Convex Corson compacta and Radon measures

Grzegorz Plebanek

Displaying similar documents to “Convex Corson compacta and Radon measures”

Continuous linear functionals on the space of Borel vector measures

Pola Siwek (2008)

Annales Polonici Mathematici

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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 ${| | \cdot | |}_{ℱ}$ 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 $(ℳ, | | \cdot | |_{ℱ}) *$ is contained in but not equal to the space (ℳ,||·||(X))*. We obtain a representation of the continuous functionals...

On a decomposition of non-negative Radon measures

Bérenger Akon Kpata (2019)

Archivum Mathematicum

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We establish a decomposition of non-negative Radon measures on $ℝ^{d}$ which extends that obtained by Strichartz [6] in the setting of $α$ -dimensional measures. As consequences, we deduce some well-known properties concerning the density of non-negative Radon measures. Furthermore, some properties of non-negative Radon measures having their Riesz potential in a Lebesgue space are obtained.

Simple fractions and linear decomposition of some convolutions of measures

Jolanta K. Misiewicz, Roger Cooke (2001)

Discussiones Mathematicae Probability and Statistics

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Every characteristic function φ can be written in the following way: φ(ξ) = 1/(h(ξ) + 1), where h(ξ) = ⎧ 1/φ(ξ) - 1 if φ(ξ) ≠ 0 ⎨ ⎩ ∞ if φ(ξ) = 0 This simple remark implies that every characteristic function can be treated as a simple fraction of the function h(ξ). In the paper, we consider a class C(φ) of all characteristic functions of the form $φ_{a} (ξ) = [a / (h (ξ) + a)]$ , where φ(ξ) is a fixed characteristic function. Using the well known theorem on simple fraction decomposition of rational functions we obtain...

On the isotropic constant of marginals

Grigoris Paouris (2012)

Studia Mathematica

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We show that if μ₁, ..., μₘ are log-concave subgaussian or supergaussian probability measures in $ℝ^{n_{i}}$ , i ≤ m, then for every F in the Grassmannian $G_{N, n}$ , where N = n₁ + ⋯ + nₘ and n< N, the isotropic constant of the marginal of the product of these measures, $π_{F} (μ ₁ \otimes \dots \otimes μ ₘ)$ , is bounded. This extends known results on bounds of the isotropic constant to a larger class of measures.

Limit theorems for random fields

Nguyen van Thu

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CONTENTSIntroduction............................................................................................................................................................................ 51. Notation and preliminaries............................................................................................................................................ 52. Statement of the problem..................................................................................................................................................

A note on almost strong liftings

C. Ionescu-Tulcea, R. Maher (1971)

Annales de l'institut Fourier

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Let $X$ be a locally compact space. A lifting $ρ$ of $M_{R}^{\infty} (X, μ)$ where $μ$ is a positive measure on $X$ , is almost strong if for each bounded, continuous function $f$ , $ρ (f)$ and $f$ coincide locally almost everywhere. We prove here that the set of all measures $μ$ on $X$ such that there exists an almost strong lifting of $M_{R}^{\infty} (X, | μ |)$ is a band.

Finitely-additive, countably-additive and internal probability measures

Haosui Duanmu, William Weiss (2018)

Commentationes Mathematicae Universitatis Carolinae

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We discuss two ways to construct standard probability measures, called push-down measures, from internal probability measures. We show that the Wasserstein distance between an internal probability measure and its push-down measure is infinitesimal. As an application to standard probability theory, we show that every finitely-additive Borel probability measure $P$ on a separable metric space is a limit of a sequence of countably-additive Borel probability measures ${P_{n}}_{n \in ℕ}$ in the sense that $\int f d P = {lim}_{n \to \infty} \int f d P_{n}$ ...

Integral representation and relaxation for functionals defined on measures

Ennio De Giorgi, Luigi Ambrosio, Giuseppe Buttazzo (1987)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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Given a separable metric locally compact space $\Omega$ , and a positive finite non-atomic measure $\lambda$ on $\Omega$ , we study the integral representation on the space of measures with bounded variation $\Omega$ of the lower semicontinuous envelope of the functional $F(u)=\int_{\Omega}f(x,u)d\lambda\qquad u\in L^{1}(\Omega,\lambda,\mathbb{R}^{n})$ with respect to the weak convergence of measures.

A convolution property of some measures with self-similar fractal support

Denise Szecsei (2007)

Colloquium Mathematicae

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We define a class of measures having the following properties: (1) the measures are supported on self-similar fractal subsets of the unit cube $I^{M} = {[0, 1)}^{M}$ , with 0 and 1 identified as necessary; (2) the measures are singular with respect to normalized Lebesgue measure m on $I^{M}$ ; (3) the measures have the convolution property that $μ * L^{p} \subseteq L^{p + ε}$ for some ε = ε(p) > 0 and all p ∈ (1,∞). We will show that if (1/p,1/q) lies in the triangle with vertices (0,0), (1,1) and (1/2,1/3), then $μ * L^{p} \subseteq L^{q}$ for any measure μ in our...

Continuous version of the Choquet integral representation theorem

Piotr Puchała (2005)

Studia Mathematica

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Let E be a locally convex topological Hausdorff space, K a nonempty compact convex subset of E, μ a regular Borel probability measure on E and γ > 0. We say that the measure μ γ-represents a point x ∈ K if $s u p_{| | f | | \leq 1} | f (x) - \int_{K} f d μ | < γ$ for any f ∈ E*. In this paper a continuous version of the Choquet theorem is proved, namely, if P is a continuous multivalued mapping from a metric space T into the space of nonempty, bounded convex subsets of a Banach space X, then there exists a weak* continuous family $(μ_{t})$ of...

Self-affine measures that are $L^{p}$ -improving

Kathryn E. Hare (2015)

Colloquium Mathematicae

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A measure is called $L^{p}$ -improving if it acts by convolution as a bounded operator from $L^{q}$ to L² for some q < 2. Interesting examples include Riesz product measures, Cantor measures and certain measures on curves. We show that equicontractive, self-similar measures are $L^{p}$ -improving if and only if they satisfy a suitable linear independence property. Certain self-affine measures are also seen to be $L^{p}$ -improving.

On Ordinary and Standard Lebesgue Measures on $ℝ^{\infty}$

Gogi Pantsulaia (2009)

Bulletin of the Polish Academy of Sciences. Mathematics

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New concepts of Lebesgue measure on $ℝ^{\infty}$ are proposed and some of their realizations in the ZFC theory are given. Also, it is shown that Baker’s both measures [1], [2], Mankiewicz and Preiss-Tišer generators [6] and the measure of [4] are not α-standard Lebesgue measures on $ℝ^{\infty}$ for α = (1,1,...).

A unified Lorenz-type approach to divergence and dependence

Teresa Kowalczyk

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AbstractThe paper deals with function-valued and numerical measures of absolute and directed divergence of one probability measure from another. In case of absolute divergence, some new results are added to the known ones to form a unified structure. In case of directed divergence, new concepts are introduced and investigated. It is shown that the notions of absolute and directed divergences complement each other and provide a good insight into the extent and the type of discrepancy...

Integral representation and relaxation for Junctionals defined on measures

Ennio De Giorgi, Luigi Ambrosio, Giuseppe Buttazzo (1987)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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Given a separable metric locally compact space $\Omega$ , and a positive finite non-atomic measure $\lambda$ on $\Omega$ , we study the integral representation on the space of measures with bounded variation $\Omega$ of the lower semicontinuous envelope of the functional $F(u)=\int_{\Omega}f(x,y)d\lambda\qquad u\in L^{1}(\Omega,\lambda,\mathbb{R}^{n})$ with respect to the weak convergence of measures.

Lineability and spaceability on vector-measure spaces

Giuseppina Barbieri, Francisco J. García-Pacheco, Daniele Puglisi (2013)

Studia Mathematica

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It is proved that if X is infinite-dimensional, then there exists an infinite-dimensional space of X-valued measures which have infinite variation on sets of positive Lebesgue measure. In term of spaceability, it is also shown that $c a (ℬ, λ, X) ∖ M_{σ}$ , the measures with non-σ-finite variation, contains a closed subspace. Other considerations concern the space of vector measures whose range is neither closed nor convex. All of those results extend in some sense theorems of Muñoz Fernández et al. [Linear...

On inhomogeneous self-similar measures and their $L^{q}$ spectra

Przemysław Liszka (2013)

Annales Polonici Mathematici

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Let $S_{i} : ℝ^{d} \to ℝ^{d}$ for i = 1,..., N be contracting similarities, let $(p ₁, . . ., p_{N}, p)$ be a probability vector and let ν be a probability measure on $ℝ^{d}$ with compact support. It is well known that there exists a unique inhomogeneous self-similar probability measure μ on $ℝ^{d}$ such that $μ = \sum_{i = 1}^{N} p_{i} μ \circ S_{i}^{- 1} + p ν$ . We give satisfactory estimates for the lower and upper bounds of the $L^{q}$ spectra of inhomogeneous self-similar measures. The case in which there are a countable number of contracting similarities and probabilities is considered. In particular,...

Erratum to “Geometric rigidity of $\times m$ invariant measures” (J. Eur. Math. Soc. 14, 1539–1563 (2012))

Michael Hochman (2013)

Journal of the European Mathematical Society

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Weak compactness in the space of operator valued measures $M_{b} a (Σ, (X, Y))$ and its applications

N.U. Ahmed (2011)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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In this note we present necessary and sufficient conditions characterizing conditionally weakly compact sets in the space of (bounded linear) operator valued measures $M_{b a} (Σ, (X, Y))$ . This generalizes a recent result of the author characterizing conditionally weakly compact subsets of the space of nuclear operator valued measures $M_{b a} (Σ, ₁ (X, Y))$ . This result has interesting applications in optimization and control theory as illustrated by several examples.

Level by level equivalence and the number of normal measures over $P_{κ} (λ)$

Arthur W. Apter (2007)

Fundamenta Mathematicae

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We construct two models for the level by level equivalence between strong compactness and supercompactness in which if κ is λ supercompact and λ ≥ κ is regular, we are able to determine exactly the number of normal measures $P_{κ} (λ)$ carries. In the first of these models, $P_{κ} (λ)$ carries $2^{2^{{[λ]}^{< κ}}}$ many normal measures, the maximal number. In the second of these models, $P_{κ} (λ)$ carries $2^{2^{{[λ]}^{< κ}}}$ many normal measures, except if κ is a measurable cardinal which is not a limit of measurable cardinals. In this case, κ (and...

Denseness and Borel complexity of some sets of vector measures

Zbigniew Lipecki (2004)

Studia Mathematica

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Let ν be a positive measure on a σ-algebra Σ of subsets of some set and let X be a Banach space. Denote by ca(Σ,X) the Banach space of X-valued measures on Σ, equipped with the uniform norm, and by ca(Σ,ν,X) its closed subspace consisting of those measures which vanish at every ν-null set. We are concerned with the subsets $_{ν} (X)$ and $_{ν} (X)$ of ca(Σ,X) defined by the conditions |φ| = ν and |φ| ≥ ν, respectively, where |φ| stands for the variation of φ ∈ ca(Σ,X). We establish necessary and sufficient...

On the duality between $p$ -modulus and probability measures

Luigi Ambrosio, Simone Di Marino, Giuseppe Savaré (2015)

Journal of the European Mathematical Society

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Motivated by recent developments on calculus in metric measure spaces $(X, d, m)$ , we prove a general duality principle between Fuglede’s notion [15] of $p$ -modulus for families of finite Borel measures in $(X, d)$ and probability measures with barycenter in $L^{q} (X, m)$ , with $q$ dual exponent of $p \in (1, \infty)$ . We apply this general duality principle to study null sets for families of parametric and non-parametric curves in $X$ . In the final part of the paper we provide a new proof, independent of optimal transportation, of the...

$L^{p}$ -improving properties of measures of positive energy dimension

Kathryn E. Hare, Maria Roginskaya (2005)

Colloquium Mathematicae

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A measure is called $L^{p}$ -improving if it acts by convolution as a bounded operator from $L^{p}$ to $L^{q}$ for some q > p. Positive measures which are $L^{p}$ -improving are known to have positive Hausdorff dimension. We extend this result to complex $L^{p}$ -improving measures and show that even their energy dimension is positive. Measures of positive energy dimension are seen to be the Lipschitz measures and are characterized in terms of their improving behaviour on a subset of $L^{p}$ -functions.

Some singular measures on the circle which improve $L^{p}$ spaces

David L. Ritter (1987)

Colloquium Mathematicae

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Estimates of capacity of self-similar measures

Jozef Myjak, Tomasz Szarek (2002)

Annales Polonici Mathematici

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We give lower and upper estimates of the capacity of self-similar measures generated by iterated function systems $(S_{i}, p_{i}) : i = 1, . . ., N$ where $S_{i}$ are bi-lipschitzean transformations.

Exact $^{\infty}$ covering maps of the circle without (weak) limit measure

Roland Zweimüller (2002)

Colloquium Mathematicae

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We construct $^{\infty}$ maps T on the interval and on the circle which are Lebesgue exact preserving an absolutely continuous infinite measure μ ≪ λ, such that for any probability measure ν ≪ λ the sequence ${(n^{- 1} \sum_{k = 0}^{n - 1} ν \circ T^{- k})}_{n \geq 1}$ of arithmetical averages of image measures does not converge weakly.

A characterization of partition polynomials and good Bernoulli trial measures in many symbols

Andrew Yingst (2014)

Colloquium Mathematicae

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Consider an experiment with d+1 possible outcomes, d of which occur with probabilities $x ₁, . . ., x_{d}$ . If we consider a large number of independent occurrences of this experiment, the probability of any event in the resulting space is a polynomial in $x ₁, . . ., x_{d}$ . We characterize those polynomials which arise as the probability of such an event. We use this to characterize those x⃗ for which the measure resulting from an infinite sequence of such trials is good in the sense of Akin.

Wasserstein metric and subordination

Philippe Clément, Wolfgang Desch (2008)

Studia Mathematica

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Let $(X, d_{X})$ , $(Ω, d_{Ω})$ be complete separable metric spaces. Denote by (X) the space of probability measures on X, by $W_{p}$ the p-Wasserstein metric with some p ∈ [1,∞), and by $_{p} (X)$ the space of probability measures on X with finite Wasserstein distance from any point measure. Let $f : Ω \to_{p} (X)$ , $ω \mapsto f_{ω}$ , be a Borel map such that f is a contraction from $(Ω, d_{Ω})$ into $(_{p} (X), W_{p})$ . Let ν₁,ν₂ be probability measures on Ω with $W_{p} (ν ₁, ν ₂)$ finite. On X we consider the subordinated measures $μ_{i} = \int_{Ω} f_{ω} d ν_{i} (ω)$ . Then $W_{p} (μ ₁, μ ₂) \leq W_{p} (ν ₁, ν ₂)$ . As an application we show that the solution measures $ϱ_{α} (t)$ ...

Osgood type conditions for an m th-order differential equation

Stanisaw Szufla (1998)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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We present a new theorem on the differential inequality $u^{(m)} \leq w (u)$ . Next, we apply this result to obtain existence theorems for the equation $x^{(m)} = f (t, x)$ .

Path functionals over Wasserstein spaces

Alessio Brancolini, Giuseppe Buttazzo, Filippo Santambrogio (2006)

Journal of the European Mathematical Society

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Given a metric space $X$ we consider a general class of functionals which measure the cost of a path in $X$ joining two given points $x_{0}$ and $x_{1}$ , providing abstract existence results for optimal paths. The results are then applied to the case when $X$ is aWasserstein space of probabilities on a given set $Ω$ and the cost of a path depends on the value of classical functionals over measures. Conditions for linking arbitrary extremal measures $μ_{0}$ and $μ_{1}$ by means of finite cost paths are given. ...