Uniform measures are defined as the functionals on the space of bounded uniformly continuous functions that are continuous on bounded uniformly equicontinuous sets. If every cardinal has measure zero then every countably additive measure is a uniform measure. The functionals sequentially continuous on bounded uniformly equicontinuous sets are exactly uniform measures on the separable modification of the underlying uniform space.
We derive two types of representation results for increasing convex functionals in terms of countably additive measures. The first is a max-representation of functionals defined on spaces of real-valued continuous functions and the second a sup-representation of functionals defined on spaces of real-valued Borel measurable functions. Our assumptions consist of sequential semicontinuity conditions which are easy to verify in different applications.
In this paper, we study Markov chains (MC) on topological spaces within the framework of the operator approach. We extend the Markov operator from the space of countably additive measures to the space of finitely additive measures. Cesaro means for a Markov sequence of measures and their asymptotic behavior in the weak topology are considered. It is proved ergodic theorem that in order for the Cesaro means to converge weakly to some bounded regular finitely additive (or countably additive) measure it is necessary and sufficient that all invariant finitely additive measures are not separable fr…
General Markov chains in an arbitrary phase space are considered in the framework of the operator treatment. Markov operators continue from the space of countably additive measures to the space of finitely additive measures. Cycles of measures generated by the corresponding operator are constructed, and algebraic operations on them are introduced. One of the main results obtained is that any cycle of finitely additive measures can be uniquely decomposed into the coordinate-wise sum of a cycle of countably additive measures and a cycle of purely finitely additive measures.We have proved theorem…
We study some aspects of countably additive vector measures with values in $\ell_\infty$ and the Banach lattices of real-valued functions that are integrable with respect to such a vector measure. On the one hand, we prove that if $W \subseteq \ell_\infty^*$ is a total set not containing sets equivalent to the canonical basis of $\ell_1(\mathfrak{c})$, then there is a non-countably additive $\ell_\infty$-valued map $ν$ defined on a $σ$-algebra such that the composition $x^* \circ ν$ is countably additive for every $x^*\in W$. On the other hand, we show that a Banach lattice $E$ is separable wh…
In this note the result by A. Swift concerning the embeddability of countably branching bundle graphs into Banach spaces is extended from the context of reflexive spaces with an unconditional asymptotic structure to the context of dual spaces with a weak$^*$ unconditional asymptotic structure. We also investigate weak$^*$ asymptotic midpoint uniform convexity in dual spaces which turns out to be equivalent to its weak version in general and to the standard weak$^*$ asymptotic uniform convexity up to renorming in dual spaces with a weak$^*$ unconditional asymptotic structure.
We propose a family of extensions of the Kantorovich-Rubinstein norm from the space of zero-charge countably additive measures on a compact metric space to the space of all countably additive measures, and a family of extensions of the Lipschitz norm from the quotient space of Lipschitz functions on a compact metric space to the space of all Lipschitz functions. These families are parameterized by $p,q \in [1,\infty]$, and if $p,q$ are Hölder conjugates, then the dual of the resulting $p$-Kantorovich space is isometrically isomorphic to the resulting $q$-Lipschitz space.
In a countably normed space which is a linear space equipped with a countable number of pair-wise compatible norms, we prove the existence of a common nearest point (in all norms) from a point outside a nonempty subset if this subset is compact with respect to all norms. We also prove the uniqueness of that common nearest point if the completion of the space equipped with only one of its norms is uniformly convex.
In this article, the bi-Lipschitz embeddability of the sequence of countably branching diamond graphs $(D_k^ω)_{k\in\mathbb{N}}$ is investigated. In particular it is shown that for every $\varepsilon>0$ and $k\in\mathbb{N}$, $D_k^ω$ embeds bi-Lipschiztly with distortion at most $6(1+\varepsilon)$ into any reflexive Banach space with an unconditional asymptotic structure that does not admit an equivalent asymptotically uniformly convex norm. On the other hand it is shown that the sequence $(D_k^ω)_{k\in\mathbb{N}}$ does not admit an equi-bi-Lipschitz embedding into any Banach space that has an …
The Cesàro limit - the asymptotic average of a sequence of real numbers - is an operator of fundamental importance in probability, statistics and mathematical analysis. To better understand sequences with Cesàro limits, this paper considers the space $\mathcal{F}$ comprised of all binary sequences with a Cesàro limit, and the associated functional $ν: \mathcal{F} \rightarrow [0,1]$ mapping each such sequence to its Cesàro limit. The basic properties of $\mathcal{F}$ and $ν$ are enumerated, and chains (totally ordered sets) in $\mathcal{F}$ on which $ν$ is countably additive are studied in deta…
We consider general Markov chains with discrete time in an arbitrary measurable (phase) space and homogeneous in time. Markov chains are defined by the classical transition function which within the framework of the operator treatment generates a conjugate pair of linear Markov operators in the Banach space of measurable bounded functions and in the Banach space of bounded finite additive measures. It is proved that the well-known Doeblin condition $ (D) $ of ergodicity (quasi\-compactness) of the Markov chain is equivalent to the condition $ (*) $: all finitely additive invariant measures of …
We develop tools to produce equivalent norms with specific local geometry around infinitely many points in the sphere of a Banach space via an inductive procedure. We combine this process with smoothness results and techniques to solve two open problems posed in the recently published monograph [GMZ22] by A. J. Guirao, V. Montesinos and V. Zizler. Specifically, on the one hand we construct in every separable Banach space admitting a $C^k$-smooth norm an equivalent norm which is $C^k$-smooth but fails to be uniformly Gâteaux in any direction; and on the other hand we produce in $c_0(Γ)$ for any…
We introduce Kuelbs-Steadman-type spaces for real-valued functions, with respect to countably additive measures, taking values in Banach spaces. We investigate their main properties and embeddings in $L^p$-type spaces, considering both the norm associated to norm convergence of the involved integrals and that related to weak convergence of the integrals.
We consider symmetric non-negative definite bilinear forms on algebras of bounded real valued functions and investigate closability with respect to the supremum norm. In particular, any Dirichlet form gives rise to a sup-norm closable bilinear form. Under mild conditions a sup-norm closable bilinear form admits finitely additive energy measures. If, in addition, there exists a (countably additive) energy dominant measure, then a sup-norm closable bilinear form can be turned into a Dirichlet form admitting a carré du champ. Moreover, we can always transfer the bilinear form to an isometrically …
In this note, following \cite{Chitescuetal2014}, we show that the Monge-Kantorovich norm on the vector space of countably additive measures on a compact metric space has a primal representation analogous to the Hanin norm, meaning that similarly to the Hanin norm, the Monge-Kantorovich norm can be seen as an extension of the Kantorovich-Rubinstein norm from the vector subspace of zero-charge measures, implying a number of novel results, such as the equivalence of the Monge-Kantorovich and Hanin norms.
This paper concerns the dilations of Banach space operator-valued quantum measures. While the recently developed general dilation theory can lead to a projection (idempotent) valued dilation for any quantum measure over the projection lattice for a von Neumann algebra that dose not contain type $I_{2}$ direct summand, such a dilation does not necessarily guarantee the preservation of countable additivity of the quantum measure. So it remain an open question whether every countably additive $B(X)$-valued quantum measure can be dilated to a countably additive projection-valued measure.The main p…
Given a Banach space we consider the $σ$-ideal of all of its subsets which are covered by countably many hyperplanes and investigate its standard cardinal characteristics as the additivity, the covering number, the uniformity, the cofinality. We determine their values for separable Banach spaces, and approximate them for nonseparable Banach spaces. The remaining questions reduce to deciding if the following can be proved in ZFC for every nonseparable Banach space $X$: (1) $X$ can be covered by $ω_1$-many of its hyperplanes; (2) All subsets of $X$ of cardinalities less than ${\rm cf}([{\rm dens…
We provide two new characterizations of bounded orthogonally additive polynomials from a uniformly complete vector lattice into a convex bornological space using separately two polynomial identities of Kusraeva involving the root mean power and the geometric mean. Furthermore, it is shown that a polynomial on a vector lattice is orthogonally additive whenever it is orthogonally additive on the positive cone. These results improve recent characterizations of bounded orthogonally additive polynomials by G. Buskes and the author.
We consider a countably generated and uniformly closed algebra of bounded functions. We assume that there is a lower semicontinuous, with respect to the supremum norm, quadratic form and that normal contractions operate in a certain sense. Then we prove that a subspace of the effective domain of the quadratic form is naturally isomorphic to a core of a regular Dirichlet form on a locally compact separable metric space. We also show that any Dirichlet form on a countably generated measure space can be approximated by essentially discrete Dirichlet forms, i.e. energy forms on finite weighted gra…
The aim of this article is to prove a representation theorem for orthogonally additive polynomials in the spirit of the recent theorem on representation of orthogonally additive polynomials on Banach lattices but for the setting of Riesz spaces. To this purpose the notion of $p$--orthosymmetric multilinear form is introduced and it is shown to be equivalent to the orthogonally additive property of the corresponding polynomial. Then the space of positive orthogonally additive polynomials on an Archimedean Riesz space taking values on an uniformly complete Archimedean Riesz space is shown to be …
We study the space of orthogonally additive $n$-homogeneous polynomials on $C(K)$. There are two natural norms on this space. First, there is the usual supremum norm of uniform convergence on the closed unit ball. As every orthogonally additive $n$-homogeneous polynomial is regular with respect to the Banach lattice structure, there is also the regular norm. These norms are equivalent, but have significantly different geometric properties. We characterise the extreme points of the unit ball for both norms, with different results for even and odd degrees. As an application, we prove a Banach-St…
We study holomorphic maps between C$^*$-algebras $A$ and $B$. When $f:B_A (0,\varrho) \longrightarrow B$ is a holomorphic mapping whose Taylor series at zero is uniformly converging in some open unit ball $U=B_{A}(0,δ)$ and we assume that $f$ is orthogonality preserving on $A_{sa}\cap U$, orthogonally additive on $U$ and $f(U)$ contains an invertible element in $B$, then there exist a sequence $(h_n)$ in $B^{**}$ and Jordan $^*$-homomorphisms $Θ, \widetildeΘ : M(A) \to B^{**}$ such that $$ f(x) = \sum_{n=1}^\infty h_n \widetildeΘ (a^n)= \sum_{n=1}^\infty Θ (a^n) h_n,$$ uniformly in $a\in U$. W…
We introduce a uniform structure on any Hilbert $C^*$-module $\mathcal N$ and prove the following theorem: suppose, $F:{\mathcal M}\to {\mathcal N}$ is a bounded adjointable morphism of Hilbert $C^*$-modules over $\mathcal A$ and $\mathcal N$ is countably generated. Then $F$ belongs to the Banach space generated by operators $θ_{x,y}$, $θ_{x,y}(z):=x\langle y,z\rangle$, $x\in {\mathcal N}$, $y,z\in {\mathcal M}$ (i.e. $F$ is ${\mathcal A}$-compact, or "compact") if and only if $F$ maps the unit ball of ${\mathcal M}$ to a totally bounded set with respect to this uniform structure (i.e. $F$ is …
