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Proceedings of the American Mathematical Society

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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
DOI: https://doi.org/10.1090/S0002-9939-1986-0845977-8
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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Keywords: Nondecreasing rearrangement, Mackey topology
Article copyright: © Copyright 1986 American Mathematical Society