On continuous collections of measures

Robert M. Blumenthal; Harry H. Corson

Displaying similar documents to “On continuous collections of measures”

Mean values and associated measures of $δ$ -subharmonic functions

Neil A. Watson (2002)

Mathematica Bohemica

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Let $u$ be a $δ$ -subharmonic function with associated measure $μ$ , and let $v$ be a superharmonic function with associated measure $ν$ , on an open set $E$ . For any closed ball $B (x, r)$ , of centre $x$ and radius $r$ , contained in $E$ , let $ℳ (u, x, r)$ denote the mean value of $u$ over the surface of the ball. We prove that the upper and lower limits as $s, t \to 0$ with $0 < s < t$ of the quotient $(ℳ (u, x, s) - ℳ (u, x, t)) / (ℳ (v, x, s) - ℳ (v, x, t))$ , lie between the upper and lower limits as $r \to 0 +$ of the quotient $μ (B (x, r)) / ν (B (x, r))$ . This enables us to use some well-known measure-theoretic results to prove new variants...

On the existence of probability measures with given marginals

David Alan Edwards (1978)

Annales de l'institut Fourier

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Let $X$ be a compact ordered space and let $μ, ν$ be two probabilities on $X$ such that $μ (f) \leq ν (f)$ for every increasing continuous function $f : X \to R$ . Then we show that there exists a probability $θ$ on $X \times X$ such that (i) $θ (R) = 1$ , where $R$ is the graph of the order in $X$ , (ii) the projections of $θ$ onto $X$ are $μ$ and $ν$ . This theorem is generalized to the completely regular ordered spaces of Nachbin and also to certain infinite products. We show how these theorems are related to certain results...

On the complexity of sums of Dirichlet measures

Sylvain Kahane (1993)

Annales de l'institut Fourier

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Let $M$ be the set of all Dirichlet measures on the unit circle. We prove that $M + M$ is a non Borel analytic set for the weak* topology and that $M + M$ is not norm-closed. More precisely, we prove that there is no weak* Borel set which separates $M + M$ from $D^{⊥}$ (or even $L_{0}^{⊥})$ , the set of all measures singular with respect to every measure in $M$ . This extends results of Kaufman, Kechris and Lyons about $D^{⊥}$ and $H^{⊥}$ and gives many examples of non Borel analytic sets.

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 $\int f d P = {lim}_{n \to \infty} \int f d P_{n}$ ...

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...

Conical measures and vector measures

Igor Kluvánek (1977)

Annales de l'institut Fourier

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Every conical measure on a weak complete space $E$ is represented as integration with respect to a $σ$ -additive measure on the cylindrical $σ$ -algebra in $E$ . The connection between conical measures on $E$ and $E$ -valued measures gives then some sufficient conditions for the representing measure to be finite.

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.

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...