<abstract xmlns="http://www.w3.org/1999/xhtml">Let (<italic>X,</italic> ℱ) be a measurable space with a nonatomic vector measure <italic>µ</italic> =(<italic>µ</italic>1, <italic>µ</italic>2). Denote by <italic>R</italic>(<italic>Y</italic>) the subrange <italic>R</italic>(<italic>Y</italic>)= <italic>{µ</italic>(<italic>Z</italic>): <italic>Z</italic> ∈ ℱ, <italic>Z</italic> ⊆ <italic>Y }</italic>. For a given <italic>p</italic> ∈ <italic>µ</italic>(ℱ) consider a family of measurable subsets ℱ<italic>p</italic> = <italic>{Z</italic> ∈ ℱ : <italic>µ</italic>(<italic>Z</italic>)= <italic>p}.</italic> Dai and Feinberg proved the existence of a maximal subset <italic>Z*</italic> ∈ <italic>Fp</italic> having the maximal subrange <italic>R</italic>(<italic>Z*</italic>) and also a minimal subset <italic>M*</italic> ∈ ℱ<italic>p</italic> with the minimal subrange <italic>R</italic>(<italic>M*</italic>). We present a method of obtaining the maximal and the minimal subsets. Hence, we get simple proofs of the results of Dai and Feinberg.</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.

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

Tatra Mountains Mathematical Publications

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

{"article-title":"How to Obtain Maximal and Minimal Subranges of Two-Dimensional Vector Measures"}

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)...

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

Published Online: Nov 15, 2019

Page range: 85 - 90

Received: Feb 05, 2018

DOI: https://doi.org/10.2478/tmmp-2019-0022

Keywords
Lyapunov convexity theorem, range of a vector measure, maximal subset, maximal subrange of a vector measure

© 2019 Jerzy Legut et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

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

Published Online: Nov 15, 2019

Page range: 85 - 90

Received: Feb 05, 2018

DOI: https://doi.org/10.2478/tmmp-2019-0022

KeywordsLyapunov convexity theorem, range of a vector measure, maximal subset, maximal subrange of a vector measure

© 2019 Jerzy Legut et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Keywords
Lyapunov convexity theorem, range of a vector measure, maximal subset, maximal subrange of a vector measure