### A Choquet Ordering and Unique Decompositions in Convex sets of Tight Measures

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A new criterion of asymptotic periodicity of Markov operators on ${L}^{1}$, established in [3], is extended to the class of Markov operators on signed measures.

It is proved (independently of the result of Holmes [Fund. Math. 140 (1992)]) that the dual space of the uniform closure $CFL{(}_{r})$ of the linear span of the maps x ↦ d(x,a) - d(x,b), where d is the metric of the Urysohn space ${}_{r}$ of diameter r, is (isometrically if r = +∞) isomorphic to the space $LIP{(}_{r})$ of equivalence classes of all real-valued Lipschitz maps on ${}_{r}$. The space of all signed (real-valued) Borel measures on ${}_{r}$ is isometrically embedded in the dual space of $CFL{(}_{r})$ and it is shown that the image of the embedding...

We study properties of the space ℳ of Borel vector measures on a compact metric space X, taking values in a Banach space E. The space ℳ is equipped with the Fortet-Mourier norm ${\left|\right|\xb7\left|\right|}_{\mathcal{F}}$ and the semivariation norm ||·||(X). The integral introduced by K. Baron and A. Lasota plays the most important role in the paper. Investigating its properties one can prove that in most cases the space $(\mathcal{M},|\left|\xb7\right|{|}_{\mathcal{F}})*$ is contained in but not equal to the space (ℳ,||·||(X))*. We obtain a representation of the continuous functionals on...

Given a complete and separable metric space $X$, we study the weak convergence of sequences of measures defined on the space $\mathcal{S}(X)$ of all real-valued lower semicontinuous functions on $X$ as well as on the space $\mathcal{F}(X)$ of all closed subsets of $X$.

Assuming the continuum hypothesis, we show that (i) there is a compact convex subset L of $\Sigma \left({\mathbb{R}}^{\omega \u2081}\right)$, and a probability Radon measure on L which has no separable support; (ii) there is a Corson compact space K, and a convex weak*-compact set M of Radon probability measures on K which has no ${G}_{\delta}$-points.

Let (Ω,Σ,μ) be a complete finite measure space and X a Banach space. We show that the space of all weakly μ-measurable (classes of scalarly equivalent) X-valued Pettis integrable functions with integrals of finite variation, equipped with the variation norm, contains a copy of ${\ell}_{\infty}$ if and only if X does.