Borel Measure Research Articles

On Almost Shadowable Measures

In this paper we study the almost shadowable measures for homeomorphisms on compact metric spaces. First, we give examples of measures that are not shadowable. Next, we show that almost shadowable measures are weakly shadowable, namely, that there are Borelians with a measure close to $1$ such that every pseudo-orbit through it can be shadowed. Afterwards, the set of weakly shadowable measures is shown to be an $F_{\sigma\delta}$ subset of the space of Borel probability measures. Also, we show that the weakly shadowable measures can be weakly* approximated by shadowable ones. Furthermore, the closure of the set of shadowable points has full measure with respect to any weakly shadowable measure. We show that the notions of shadowableness, almost shadowableness and weak shadowableness coincide for finitely supported measures, or, for every measure when the set of shadowable points is closed. We investigate the stability of weakly shadowable expansive measures for homeomorphisms on compact metric spaces.

Free noncommutative hereditary kernels: Jordan decomposition, Arveson extension, kernel domination

We discuss (i) a quantized version of the Jordan decomposition theorem for a complex Borel measure on a compact Hausdorff space, namely, the more general problem of decomposing a general noncommutative kernel (a quantization of the standard notion of kernel function) as a linear combination of completely positive noncommutative kernels (a quantization of the standard notion of positive definite kernel). Other special cases of (i) include: the problem of decomposing a general operator-valued kernel function as a linear combination of positive kernels (not always possible), of decomposing a general bounded linear Hilbert-space operator as a linear combination of positive linear operators (always possible), of decomposing a completely bounded linear map from a C^* -algebra \mathcal{A} to an injective C^* -algebra \mathcal{L}(\mathcal{Y}) as a linear combination of completely positive maps from \mathcal{A} to \mathcal{L}(\mathcal{Y}) (always possible). We also discuss (ii) a noncommutative kernel generalization of the Arveson extension theorem (any completely positive map \phi from an operator system \mathbb{S} to an injective C^* -algebra \mathcal{L}(\mathcal{Y}) can be extended to a completely positive map \phi_e from a C^* -algebra containing \mathbb{S} to \mathcal{L}(\mathcal{Y})) , and (iii) a noncommutative kernel version of a Positivstellensatz (i.e., finding a certificate to explain why one kernel is positive at points where another given kernel is strictly positive).

Tail measures and regular variation

A general framework for the study of regular variation (RV) is that of Polish star-shaped metric spaces, while recent developments in [1] have discussed RV with respect to some properly localised boundedness $\mathcal{B}$ imposing weak assumptions on the structure of Polish space. Along the lines of the latter approach, we discuss the RV of Borel measures and random processes on general Polish metric spaces. Tail measures introduced in [2] appear naturally as limiting measures of regularly varying time series. We define tail measures on a measurable space indexed by $\mathcal{H}(D)$, a countable family of homogeneous coordinate maps, and show some tractable instances for the investigation of RV when $\mathcal{B}$ is determined by $\mathcal{H}(D)$. This allows us to study the regular variation of cadlag processes on $D(R^l, R^d)$ retrieving in particular results obtained in [1] for RV of stationary cadlag processes on the real line removing $l=1$ therein. Further, we discuss potential applications and open questions.

Simple halfspace depth

The halfspace depth is a prominent tool of nonparametric inference for multivariate data. We consider it in the general context of finite Borel measures μ on Rd. The halfspace depth of a point x∈Rd is defined as the infimum of the μ-masses of halfspaces that contain x. We say that a measure μ has a simple (halfspace) depth if the set of all attained halfspace depth values of μ on Rd is finite. We give a complete description of measures with simple depths by showing that the halfspace depth of μ is simple if and only if μ is atomic with finitely many atoms. This result completely resolves the halfspace depth characterization problem for the particular situation of simple halfspace depths and datasets. We also discuss the cardinality of the set of the attained halfspace depth values.

An abstract and generalized formulation of a theorem by Pelc and Prikry on invariant extension of Borel measure

There are certain countably generated ?-algebras of sets in the real line which do not admit any non-zero, ?-finite, diffused (or, continuous) measure. Such countably generated ?-algebras can be obtained by the use of some special types of infinite matrix known as the Banach-Kuratowski matrix and the same may be used in deriving a generalized version of Pelc and Prikry?s theoremas shown by Kharazishvili. Here, in this paper, we develop an abstract and generalized formulation of Pelc and Prikry?s theorem in spaces with transformation groups, where instead of using measure type functionals as done by Kharazishvili, we utilize a newly introduced concept which is that of an admissible, diffused k-additive algebra where k is an arbitrary infinite cardinal.

Multifractal dimensions for projections of measures

In this paper, we calculate the multifractal Hausdorff and packing dimensions of Borel probability measures and study their behaviors under orthogonal projections. In particular, we try through these results to improve the main result of M. Dai in \cite{D} about the multifractal analysis of a measure of multifractal exact dimension.

Dimensions of ‘self-affine sponges’ invariant under the action of multiplicative integers

AbstractLet $m_1 \geq m_2 \geq 2$ be integers. We consider subsets of the product symbolic sequence space $(\{0,\ldots ,m_1-1\} \times \{0,\ldots ,m_2-1\})^{\mathbb {N}^*}$ that are invariant under the action of the semigroup of multiplicative integers. These sets are defined following Kenyon, Peres, and Solomyak and using a fixed integer $q \geq 2$ . We compute the Hausdorff and Minkowski dimensions of the projection of these sets onto an affine grid of the unit square. The proof of our Hausdorff dimension formula proceeds via a variational principle over some class of Borel probability measures on the studied sets. This extends well-known results on self-affine Sierpiński carpets. However, the combinatoric arguments we use in our proofs are more elaborate than in the self-similar case and involve a new parameter, namely $j = \lfloor \log _q ( {\log (m_1)}/{\log (m_2)} ) \rfloor $ . We then generalize our results to the same subsets defined in dimension $d \geq 2$ . There, the situation is even more delicate and our formulas involve a collection of $2d-3$ parameters.

Embedding weighted Hardy spaces into Lebesgue spaces and generalized area operators

In this paper, we consider the weighted Hardy space induced by an weight ω. We characterize the positive Borel measures μ such that the identity operator is bounded when As an application, we obtain necessary and sufficient conditions for the boundedness of the generalized area operator from to

Integral Representation and Supports of Functionals on Lipschitz Spaces

AbstractWe analyze the relationship between Borel measures and continuous linear functionals on the space $\textrm{Lip}_0(M)$ of Lipschitz functions on a complete metric space $M$. In particular, we describe continuous functionals arising from measures and vice versa. In the case of weak$^\ast $ continuous functionals, that is, members of the Lipschitz-free space $\mathcal{F}(M)$, measures on $M$ are considered. For the general case, we show that the appropriate setting is rather the uniform (or Samuel) compactification of $M$ and that it is consistent with the treatment of $\mathcal{F}(M)$. This setting also allows us to give a definition of support for all elements of $\textrm{Lip}_0(M)^\ast $ with similar properties to those in $\mathcal{F}(M)$, and we show that it coincides with the support of the representing measure when such a measure exists. We deduce that the members of $\textrm{Lip}_0(M)^\ast $ that can be expressed as the difference of two positive functionals admit a Jordan-like decomposition into a positive and a negative part.

More Bisections by Hyperplane Arrangements

A union of an arrangement of affine hyperplanes $H$ in $R^d$ is the real algebraic variety associated to the principal ideal generated by the polynomial $p_{H}$ given as the product of the degree one polynomials which define the hyperplanes of the arrangement. A finite Borel measure on $R^d$ is bisected by the arrangement of affine hyperplanes $H$ if the measure on the "non-negative side" of the arrangement $\{x\in R^d : p_{H}(x)\ge 0\}$ is the same as the measure on the "non-positive" side $\{x\in R^d : p_{H}(x)\le 0\}$. In 2017 Barba, Pilz \& Schnider considered special cases of the following measure partition hypothesis: For a given collection of $j$ finite Borel measures on $R^d$ there exists a $k$-element affine hyperplane arrangement that bisects each of the measures into equal halves simultaneously. They showed that there are simultaneous bisections in the case when $d=k=2$ and $j=4$. They conjectured that every collection of $j$ measures on $R^d$ can be simultaneously bisected with a $k$-element affine hyperplane arrangement provided that $d\ge \lceil j/k \rceil$. The conjecture was confirmed in the case when $d\ge j/k=2^a$ by Hubard and Karasev in 2018. In this paper we give a different proof of the Hubard and Karasev result using the framework of Blagojevi\'c, Frick, Haase \& Ziegler (2016), based on the equivariant relative obstruction theory of tom Dieck, which was developed for handling the Gr\"unbaum--Hadwiger--Ramos hyperplane measure partition problem. Furthermore, this approach allowed us to prove even more, that for every collection of $2^a(2h+1)+\ell$ measures on $R^{2^a+\ell}$, where $1\leq \ell\leq 2^a-1$, there exists a $(2h+1)$-element affine hyperplane arrangement that bisects all of them simultaneously. Our result was extended to the case of spherical arrangements and reproved by alternative methods in a beautiful way by Crabb in 2020.

Isometries of probability measures with respect to the total variation distance

We characterize surjective isometries of the total variation distance between various subsets of Borel probability measures. In the case when isometries are w⁎-continuous or in the case when their domain consists of measures which are absolutely continuous with respect to a fixed measure, such isometries are push-forwards, i.e., compositum of a measure with a bijection. When the domain consists of all Borel probability measures, their classification is more involved.

Extension of Haar’s theorem

Haar’s theorem ensures a unique nontrivial regular Borel measure on a locally compact Hausdorff topological group, up to multiplication by a positive constant. In this article, we extend Haar’s theorem to the case of locally compact Hausdorff strongly topological gyrogroups. We simultaneously prove the existence and uniqueness of a Haar measure on a locally compact Hausdorff strongly topological gyrogroup, using the method of Steinlage. We then find a natural relationship between Haar measures on gyrogroups and on their related groups. As an application of this result, we study some properties of a convolution-like operation on the space of Haar integrable functions defined on a locally compact Hausdorff strongly topological gyrogroup

A Note on Variational Principle of Subsets for Nonautonomous Dynamical Systems

Abstract In this paper, we introduce measure-theoretic for Borel probability measures to characterize upper and lower Katok measure-theoretic entropies in discrete type and the measure-theoretic entropy for arbitrary Borel probability measure in nonautonomous case. Then we establish new variational principles for Bowen topological entropy for nonautonomous dynamical systems. JEL classification numbers: 37A35. Keywords: Nonautonomous, Measure-theoretical entropies, Variational principles.

The structure of continuum-wise expansive measures

In [13] it was introduced and studied the notion of continuum-wise expansive measure. Here we extend the results for expansive measures obtained in [8] to the cw-expansive measure context. More precisely, we show that this set is a Gδσ subset of the space of Borel probability measures of the underlying space and that the cw-expansive measures with invariant support are weak* dense on it. We also study the homeomorphisms of compact metric spaces f:X→X for which the cw-expansive measures are weak* dense in the set of Borel probability measures. We show that the set of heteroclinic points of f has empty interior and X has no isolated points.

Packing topological entropy for amenable group actions

AbstractPacking topological entropy is a dynamical analogy of the packing dimension, which can be viewed as a counterpart of Bowen topological entropy. In the present paper we give a systematic study of the packing topological entropy for a continuous G-action dynamical system $(X,G)$ , where X is a compact metric space and G is a countable infinite discrete amenable group. We first prove a variational principle for amenable packing topological entropy: for any Borel subset Z of X, the packing topological entropy of Z equals the supremum of upper local entropy over all Borel probability measures for which the subset Z has full measure. Then we obtain an entropy inequality concerning amenable packing entropy. Finally, we show that the packing topological entropy of the set of generic points for any invariant Borel probability measure $\mu $ coincides with the metric entropy if either $\mu $ is ergodic or the system satisfies a kind of specification property.

Toeplitz operators on Bergman spaces with exponential weights for 0 < p ≤ 1

Given 0<p<∞, the Bergman space Aφp consists of all holomorphic functions on D such that‖f‖p,φ=(∫D|f(z)e−φ(z)|pdA(z))1/p<∞, where φ belongs to a large class W0 which cover those defined by Borichev, Dhuez and Kellay (2007) [4]. For 0<p<1, Aφp is neither a self-adjoint nor Banach space. By new approaches, we study the characterizations on positive Borel measures μ on D for which the induced Toeplitz operators Tμ are bounded or compact from one Bergman space Aφp to another Aφq for p or q∈(0,1].

$\mu$-statistical convergence and the space of functions $\mu$-stat continuous on the segment

In this work, the concept of a point $\mu$-statistical density is defined. Basing on this notion, the concept of $\mu$-statistical limit, generated by some Borel measure $\mu\left(\cdot \right)$, is defined at a point. We also introduce the concept of $\mu$-statistical fundamentality at a point, and prove its equivalence to the concept of $\mu$-stat convergence. The classification of discontinuity points is transferred to this case. The appropriate space of $\mu$-stat continuous functions on the segment with sup-norm is defined. It is proved that this space is a Banach space and the relationship between this space and the spaces of continuous and Lebesgue summable functions is considered.

Density topologies for strictly positive Borel measures

The aim of this paper is to describe density topologies generated by Borel complete regular measures. For any such measure μ a generalization of the classical Lebesgue differentiation theorem is proved here. Using it, we define a topology Tμ which is some kind of abstract density topology and, simultaneously, a natural generalization of the classical density topology Td.We focus on separation axioms of topologies Tμ. We show that considered topologies are regular and not normal. In some cases we prove that Tμ gives a Tychonoff space. We also consider homeomorphisms between topologies Tμ. It is shown that for any atomless measure ν the topology Tν is homeomorphic to Td. However, there are measures μ generating topologies quite different than Td, for example separable and not connected. Some cases deliver us examples of spaces which are not Lindelöf but they are separable.

Converging outer approximations to global attractors using semidefinite programming

This paper develops a method for obtaining guaranteed outer approximations for global attractors of continuous and discrete time nonlinear dynamical systems. The method is based on a hierarchy of semidefinite programming problems of increasing size with guaranteed convergence to the global attractor. The approach taken follows an established line of reasoning, where we first characterize the global attractor via an infinite dimensional linear programming problem (LP) in the space of Borel measures. The dual to this LP is in the space of continuous functions and its feasible solutions provide guaranteed outer approximations to the global attractor. For systems with polynomial dynamics, a hierarchy of finite-dimensional sum-of-squares tightenings of the dual LP provides a sequence of outer approximations to the global attractor with guaranteed convergence in the sense of volume discrepancy tending to zero. The method is very simple to use and based purely on convex optimization. Numerical examples with the code available online demonstrate the method.

Trace estimates of Toeplitz operators on Bergman spaces and applications to composition operators

Let \Omega be a subdomain of \mathbb{C} and let \mu be a positive Borel measure on \Omega . In this paper, we study the asymptotic behavior of the eigenvalues of compact Toeplitz operators T_\mu acting on Bergman spaces on \Omega . Let (\lambda _n(T_\mu)) be the decreasing sequence of the eigenvalues of T_\mu , and let \rho be an increasing function such that \rho (n)/n^A is decreasing for some A&gt;0 . We give an explicit necessary and sufficient geometric condition on \mu in order to have \lambda _n(T_\mu)\asymp 1/\rho (n) . As applications, we consider composition operators C_\varphi , acting on some standard analytic spaces on the unit disc \mathbb{D} . First, we give a general criterion ensuring that the singular values of C_\varphi satisfy s_n(C_\varphi ) \asymp 1/\rho(n) . Next, we focus our attention on composition operators with univalent symbols, where we express our general criterion in terms of the harmonic measure of \varphi (\mathbb{D}) . We finally study the case where \partial \varphi (\mathbb{D}) meets the unit circle in one point and give several concrete examples. Our method is based on upper and lower estimates of the trace of h(T_\mu) , where h is a suitable concave or convex function.