scholarly journals On the Radon-Nikodym Property for Vector Measures and Extensions of Transfunctions

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Piotr Mikusiński ◽  
John Paul Ward
Keyword(s):  
Banach Space ◽  
Bounded Variation ◽  
Vector Measure ◽  
Measurable Space ◽  
Unique Extension ◽  
Natural Conditions ◽  
Vector Measures ◽  
Nikodym Property

AbstractIf \left( {{\mu _n}} \right)_{n = 1}^\infty are positive measures on a measurable space (X, Σ) and \left( {{v_n}} \right)_{n = 1}^\infty are elements of a Banach space 𝔼 such that \sum\nolimits_{n = 1}^\infty {\left\| {{v_n}} \right\|{\mu _n}\left( X \right)} < \infty, then \omega \left( S \right) = \sum\nolimits_{n = 1}^\infty {{v_n}{\mu _n}\left( S \right)} defines a vector measure of bounded variation on (X, Σ). We show 𝔼 has the Radon-Nikodym property if and only if every 𝔼-valued measure of bounded variation on (X, Σ) is of this form. This characterization of the Radon-Nikodym property leads to a new proof of the Lewis-Stegall theorem.We also use this result to show that under natural conditions an operator defined on positive measures has a unique extension to an operator defined on 𝔼-valued measures for any Banach space 𝔼 that has the Radon-Nikodym property.

Inductive Extension of a Vector Measure Under a Convergence Condition

1968 ◽  
Vol 20 ◽  
pp. 1246-1255 ◽  
Author(s):  
Geoffrey Fox
Keyword(s):  
Banach Space ◽  
Bounded Variation ◽  
Vector Measure ◽  
Convergence Condition ◽  
Set Function

Let μ be a vector measure (countably additive set function with values in a Banach space) on a field. If μ is of bounded variation, it extends to a vector measure on the generated σ-field (2; 5; 8). Arsene and Strătilă (1) have obtained a result, which when specialized somewhat in form and context, reads as follows: “A vector measure on a field, majorized in norm by a positive, finite, subadditive increasing set function defined on the generated σ-field, extends to a vector measure on the generated σ-field”.

scholarly journals Nuclear operators on spaces of continuous vector-valued functions

1991 ◽  
Vol 33 (2) ◽  
pp. 223-230 ◽  
Author(s):  
Paulette Saab ◽  
Brenda Smith
Keyword(s):  
Banach Space ◽  
Vector Measure ◽  
Type Theorem ◽  
Representing Measure ◽  
Bounded Linear ◽  
Continuous Vector ◽  
Nuclear Operators ◽  
Nikodym Property ◽  
Vector Valued Functions ◽  
Vector Valued

Let Ω: be a compact Hausdorff space, let E be a Banach space, and let C(Ω, E) stand for the Banach space of continuous E-valued functions on Ω under supnorm. It is well known [3, p. 182] that if F is a Banach space then any bounded linear operator T:C(Ω, E)→ F has a finitely additive vector measure G defined on the σ-field of Borel subsets of Ω with values in the space ℒ(E, F**) of bounded linear operators from E to the second dual F** of F. The measure G is said to represent T. The purpose of this note is to study the interplay between certain properties of the operator T and properties of the representing measure G. Precisely, one of our goals is to study when one can characterize nuclear operators in terms of their representing measures. This is of course motivated by a well-known theorem of L. Schwartz [5] (see also [3, p. 173]) concerning nuclear operators on spaces C(Ω) of continuous scalar-valued functions. The study of nuclear operators on spaces C(Ω, E) of continuous vector-valued functions was initiated in [1], where the author extended Schwartz's result in case E* has the Radon-Nikodym property. In this paper, we will show that the condition on E* to have the Radon-Nikodym property is necessary to have a Schwartz's type theorem. This leads to a new characterization of dual spaces E* with the Radon-Nikodym property. In [2], it was shown that if T:C(Ω, E)→ F is nuclear than its representing measure G takes its values in the space (E, F) of nuclear operators from E to F. One of the results of this paper is that if T:C(Ω, E)→ F is nuclear then its representing measure G is countably additive and of bounded variation as a vector measure taking its values in (E, F) equipped with the nuclear norm. Finally, we show by easy examples that the above mentioned conditions on the representing measure G do not characterize nuclear operators on C(Ω, E) spaces, and we also look at cases where nuclear operators are indeed characterized by the above two conditions. For all undefined notions and terminologies, we refer the reader to [3].

scholarly journals How to Obtain Maximal and Minimal Subranges of Two-Dimensional Vector Measures

2019 ◽  
Vol 74 (1) ◽  
pp. 85-90
Author(s):  
Jerzy Legut ◽  
Maciej Wilczyński
Keyword(s):  
Vector Measure ◽  
Measurable Space ◽  
Two Dimensional ◽  
Dimensional Vector ◽  
Vector Measures ◽  
Maximal Subset ◽  
Minimal Subset

Abstract Let (X, ℱ) be a measurable space with a nonatomic vector measure µ =(µ1, µ2). Denote by R(Y) the subrange R(Y)= {µ(Z): Z ∈ ℱ, Z ⊆ Y }. For a given p ∈ µ(ℱ) consider a family of measurable subsets ℱp = {Z ∈ ℱ : µ(Z)= p}. Dai and Feinberg proved the existence of a maximal subset Z* ∈ Fp having the maximal subrange R(Z*) and also a minimal subset M* ∈ ℱp with the minimal subrange R(M*). We present a method of obtaining the maximal and the minimal subsets. Hence, we get simple proofs of the results of Dai and Feinberg.

ON SOME ASPECTS OF VECTOR MEASURES

2022 ◽  
Vol 11 (1) ◽  
pp. 17-23
Author(s):  
S.O. Hazoume ◽  
Y. Mensah
Keyword(s):  
Banach Space ◽  
Bounded Variation ◽  
Vector Measures

This paper addresses some properties of vector measures (Banach space valued measures) as well as topological results on some spaces of vector measures of bounded variation.

scholarly journals On copies of the null sequence Banach space in some vector measure spaces

1999 ◽  
Vol 59 (3) ◽  
pp. 443-447
Author(s):  
J.C. Ferrando ◽  
J.M. Amigó
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Vector Measure ◽  
Null Sequence ◽  
Vector Measures ◽  
Measure Spaces ◽  
Additive Vector

In this note we extend a result of Drewnowski concerning copies of C0 in the Banach space of all countably additive vector measures and study some properties of complemented copies of C0 in several Banach spaces of vector measures.

Integral Representation by Boundary Vector Measures

1982 ◽  
Vol 25 (2) ◽  
pp. 164-168 ◽  
Author(s):  
Paulette Saab
Keyword(s):  
Banach Space ◽  
Integral Representation ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Linear Subspace ◽  
Vector Measure ◽  
Vector Measures ◽  
Boundary Vector

AbstractIn this paper we show that if X is a compact Hausdorff space, A is an arbitrary linear subspace of C(X, C), and if E is a Banach space, then each element L of (A ⊗ E)* can be represented by a boundary E*-valued vector measure of the same norm as L.

Extension of a Bounded Vector Measure with Values in a Reflexive Banach Space

1967 ◽  
Vol 10 (4) ◽  
pp. 525-529 ◽  
Author(s):  
Geoffrey Fox
Keyword(s):  
Banach Space ◽  
Bounded Variation ◽  
Reflexive Banach Space ◽  
Vector Measure ◽  
Extension Theorem ◽  
Set Function ◽  
General Conditions

A vector measure (countable additive set function with values in a Banach space) on a field may be extended to a vector measure on the generated σ- field, under certain hypotheses. For example, the extension is established for the bounded variation case [2, 5, 8], and there are more general conditions under which the extension exists [ 1 ]. The above results have as hypotheses fairly strong boundedness conditions on the n o rm of the measure to be extended. In this paper we prove an extension theorem of the same type with a restriction on the range, supposing further that the measure is merely bounded. In fact a vector measure on a σ- field is bounded (III. 4. 5 of [3]) but it is conceivable that a vector measure on a field could be unbounded.

scholarly journals Convolution structures for an Orlicz space with respect to vector measures on a compact group

2021 ◽  
Vol 64 (1) ◽  
pp. 87-98
Author(s):  
Manoj Kumar ◽  
N. Shravan Kumar
Keyword(s):  
Abelian Group ◽  
Compact Group ◽  
Orlicz Space ◽  
Vector Measure ◽  
Young Function ◽  
Natural Conditions ◽  
Vector Measures ◽  
Convolution Structure

The aim of this paper is to present some results about the space $L^{\varPhi }(\nu ),$ where $\nu$ is a vector measure on a compact (not necessarily abelian) group and $\varPhi$ is a Young function. We show that under natural conditions, the space $L^{\varPhi }(\nu )$ becomes an $L^{1}(G)$-module with respect to the usual convolution of functions. We also define one more convolution structure on $L^{\varPhi }(\nu ).$

scholarly journals Weak Precompactness in the Space of Vector-Valued Measures of Bounded Variation

2015 ◽  
Vol 2015 ◽  
pp. 1-6
Author(s):  
Ioana Ghenciu
Keyword(s):  
Banach Space ◽  
Total Variation ◽  
Bounded Variation ◽  
Measure Space ◽  
Total Variation Norm ◽  
Variation Norm ◽  
Additive Measures ◽  
Vector Valued

For a Banach spaceXand a measure space(Ω,Σ), letM(Ω,X)be the space of allX-valued countably additive measures on(Ω,Σ)of bounded variation, with the total variation norm. In this paper we give a characterization of weakly precompact subsets ofM(Ω,X).

Points of Weak*-Norm Continuity in the Unit Ball of the Space WC(K; X)*

1999 ◽  
Vol 42 (1) ◽  
pp. 118-124 ◽  
Author(s):  
T. S. S. R. K. Rao
Keyword(s):  
Banach Space ◽  
Unit Ball ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Vector Measures ◽  
Weak Norm ◽  
Norm Continuity ◽  
Nikodym Property ◽  
Isolated Points ◽  
Dense Set

AbstractFor a compact Hausdorff space with a dense set of isolated points, we give a complete description of points of weak*-norm continuity in the dual unit ball of the space of Banach space valued functions that are continuous when the range has the weak topology. As an application we give a complete description of points of weak-norm continuity of the unit ball of the space of vector measures when the underlying Banach space has the Radon-Nikodym property.

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