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On vector valued Orlicz spaces. (English) Zbl 0531.46022
In the paper [M. S. Skaff, Pac. J. Math. 28, 413-430 (1969; Zbl 0176.110)], two main results are stated as follows:
(1) The Orlicz class \(L_ M\) is a vector space iff M(t,x) satisfies a \(\Delta\)-condition;
(2) If mes T\(<\infty\), then modular convergence and norm convergence in Orlicz space \(L_ M\) are equivalent iff M(t,x) satisfies a \(\Delta\)- condition.
In this article, the author gives a counterexample to show the incorrectness of proofs of those two results, and shows that the results still hold even without the restriction mes T\(<\infty\).
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)