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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Decreasing rearrangements and doubly stochastic operators

Author: Peter W. Day
Journal: Trans. Amer. Math. Soc. 178 (1973), 383-392
MSC: Primary 47B99; Secondary 46E30
MathSciNet review: 0318962
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Abstract: In this paper generalizations to measurable functions on a finite measure space $ (X,\Lambda ,\mu )$ of some characterizations of the Hardy-Littlewood-Pólya preorder relation $ \prec $ are considered. Let $ \rho $ be a saturated, Fatou function norm such that $ {L^\infty } \subset {L^\rho } \subset {L^1}$, and let $ {L^\rho }$ be universally rearrangement invariant. The following equivalence is shown to hold for all $ f \in {L^\rho }$ iff $ (X,\Lambda ,\mu )$ is nonatomic or discrete: $ g \prec f$ iff g is in the $ \rho $-closed convex hull of the set of all rearrangements of f. Finally, it is shown that $ g \prec f \in {L^1}$ iff g is the image of f by a doubly stochastic operator.

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Keywords: Doubly stochastic operator, decreasing rearrangement, measure preserving transformation, nonatomic measure, Riesz space, equimeasurability, finitely additive measure
Article copyright: © Copyright 1973 American Mathematical Society