In Analysis two modes of non-topological convergence are interesting: order
convergence and convergence almost everywhere. It is proved here that oder
convergence of sequences can be induced by a limit structure, even a finest
one, whenever it is considered in sigma-distributive lattices. Since
convergence almost everywhere can be regarded as order convergence in a
certain sigma-distributive lattice, this result can be applied to convergence
of sequences almost everywhere and thus generalizing a former result of U.
Höhle obtained in a more indirect way by using fuzzy topologies
