines a homeomorphism of the unit circle on itself, and the problem arises, which mappings f can occur as boundary mappings?
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