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Transactions of the American Mathematical Society

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

 
 

 

Necessary and sufficient conditions for the derivation of integrals of $ L\sb{\psi }$-functions


Author: C. A. Hayes
Journal: Trans. Amer. Math. Soc. 223 (1976), 385-395
MSC: Primary 26A24; Secondary 28A15
DOI: https://doi.org/10.1090/S0002-9947-1976-0427554-4
MathSciNet review: 0427554
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Abstract: It has been shown recently that a necessary and sufficient condition for a derivation basis to derive the $ \mu $-integrals of all functions in $ {L^{(q)}}(\mu )$, where $ 1 < q < + \infty $, and $ \mu $ is a $ \sigma $-finite measure, is that the basis possess Vitali-like covering properties, with covering families having arbitrarily small $ {L^{(p)}}(\mu )$-overlap, where $ {p^{ - 1}} + {q^{ - 1}} = 1$. The corresponding theorem for the case $ p = 1,q = + \infty $ was established by R. de Possel in 1936.

The present paper extends these results to more general dual Orlicz spaces. Under suitable restrictions on the dual Orlicz functions $ \Phi $ and $ \Psi $, it is shown that a necessary and sufficient condition for a basis to derive the $ \mu $-integrals of all functions in $ {L_\Psi }(\mu )$ is that the basis possess Vitali-like covering families whose $ {L_\Phi }(\mu )$-overlap is arbitrarily small. Certain other conditions relating $ {L_\Phi }(\mu )$-strength and derivability are also discussed.


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DOI: https://doi.org/10.1090/S0002-9947-1976-0427554-4
Article copyright: © Copyright 1976 American Mathematical Society

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