<abstract xmlns="http://www.w3.org/1999/xhtml"><p>If <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_amsil-2020-0022_eq_001.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msubsup><mrow><mrow><mrow><mo>(</mo><mrow><msub><mrow><mi>μ</mi></mrow><mi>n</mi></msub></mrow><mo>)</mo></mrow></mrow></mrow><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mo>∞</mo></msubsup></mrow></math><tex-math>\left( {{\mu _n}} \right)_{n = 1}^\infty</tex-math></alternatives></inline-formula> are positive measures on a measurable space (<italic>X,</italic> Σ) and <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_amsil-2020-0022_eq_002.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msubsup><mrow><mrow><mrow><mo>(</mo><mrow><msub><mrow><mi>v</mi></mrow><mi>n</mi></msub></mrow><mo>)</mo></mrow></mrow></mrow><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mo>∞</mo></msubsup></mrow></math><tex-math>\left( {{v_n}} \right)_{n = 1}^\infty</tex-math></alternatives></inline-formula> are elements of a Banach space 𝔼 such that <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_amsil-2020-0022_eq_003.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msubsup><mo>∑</mo><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mo>∞</mo></msubsup><mrow><mrow><mo>‖</mo><mrow><msub><mrow><mi>v</mi></mrow><mi>n</mi></msub></mrow><mo>‖</mo></mrow><msub><mrow><mi>μ</mi></mrow><mi>n</mi></msub><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow></mrow><mo>&lt;</mo><mo>∞</mo></mrow></math><tex-math>\sum\nolimits_{n = 1}^\infty {\left\| {{v_n}} \right\|{\mu _n}\left( X \right)} &lt; \infty</tex-math></alternatives></inline-formula>, then <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_amsil-2020-0022_eq_004.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>ω</mi><mrow><mo>(</mo><mi>S</mi><mo>)</mo></mrow><mo>=</mo><msubsup><mo>∑</mo><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mo>∞</mo></msubsup><mrow><msub><mrow><mi>v</mi></mrow><mi>n</mi></msub><msub><mrow><mi>μ</mi></mrow><mi>n</mi></msub><mrow><mo>(</mo><mi>S</mi><mo>)</mo></mrow></mrow></mrow></math><tex-math>\omega \left( S \right) = \sum\nolimits_{n = 1}^\infty {{v_n}{\mu _n}\left( S \right)}</tex-math></alternatives></inline-formula> defines a vector measure of bounded variation on (<italic>X,</italic> Σ). We show 𝔼 has the Radon-Nikodym property if and only if every 𝔼-valued measure of bounded variation on (<italic>X,</italic> Σ) is of this form. This characterization of the Radon-Nikodym property leads to a new proof of the Lewis-Stegall theorem.</p><p>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.</p></abstract>

If (μn)n=1∞\left( {{\mu _n}} \right)_{n = 1}^\infty are positive measures on a measurable space (X, Σ) and (vn)n=1∞\left( {{v_n}} \right)_{n = 1}^\infty are elements of a Banach space 𝔼 such that ∑n=1∞‖vn‖μn(X)<∞\sum\nolimits_{n = 1}^\infty {\left\| {{v_n}} \right\|{\mu _n}\left( X \right)} < \infty, then ω(S)=∑n=1∞vnμn(S)\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.

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

Annales Mathematicae Silesianae

This work is licensed under the Creative Commons Attribution 4.0 International License.

If (μn)n=1∞\left( {{\mu _n}} \right)_{n = 1}^\infty are positive measures on a measurable space (X, Σ) and (vn)n=1∞\left( {{v_n}} \right)_{n = 1}^\infty are...

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

Published Online: Oct 06, 2020

Page range: 77 - 89

Received: Apr 03, 2020

Accepted: Aug 29, 2020

DOI: https://doi.org/10.2478/amsil-2020-0022

Keywords
vector measures, Radon-Nikodym property, transfunctions

© 2020 Piotr Mikusiński et al., published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

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

Published Online: Oct 06, 2020

Page range: 77 - 89

Received: Apr 03, 2020

Accepted: Aug 29, 2020

DOI: https://doi.org/10.2478/amsil-2020-0022

Keywordsvector measures, Radon-Nikodym property, transfunctions

© 2020 Piotr Mikusiński et al., published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

Keywords
vector measures, Radon-Nikodym property, transfunctions