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On absolute continuity of a modular connected with strong summability. (English) Zbl 0832.46020
Summary: There are given sufficient conditions in order that the modular $${\mathcal A}_\Phi$$, where ${\mathcal A}_\Phi(f)= \sup_{w\in {\mathcal W}} \int^b_a a_w(x) {\mathcal J}_\Phi(x, f) dm(x),$ with ${\mathcal J}_\Phi(x, f)= \int_\Omega \Phi(x, |f(t)|) d\mu(t),$ be absolutely continuous and absolutely finite.

##### MSC:
 46E30 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 46A80 Modular spaces 41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces) 40F05 Absolute and strong summability 45P05 Integral operators
##### Keywords:
absolute continuity; modular space; summability method; modular