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

HTML articles powered by AMS MathViewer

- by C. A. Hayes
- Trans. Amer. Math. Soc.
**223**(1976), 385-395 - DOI: https://doi.org/10.1090/S0002-9947-1976-0427554-4
- PDF | Request permission

## 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.## References

- Antonio Cordoba,
*On the Vitali covering properties of a differentiation basis*, Studia Math.**57**(1976), no. 1, 91–95. MR**419714**, DOI 10.4064/sm-57-1-91-95 - Miguel de Guzmán,
*On the derivation and covering properties of a differentiation basis*, Studia Math.**44**(1972), 359–364. MR**330373**, DOI 10.4064/sm-44-4-359-364 - C. A. Hayes Jr.,
*Derivation in Orlicz spaces*, J. Reine Angew. Math.**253**(1972), 162–174. MR**306431**, DOI 10.1515/crll.1972.253.162
—, - Adriaan Cornelis Zaanen,
*Linear analysis. Measure and integral, Banach and Hilbert space, linear integral equations*, Interscience Publishers Inc., New York; North-Holland Publishing Co., Amsterdam; P. Noordhoff N. V., Groningen, 1953. MR**0061752**

*Derivation of the integrals of*${L^{(q)}}$-

*functions*, Pacific J. Math. (to appear). C. Hayes and C. Y. Pauc,

*Derivation and martingales*, Ergebnisse Math. Grenzgebiete, Bd. 49, Springer-Verlag, Berlin, 1970. R. de Posses,

*Sur la derivation abstraite des fonctions d’ensemble*, J. Math. Pures Appl.

**15**(1936), 391-409.

## Bibliographic Information

- © Copyright 1976 American Mathematical Society
- 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