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On Riesz product measures ; mutual absolute continuity and singularity

Shelby J. Kilmer;  Sadahiro Saeki — 1988

Annales de l'institut Fourier

We give some criteria for mutual absolute continuity and for singularity of Riesz product measures on locally compact abelian groups. The first section gives the definition of such a measure which is more general than the usual definition. The second section provides three sufficient conditions for one Riesz product measure to be absolutely continuous with respect to another. One of our results contains a theorem of Brown-Moran-Ritter as a special case. The final section deals with random Riesz...

Sigma order continuity and best approximation in L ϱ -spaces

Shelby J. Kilmer;  Wojciech M. Kozƚowski;  Grzegorz Lewicki — 1991

Commentationes Mathematicae Universitatis Carolinae

In this paper we give a characterization of σ -order continuity of modular function spaces L ϱ in terms of the existence of best approximants by elements of order closed sublattices of L ϱ . We consider separately the case of Musielak–Orlicz spaces generated by non- σ -finite measures. Such spaces are not modular function spaces and the proofs require somewhat different methods.

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