On the Radon-Nikodym property for vector measures and extensions of transfunctions
Abstract
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 Σn=1∞‖vn‖μn(X)<∞, then ω(S) = Σn=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.
Keywords
vector measures; Radon-Nikodym property; transfunctions
References
2. J. Bentley and P. Mikusiński, Transfunctions and their connections to plans, Markov operators and optimal transport, arXiv preprint. Avaliable at arXiv:1810.08349.
3. Y. Benyamini and J. Lindenstrauss, Geometric Nonlinear Functional Analysis. Vol. 1, American Mathematical Society Colloquium Publications 48, American Mathematical Society, Providence, 2000.
4. J. Diestel and J.J. Uhl, Jr., Vector Measures, American Mathematical Society, Providence, 1977.
5. D.R. Lewis and C. Stegall, Banach spaces whose duals are isomorphic to l_1(Γ), J. Functional Analysis 12 (1973), 177–187.
6. J. Mikusiński, The Bochner Integral, Lehrbücher und Monographien aus dem Gebiete der exakten Wissenschaften, Mathematische Reihe, Band 55. Birkhäuser Verlag, Basel-Stuttgart, 1978.
7. P. Mikusiński, Transfunctions, arXiv preprint. Avaliable at arXiv:1507.03441.
8. P. Mikusiński, Integrals with values in Banach spaces and locally convex spaces, arXiv preprint. Avaliable at arXiv:1403.5209.
9. H.P. Rosenthal, The Banach spaces C(K) and L_p(μ), Bull. Amer. Math. Soc. 81 (1975), 763–781.
10. R.A. Ryan, Introduction to Tensor Products of Banach Spaces, Springer Monographs in Mathematics, Springer-Verlag London, Ltd., London, 2002.
Department of Mathematics, University of Central Florida, USA United States
https://orcid.org/0000-0002-1890-8039
Department of Mathematics, North Carolina A&T State University, USA United States
The Copyright Holders of the submitted text are the Author and the Journal. The Reader is granted the right to use the pdf documents under the provisions of the Creative Commons 4.0 International License: Attribution (CC BY). The user can copy and redistribute the material in any medium or format and remix, transform, and build upon the material for any purpose.
- License
This journal provides immediate open access to its content under the Creative Commons BY 4.0 license (http://creativecommons.org/licenses/by/4.0/). Authors who publish with this journal retain all copyrights and agree to the terms of the above-mentioned CC BY 4.0 license. - Author’s Warranties
The author warrants that the article is original, written by stated author/s, has not been published before, contains no unlawful statements, does not infringe the rights of others, is subject to copyright that is vested exclusively in the author and free of any third party rights, and that any necessary written permissions to quote from other sources have been obtained by the author/s. - User Rights
Under the Creative Commons Attribution license, the users are free to share (copy, distribute and transmit the contribution) and adapt (remix, transform, and build upon the material) the article for any purpose, provided they attribute the contribution in the manner specified by the author or licensor. - Co-Authorship
If the article was prepared jointly with other authors, the signatory of this form warrants that he/she has been authorized by all co-authors to sign this agreement on their behalf, and agrees to inform his/her co-authors of the terms of this agreement.
