Abstract: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.
Keywords: best approximation, lattices, modular function spaces, $L_\varrho $-spaces, Orlicz spaces
AMS Subject Classification: Primary 46E30, 41A50