Remote Access Transactions of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9947-1973-0318962-8
MathSciNet review: 0318962
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 47B99, 46E30

Retrieve articles in all journals with MSC: 47B99, 46E30


Additional Information

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