Choquet Theory Research Articles

An application of the Choquet theorem to the study of randomly-superinvariant measures

Given a real valued random variable we consider Borel measures onB(R), which satisfy the inequality (B) E (B ) ( B2B(R)) (or the integral inequality (B) R 1 1 (B h) (dh)). We apply the Choquet theorem to obtain an integral representation of measures satisfying this inequality. We give integral representations of these measures in the particular cases of the random variable .

Algorithmic Randomness and Capacity of Closed Sets

We investigate the connection between measure, capacity and algorithmic randomness for the space of closed sets. For any computable measure m, a computable capacity T may be defined by letting T(Q) be the measure of the family of closed sets K which have nonempty intersection with Q. We prove an effective version of Choquet's capacity theorem by showing that every computable capacity may be obtained from a computable measure in this way. We establish conditions on the measure m that characterize when the capacity of an m-random closed set equals zero. This includes new results in classical probability theory as well as results for algorithmic randomness. For certain computable measures, we construct effectively closed sets with positive capacity and with Lebesgue measure zero. We show that for computable measures, a real q is upper semi-computable if and only if there is an effectively closed set with capacity q.

$G_\delta$ ideals of compact sets

We investigate the structure of Gδ ideals of compact sets. We define a class of Gδ ideals of compact sets that, on the one hand, avoids certain phenomena present among general Gδ ideals of compact sets and, on the other hand, includes all naturally occurring Gδ ideals of compact sets. We prove structural theorems for ideals in this class, and we describe how this class is placed among all Gδ ideals. In particular, we establish a result representing ideals in this class via the meager ideal. This result is analogous to Choquet's theorem representing alternating capacities of order ∞ via Borel probability measures. Methods coming from the structure theory of Banach spaces are used in constructing important to us examples of Gδ ideals outside of our class.

Effective Capacity and Randomness of Closed Sets

We investigate the connection between measure and capacity for the space of nonempty closed subsets of {0,1}*. For any computable measure, a computable capacity T may be defined by letting T(Q) be the measure of the family of closed sets which have nonempty intersection with Q. We prove an effective version of Choquet's capacity theorem by showing that every computable capacity may be obtained from a computable measure in this way. We establish conditions that characterize when the capacity of a random closed set equals zero or is >0. We construct for certain measures an effectively closed set with positive capacity and with Lebesgue measure zero.

The Hermite–Hadamard inequality for convex functions on a global NPC space

We prove an extension of Choquet's theorem to the framework of compact metric spaces with a global nonpositive curvature. Together with Sturm's extension [K.T. Sturm, Probability measures on metric spaces of nonpositive curvature, in: Pascal Auscher, et al. (Eds.), Heat Kernels and Analysis on Manifolds, Graphs, and Metric Spaces, Lecture Notes from a Quarter Program on Heat Kernels, Random Walks, and Analysis on Manifolds and Graphs April 16–July 13, 2002, Paris, France, in: Contemp. Math., vol. 338, Amer. Math. Soc., Providence, RI, 2003, pp. 357–390] of Jensen's inequality, this provides a full analogue of the Hermite–Hadamard inequality for the convex functions defined on such spaces.

Elliptic systems and material interpenetration

We classify the second order, linear, two by two systems for which the two fundamental theorems for planar harmonic mappings, the Rado--Kneser--Choquet theorem and the H. Lewy theorem, hold. They are those which, up to a linear change of variable, can be written in diagonal form with \emph{the same} operator on both diagonal blocks. In particular, we prove that the aforementioned theorems cannot be extended to solutions of either the Lame system of elasticity, or of elliptic systems in diagonal form, even with just slightly different operators for the two components.

Characterizations of Commutative POV Measures

Two different characterizations of POV measures with commutative range are compared using a representation of some stochastic operators by (weak) Markov kernels. A representation by Choquet theorem is obtained as an integral over functions of a sharp observable appearing in one of the characterizations. A Naimark extension is constructed.

On Choquet theorem for random upper semicontinuous functions

Motivated by the problem of modeling of coarse data in statistics, we investigate in this paper topologies of the space of upper semicontinuous functions with values in the unit interval [0, 1] to define rigorously the concept of random fuzzy closed sets. Unlike the case of the extended real line by Salinetti and Wets, we obtain a topological embedding of random fuzzy closed sets into the space of closed sets of a product space, from which Choquet theorem can be investigated rigorously. Precise topological and metric considerations as well as results of probability measures and special properties of capacity functionals for random u.s.c. functions are also given.

Absolutely Summing Processes

Absolutely summing processes are defined, which form a subclass of weakly operator harmonizable processes. When the parameter space is the set of real numbers, it is proved that an absolutely summing process is represented as an integral of operator stationary processes with respect to an appropriate probability measure. To do this, weak convergence of scalar and vector measures is considered. Then we prove compactness of the unit ball of vector measures under certain topologies, and we apply the Choquet theorem to derive an integral representation.

Harmonic mappings onto stars

A general version of the Radó–Kneser–Choquet theorem implies that a piecewise constant sense-preserving mapping of the unit circle onto the vertices of a convex polygon extends to a univalent harmonic mapping of the unit disk onto the polygonal domain. This paper discusses similarly generated harmonic mappings of the disk onto nonconvex polygonal regions in the shape of regular stars. Calculation of the Blaschke product dilatation allows a determination of the exact range of parameters that produce univalent mappings.

Bochner and Bernstein theorems via the nuclear integral representation theorem

We show how, using the nuclear integral representation theorem, the Bernstein–Choquet theorem and Bochner–Schwartz theorem may be derived. In the case of the Bernstein–Choquet theorem we give an example, determining the representing measure explicitly.

Relative compactness, cotopology and some other notions from the bitopological point of view

The paper consists of two sections. Section 1 is the introduction which, in addition to the auxiliary information, contains some interesting results on Baire-like properties. Section 2 deals with the bitopological essence of the notions of relative compactness and cotopology in general topology, C-relation, subordination of topologies and closed neighborhoods condition in analysis. A generalization of Choquet's theorem on Baire spaces is given and the sufficient conditions for families of ( i, j)-nowhere dense sets to coincide with families of ( i, j)-first category sets are established using a finite measure. A bitopological solution of one of Ulam's problems is obtained. The corresponding relations are almost always studied using essentially the bitopological modifications of regularity, which, as seen in various problems of general topology, analysis and potential theory, are the most natural forms of relations of two topologies defined on the same set.

Choquet like sets in function spaces

In convex analysis when studying function spaces of continuous affine functions, notions of a geometrical character like faces, split and parallel faces, exposed or Archimedean faces were investigated in detail by many authors. In this paper we transfer these notions to a more general setting of Choquet theory of abstract function spaces. We prefer a direct functional analytic approach to the treatment of problems instead of using a transfer of a function space to its state space. Methods invoked are based mainly on a measure theory and basic tools of functional analysis and are different from ones using a geometric visualization.

Choquet's theory and the Dirichlet problem

Choquet's theory and the Dirichlet problem

Choquet theory for signed measures

We introduce the notion of barycenter for a class of non-necessarily positive Radon measures and prove on this basis several inequalities which extend classical results such as Steffensen’s inequality, Fink’s version of the Hermite-Hadamard inequality, Fuchs ’extension of the majorization inequality of Hardy-Littlewood-Polýa etc. Mathematics subject classification (2000): 26D15, 46A55, 26A51, 26B55.

The Shilov Boundary of an Operator Space and the Characterization Theorems

We study operator spaces, operator algebras, and operator modules from the point of view of the noncommutative Shilov boundary. In this attempt to utilize some noncommutative Choquet theory, we find that Hilbert C*-modules and their properties, which we studied earlier in the operator space framework, replace certain topological tools. We introduce certain multiplier operator algebras and C*-algebras of an operator space, which generalize the algebras of adjointable operators on a C*-module and the imprimitivity C*-algebra. It also generalizes a classical Banach space notion. This multiplier algebra plays a key role here. As applications of this perspective, we unify and strengthen several theorems characterizing operator algebras and modules. We also include some general notes on the commutative case of some of the topics we discuss, coming in part from joint work with Christian Le Merdy, about function modules.

On Ideals in Banach Spaces

In this paper we study the notion of an ideal, which was introduced by Godefroy, Kalton and Saphar in [7] and was called locally one complemented in [11], for injective and projective tensor products of Banach spaces. For a Banach space X and an ideal Y in X, we show that the injective tensor product space Y ⊗ e Z is an ideal in X ⊗ e Z for any Banach space Z. This as a consequence gives us a way of proving some known results about intersection properties of balls and extensions of operators on injective tensor product spaces in a unified way that does not involve any vector-valued Choquet theory. We also exhibit classes of Banach spaces in which every ideal is the range of a norm one projection.

Geometric Theory of Spaces of Integral Polynomials and Symmetric Tensor Products

We investigate the nature of extreme, (weak*-) exposed, and (weak*-) strongly exposed points of the unit ball of spaces of n-homogeneous integral polynomials on, and n-fold symmetric products of, a Banach space E. For the space of integral polynomials we show the set of extreme points is contained in the set {±φn:φ∈E′, ‖φ‖=1}. We give Šmul'yan type theorems for spaces of n-homogeneous polynomials and n-fold symmetric tensors that characterise weak*-exposed (resp. weak*-strongly exposed) points in terms of Gâteaux (resp. Fréchet) differentiability of the norm on various spaces of tensor products and polynomials. Our study of the geometry of these spaces has many applications: When E has the Radon–Nikodým property we show that the spaces of n-homogeneous integral and nuclear polynomials are isomerically isomorphic for each integer n. When the dimensions of E and n are both at least 2 then the space of n-homogeneous polynomials on E is neither smooth nor rotund. For a certain class of reflexive Banach space the space of n-homogeneous approximable polynomials on E is either reflexive or is not isometric to a dual Banach space. We conclude with a Choquet Theorem for a space of homogeneous polynomials.

Choquet Theory in Metric Spaces

This paper deals with a generalization of the classical Choquet theorem. We consider metric spaces which are endowed with an abstract notion of convexity. Convex combinations are obtained by the solutions of variational inequalities. A generalized Krein-Milman theorem is derived from our Choquet theorem. We end with an example based on hyperbolic geometry.

Some Applications of Choquet Theory to Harmonic Analysis

Some Applications of Choquet Theory to Harmonic Analysis