Geodesic
Sigma order continuity and best approximation in $L_\varrho$-spaces
Commentationes Mathematicae Universitatis Carolinae, Tome 32 (1991) no. 2, pp. 241-250

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In this paper we give a characterization of $\sigma $-order continuity of modular function spaces $L_\varrho$ in terms of the existence of best approximants by elements of order closed sublattices of $L_\varrho\,$. We consider separately the case of Musielak--Orlicz spaces generated by non-$\sigma $-finite measures. Such spaces are not modular function spaces and the proofs require somewhat different methods.
Classification : 41A50, 41A65, 46E30
Keywords: best approximation; lattices; modular function spaces; $L_\varrho $-spaces; Orlicz spaces 
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Kilmer, Shelby J.; Kozƚowski, Wojciech M.; Lewicki, Grzegorz. Sigma order continuity and best approximation in $L_\varrho$-spaces. Commentationes Mathematicae Universitatis Carolinae, Tome 32 (1991) no. 2, pp. 241-250. http://geodesic.mathdoc.fr/item/CMUC_1991__32_2_a5/