Regular Borel Probability Measure Research Articles

On Focal Borel Probability Measures

The novel concept of focality is introduced for Borel probability measures on compact Hausdorff topological spaces. We characterize focal Borel probability measures as those Borel probability measures that are strictly positive on every nonempty open subset. We also prove the existence of focal Borel probability measures on compact metric spaces. Lastly, we prove that the set of focal (regular) Borel probability measures is convex but not extremal in the set of all (regular) Borel probability measures.

Topological properties in tensor products of Banach spaces

Given two Banach spaces X and Y, we analyze when the projective tensor product X⊗ˆπY has Corson's property (C) or is weakly Lindelöf determined (WLD), subspace of a weakly compactly generated (WCG) space or subspace of a Hilbert generated space. For instance, we show that: (i) X⊗ˆπY is WLD if and only if both X and Y are WLD and all operators from X to Y⁎ and from Y to X⁎ have separable range; (ii) X⊗ˆπY is subspace of a WCG space if the same holds for both X and Y under the assumption that every operator from X to Y⁎ is compact; (iii) ℓp(Γ)⊗ˆπℓq(Γ) is subspace of a Hilbert generated space for any 1<p,q<∞ such that 1/p+1/q<1 and for any infinite set Γ. We also pay attention to the injective tensor product X⊗ˆεY. In this case, the stability of property (C) and the property of being WLD turn out to be closely related to the condition that all regular Borel probability measures on the dual ball have countable Maharam type. Along this way, we generalize a result of Plebanek and Sobota that if K is a compact space such that C(K×K) has property (C), then all regular Borel probability measures on K have countable Maharam type. This generalization provides a consistent negative answer to a question of Ruess and Werner about the preservation of the w⁎-angelicity of the dual unit ball under injective tensor products.

Rota's Fubini lectures: The first problem

In his 1998 Fubini Lectures, Rota discusses twelve problems in probability that “no one likes to bring up”. The first problem calls for a revision of the notion of a sample space, guided by the belief that mention of sample points in a probabilistic argument is bad form and that a “pointless” foundation of probability should be provided by algebras of random variables.In 1958 Chang introduced MV-algebras to prove the completeness theorem of Łukasiewicz logic Ł∞. The aim of this paper is to show that MV-algebras provide a solution of Rota's first problem.The adjunction between MV-algebras and unital commutative C*-algebras equips every MV-algebra A with a natural ring structure, as advocated by Nelson for algebras of random variables. The closed compact set S(A)⊆[0,1]A of finitely additive probability measures on A (the states of A) coincides with the set of [0,1]-valued functions on A whose finite restrictions are consistent in de Finetti's sense. MV-algebras and Ł∞ thus provide the framework for a generalization (known as ŁIPSAT) of Boole's probabilistic inference problem, and its modern reformulation known as probabilistic satisfiability, PSAT. We construct an affine homeomorphism γA of S(A) onto the weakly compact space of regular Borel probability measures on the maximal spectral space μ(A). The latter is the most general compact Hausdorff space. As a consequence, for every Kolmogorov probability space (Ω,FΩ,P), with FΩ the sigma-algebra of Borel sets of a compact Hausdorff space Ω, and P a regular probability measure on FΩ, there is an MV-algebra A and a state σ of A such that (Ω,FΩ,P)≅(μ(A),Fμ(A),γA(σ)).

The Liouville property and random walks on topological groups

We study harmonic functions and Poisson boundaries for Borel probability measures on general (i.e., not necessarily locally compact) topological groups, and we prove that a second-countable topological group is amenable if and only if it admits a fully supported, regular Borel probability measure with trivial Poisson boundary. This generalizes work of Kaimanovich--Vershik and Rosenblatt, confirms a general topological version of Furstenberg's conjecture, and entails a characterization of the amenability of isometry groups in terms of the Liouville property for induced actions. Moreover, our result has non-trivial consequences concerning Liouville actions of discrete groups on countable sets

Comparison and pure infiniteness of crossed products

Let α : G ↷ X \alpha : G\curvearrowright X be a continuous action of an infinite countable group on a compact Hausdorff space. We show that, under the hypothesis that the action α \alpha is topologically free and has no G G -invariant regular Borel probability measure on X X , dynamical comparison implies that the reduced crossed product of α \alpha is purely infinite and simple. This result, as an application, shows a dichotomy between stable finiteness and pure infiniteness for reduced crossed products arising from actions satisfying dynamical comparison. We also introduce the concepts of paradoxical comparison and the uniform tower property. Under the hypothesis that the action α \alpha is exact and essentially free, we show that paradoxical comparison together with the uniform tower property implies that the reduced crossed product of α \alpha is purely infinite. As applications, we provide new results on pure infiniteness of reduced crossed products in which the underlying spaces are not necessarily zero dimensional. Finally, we study the type semigroups of actions on the Cantor set in order to establish the equivalence of almost unperforation of the type semigroup and comparison. This sheds light on a question arising in a paper of Rørdam and Sierakowski.

States on EMV-algebras

We define a state as a [0, 1]-valued, finitely additive function attaining the value 1 on an EMV-algebra, which is an algebraic structure close to MV-algebras, where the top element is not assumed. The state space of an EMV-algebra is a convex space that is not necessarily compact, and in such a case, the Krein–Mil’man theorem cannot be used. Nevertheless, we show that the set of extremal states generates the state space. We show that states always exist and the extremal states are exactly state-morphisms. Nevertheless, the state space is a convex space that is not necessarily compact; a variant of the Krein–Mil’man theorem, saying states are generated by extremal states, is proved. We define a weaker form of states, pre-states and strong pre-states, and also Jordan signed measures which form a Dedekind complete $$\ell $$-group. Finally, we show that every state can be represented by a unique regular Borel probability measure, and a variant of the Horn–Tarski theorem is proved.

Towards a probability theory for product logic: States, integral representation and reasoning

The aim of this paper is to extend probability theory from the classical to the product t-norm fuzzy logic setting. More precisely, we axiomatize a generalized notion of finitely additive probability for product logic formulas, called state, and show that every state is the Lebesgue integral with respect to a unique regular Borel probability measure. Furthermore, the relation between states and measures is shown to be one–one. In addition, we study geometrical properties of the convex set of states and show that extremal states, i.e., the extremal points of the state space, are the same as the truth-value assignments of the logic. Finally, we axiomatize a two-tiered modal logic for probabilistic reasoning on product logic events and prove soundness and completeness with respect to probabilistic spaces, where the algebra is a free product algebra and the measure is a state in the above sense.

Ergodic invariant states and irreducible representations of crossed product $C^∗$-algebras

Given a C∗-dynamical system consisting of a unital C∗-algebra A and a discrete group Γ acting on A as automorphisms, we show that the space of (ergodic) Γ-invariant states on A is homeomorphic to a subspace of (pure) state space of A o Γ. It follows that classification of ergodic Γ-invariant regular Borel probability measures on a compact Hausdorff space X is equivalent to classification of irreducible representations of C(X) o Γ whose restriction to Γ contains the trivial representation of Γ. As an application, a reformulation of Furstenberg’s conjecture via representation theory of the semidirect product group Z[ 1 pq ] o Z 2 is obtained.

On Pietsch measures for summing operators and dominated polynomials

Let be the canonical map, where is a regular Borel probability measure on the closed unit ball of the dual of a Banach space endowed with the weak* topology. This paper has a twofold purpose: (i) to study when -summing linear operators/-dominated homogeneous polynomials on have a Pietsch measure for which the canonical map is injective; (ii) to show how these results can be used to correct the proofs of some results of the authors concerning Pietsch-type factorization of dominated polynomials.

States on Quantum Structures Versus Integrals

We show conditions when a state on a quantum structure E like an effect algebra, a pseudo effect algebra E satisfying some kind of the Riesz Decomposition Properties (RDP) or on an MV-algebra, a BL-algebra, a pseudo MV-algebra and a pseudo BL-algebra is an integral through a regular Borel probability measure defined on the Borel σ-algebra of a Choquet simplex K. In particular, if E satisfies the strongest type of (RDP), the representing Borel probability measure can be uniquely chosen to have its support in the set of the extreme points of K. The same is true for states on an MV-algebra and a BL-algebra and their noncommutative variants.

States on Pseudo Effect Algebras and Integrals

We show that every state on an interval pseudo effect algebra $E$ satisfying some kind of the Riesz Decomposition Properties (RDP) is an integral through a regular Borel probability measure defined on the Borel $\sigma$-algebra of a Choquet simplex $K$. In particular, if $E$ satisfies the strongest type of (RDP), the representing Borel probability measure can be uniquely chosen to have its support in the set of the extreme points of $K.$

Every state on interval effect algebra is integral

We show that every state on an interval effect algebra is an integral through some regular Borel probability measure defined on the Borel σ-algebra of a compact Hausdorff simplex. This is true for every effect algebra satisfying Riesz decomposition property or for every many valued (MV)-algebra. In addition, we show that each state on an effect subalgebra of an interval effect algebra E can be extended to a state on E. Our method represents also every state on the set of effect operators of a Hilbert space as an integral.

Distal actions and shifted convolution property

A locally compact group $G$ is said to have shifted convolution property (abbr. as SCP) if for every regular Borel probability measure $\mu$ on $G$, either $\sup_{x\in G} \mu ^n (Cx) \ra 0$ for all compact subsets $C$ of $G$, or there exist $x\in G$ and a compact subgroup $K$ normalised by $x$ such that $\mu^nx^{-n} \ra \omega_K$, the Haar measure on $K$. We first consider distality of factor actions of distal actions. It is shown that this holds in particular for factors under compact groups invariant under the action and for factors under the connected component of identity. We then characterize groups having SCP in terms of a readily verifiable condition on the conjugation action (point-wise distality). This has some interesting corollaries to distality of certain actions and Choquet Deny measures which actually motivated SCP and point-wise distal groups. We also relate distality of actions on groups to that of the extensions on the space of probability measures.

Compact Hankel Operators on Generalized Bergman Spaces of the Polydisc

Let \({\vartheta}\) be a measure on the polydisc \({\mathbb{D}^n}\) which is the product of n regular Borel probability measures so that \({\vartheta([r,1)^n\times\mathbb{T}^n) >0 }\) for all 0 < r < 1. The Bergman space \({A^2_{\vartheta}}\) consists of all holomorphic functions that are square integrable with respect to \({\vartheta}\). In one dimension, it is well known that if f is continuous on the closed disc \({\overline{\mathbb{D}}}\), then the Hankel operator H f is compact on \({A^2_\vartheta}\). In this paper we show that for n ≥ 2 and f a continuous function on \({{\overline{\mathbb{D}}}^n}\), H f is compact on \({A^2_\vartheta}\) if and only if there is a decomposition f = h + g, where h belongs to \({A^2_\vartheta}\) and \({\lim_{z\to\partial\mathbb{D}^n}g(z)=0}\).

On Toeplitz operators on Bergman spaces of the unit polydisk

We study Toeplitz operators on the Bergman space A 2 ϑ of the unit polydisk D n , where ϑ is a product of n rotation-invariant regular Borel probability measures. We show that if f is a bounded Borel function on D n such that F(ω) = lim f(z) exists for all ω ∈ ∂D n , then T f is compact if z∈D n and only if F = 0 a.e. with respect to a measure γ associated with ϑ on the boundary ∂D n . We also discuss the commuting problem: if g is a non-constant bounded holomorphic function on D n , then what conditions does a bounded function f need to satisfy so that T f commutes with Tg?

Continuous version of the Choquet integral representation theorem

Let $E$ be a locally convex topological Hausdorff space, $K$ a nonempty compact convex subset of $E$, $\mu$ a regular Borel probability measure on $E$ and $\gamma >0$. We say that the measure $\mu$ $\gamma $-represents a point $x\in K$ if $\sup_{\| f\|\le

Compactification of the Stationary Channel Space

Topologization of the set consisting of all channels is given. This result is applied to the proof that the entropy rates and the transmission rate are jointly upper semi-continuous on the product topological space constructed from the stationary regular Borel probability measures and the stationary channels. Finally, it is proved that there exists a combination of an input stationary measure and a stationary channel having finite memory which can make the transmission rate maximal.

On the average distance property and certain energy integrals

One of our main results is the following: LetX be a compact connected subset of the Euclidean spaceRn andr(X, d2) the rendezvous number ofX, whered2 denotes the Euclidean distance inRn. (The rendezvous numberr(X, d2) is the unique positive real number with the property that for each positive integern and for all (not necessarily distinct)x1,x2,...,xn inX, there exists somex inX such that\((1/n)\sum\nolimits_{i = 1}^n {d_2 (x_i ,x)} = r(X,d_2 )\).) Then there exists some regular Borel probability measure μ0 onX such that the value of ∫Xd2(x, y)dμ0 (y) is independent of the choicex inX, if and only ifr(X, d2) = supμ ∫X ∫Xd2(x, y)dμ(x)dμ(y), where the supremum is taken over all regular Borel probability measures μ onX.

Some remarks on countably determined measures and uniform distribution of sequences

Two topics are investigated: countably determined (regular Borel probability) measures on compact Hausdorff spaces, and uniform distribution of sequences regarding mainly this kind of measures. We prove several characterizations of countably determined measures, and apply the results in order to show the existence of a well distributed sequence in the support of a countably determined measure. We also generalize a result of Losert on the existence of uniformly distributed sequences in compact dyadic spaces.

On a measure-theoretic problem of Arveson

A probability measure ν \nu on a product space X × Y X\times Y is said to be bistochastic with respect to measures λ \lambda on X X and μ \mu on Y Y if the marginals π 1 ( ν ) \pi _1(\nu ) and π 2 ( μ ) \pi _2(\mu ) are exactly λ \lambda and μ \mu . A solution is presented to a problem of Arveson about sets which are of measure zero for all such ν \nu .