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

• Autorzy: Piotr Mikusiński , John Paul Ward
• Języki publikacji: EN

Abstrakty

EN

If (μn)n=1∞ are positive measures on a measurable space (X, Σ) and (vn)n=1∞ are elements of a Banach space 𝔼 such that ∑n=1∞‖vn‖μn(X)<∞, then ω(S)=∑n=1∞vnμn(S) 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. (original abstract)

Słowa kluczowe

EN

Mathematics; Mathematical functions

PL

Matematyka; Funkcje matematyczne

• Czasopismo: Annales Mathematicae Silesianae
• Rocznik: 2021
• Tom: 35 (1)
• Strony: 77--89

Twórcy

autor

Piotr Mikusiński

• University of Central Florida, USA

autor

John Paul Ward

• University of Central Florida, USA

Bibliografia

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• Mikusiński P., Transfunctions, arXiv preprint. Avaliable at arXiv:1507.03441.
• Mikusiński P., Integrals with values in Banach spaces and locally convex spaces, arXiv preprint. Avaliable at arXiv:1403.5209.
• Rosenthal H.P., The Banach spaces C (K) and Lp(µ), Bull. Amer. Math. Soc. 81 (1975), 763-781.
• Ryan R.A., Introduction to Tensor Products of Banach Spaces, Springer Monographs in Mathematics, Springer-Verlag London, Ltd., London, 2002.

Identyfikatory

DOI

10.2478/amsil-2020-0022