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

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.