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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)


The Mackey continuity of the monotone rearrangement

Authors: Anthony Horsley and Andrzej J. Wrobel
Journal: Proc. Amer. Math. Soc. 97 (1986), 626-628
MSC: Primary 46E30
MathSciNet review: 845977
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Abstract: Let $ (A, \mathcal{A},\mu )$ be a probability space, and let mes denote the Lebesgue measure on the Borel $ \sigma $-algebra $ \mathcal{B}$ in $ [0,1]$. The nondecreasing-rearrangement operator from the space $ {L^\infty }(\mu ) = {L^\infty }(A, \mathcal{A}, \mu)$ of real-valued essentially bounded functions into $ {L^\infty } = {L^\infty }([0,1]$, $ \mathcal{B}$, mes) is shown to be uniformly continuous in the Mackey topologies $ \tau ({L^\infty }(\mu )$, $ {L^1}(\mu ))$ and $ \tau ({L^\infty },{L^1})$ on $ {L^\infty }(\mu )$ and $ {L^\infty }$, respectively.

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PII: S 0002-9939(1986)0845977-8
Keywords: Nondecreasing rearrangement, Mackey topology
Article copyright: © Copyright 1986 American Mathematical Society