The Choquet - Bishop - de Leeuw theorem states that each element of a compact convex subset of a locally convex topological Hausdorff space is a barycenter of a probability measure supported by the set of extreme points of that set. By the Edgar - Mankiewicz result this remains true for nonempty closed bounded and convex set provided it has Radon - Nikodym property. In the paper it is shown, that Choquet - type theorem holds also for "moving" sets: they are values of a certain multifunction. Namely, the existence of a suitable weak* continuous family of probability measures "almost representing" points of such sets is proven. Both compact and noncompact cases are considered. The continuous versions of the Krein - Milman theorem are obtained as corollaries.
We introduce a~\textit{Choquet-Sugeno-like operator} generalizing many operators for bounded functions and monotone measures from the literature, e.g., Sugeno-like operator, Lovász and Owen measure extensions, $\rF$-decomposition integral with respect to a~partition decomposition system, and others. The new operator is based on the concepts of dependence relation and conditional aggregation operators, but it does not depend on $t$-level sets. We also provide conditions for which the Choquet-Sugeno-like operator coincides with some Choquet-like integrals defined on finite spaces and appeared re…
The integral representation of Choquet operators defined on a space C(X) is established by using the Choquet-Bochner integral of a real-valued function with respect to a vector capacity.
We consider integral inequalities in the sense of Choquet with respect to the Hausdorff content $\mathcal{H}_\infty^δ$. In particular, if $Ω$ is a bounded John domain in $\mathbb{R}^n$, $n\geq 2$, and $0 <δ\le n$, we prove that the corresponding $(δp/(δ-p),p)$-Poincaré-Sobolev inequalities hold for all continuously differentiable functions defined on $Ω$ whenever $δ/n < p < δ$. We prove also that the $(p,p)$-Poincaré inequality is valid for all $p>δ/n$.
A solution to the extreme point problem and other applications of Choquet theory to Lipschitz-free spaces
Instituto Universitario de Matemática Pura y Aplicada, Universitat Politècnica de València（瓦伦西亚理工大学纯粹与应用数学研究所）；Faculty of Information Technology, Czech Technical University in Prague（布拉格捷克理工大学信息技术学院）；School of Mathematics and Statistics, University College Dublin（都柏林大学学院数学与统计学院）
We present results for Choquet integrals with minimal assumptions on the monotone set function through which they are defined. They include the equivalence of sublinearity and strong subadditivity independent of regularity assumptions on the capacity, as well as various forms of standard measure theoretic convergence theorems for these non-additive integrals, e.g. Fatou's lemma and Lebesgue's dominated convergence theorem.
