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.

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.

Completeness of the L1-space of closed vector measures

1990 ◽  
Vol 33 (1) ◽  
pp. 71-78 ◽  
Author(s):  
Werner J. Ricker
Keyword(s):  
Uniform Convergence ◽  
Convex Space ◽  
Vector Measure ◽  
Sufficient Conditions ◽  
Locally Convex Space ◽  
Vector Measures ◽  
Locally Convex

The notion of a closed vector measure m, due to I. Kluv´;nek, is by now well established. Its importance stems from the fact that if the locally convex space X in which m assumes its values is sequentially complete, then m is closed if and only if its L1-space is complete for the topology of uniform convergence of indefinite integrals. However, there are important examples of X-valued measures where X is not sequentially complete. Sufficient conditions guaranteeing the completeness of L1(m) for closed X-valued measures m are presented without the requirement that X be sequentially complete.

Sign in / Sign up

Export Citation Format

Share Document