Continuous version of the Choquet Integral Reperesentation Theorem

Let C(K) be the Banach space of all continuous functions on a given compact space K. We investigate the w*-sequential closure in C(K)* of the set of all finitely supported probabilities on K. We discuss the coincidence of the Baire sigma-algebras on C(K) associated to the weak and pointwise convergence topologies.

This paper is a follow-up to the author's work "Topology of probability measure space, I" devoted to investigation of the functors $\hat P$ and $P_\tau$ of spaces of probability $\tau$-smooth and Radon measures. In this part, we study the barycenter map for spaces of Radon probability measures. The obtained results are applied to show that the functor $\hat P$ is monadic in the category of metrizable spaces. Also we show that the functors $\hat P$ and $P_\tau$ admit liftings ...

The class of $\mu$-compact sets can be considered as a natural extension of the class of compact metrizable subsets of locally convex spaces, to which the particular results well known for compact sets can be generalized. This class contains all compact sets as well as many noncompact sets widely used in applications. In this paper we give a characterization of a convex $\mu$-compact set in terms of properties of functions defined on this set. Namely, we prove that the class ...

We establish a dilation-theoretic characterization of the Choquet order on the space of measures on a compact convex set using ideas from the theory of operator algebras. This yields an extension of Cartier's dilation theorem to the non-separable setting, as well as a non-separable version of \v{S}a\v{s}kin's theorem from approximation theory. We show that a slight variant of this order characterizes the representations of a commutative C*-algebra that have the unique extensi...

Let $\mu$ be a probability measure on a separable Banach space $X$. A subset $U\subset X$ is $\mu$-continuous if $\mu(\partial U)=0$. In the paper the $\mu$-continuity and uniform $\mu$-continuity of convex bodies in $X$, especially of balls and half-spaces, is considered. The $\mu$-continuity is interesting for study of the Glivenko-Cantelli theorem in Banach spaces. Answer to a question of F. Tops{\o}e is given.

Let $(M,d)$ be a complete metric space and let $\mathcal{F}(M)$ denote the Lipschitz-free space over $M$. We develop a ``Choquet theory of Lipschitz-free spaces'' that draws from the classical Choquet theory and the De Leeuw representation of elements of $\mathcal{F}(M)$ (and its bidual) by positive Radon measures on $\beta\widetilde{M}$, where $\widetilde{M}$ is the space of pairs $(x,y) \in M \times M$, $x \neq y$. We define a quasi-order $\preccurlyeq$ on the positive Rado...

In this paper we study different aspects of the representation of weak*-compact convex sets of the bidual $X^{**}$ of a separable Banach space $X$ via a nested sequence of closed convex bounded sets of $X$.

We show that the lift zonoid concept for a probability measure on R^d, introduced in (Koshevoy and Mosler, 1997), leads naturally to a one-to one representation of any interior point of the convex hull of the support of a continuous measure as the barycenter w.r.t. to this measure of either of a half-space, or the whole space. We prove the infinite-dimensional generalization of this representation, which is based on the extension of the lift-zonoid concept for a cylindrical p...

The main result of this paper characterizes the continuity from below of monotone functionals on the space $C_b$ of bounded continuous functions on an arbitrary Polish space as lower semicontinuity in the mixed topology. In this particular situation, the mixed topology coincides with the Mackey topology for the dual pair $(C_b,{\rm ca})$, where ${\rm ca}$ denotes the space of all countably additive signed Borel measures of finite variation. Hence, lower semicontinuity in the ...

In this paper we apply ideas from the theory of Uniform Distribution of sequences to Functional Analysis and then drawing inspiration from the consequent results, we study concepts and results in Uniform Distribution itself. So let $E$ be a Banach space. Then we prove:\\ (a) If $F$ is a bounded subset of $E$ and $x \in \overline{\co}(F)$ (= the closed convex hull of $F$), then there is a sequence $(x_n) \subseteq F$ which is Ces\`{a}ro summable to $x$.\\ (b) If $E$ is separab...