Sigma order continuity and best approximation in LϱL_\varrho-spaces

summary: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