We prove in this paper a very
general measure extension theorem which has as corollaries many recent, significant
extension theorems in the literature. We apply these results to the question of when
there is a well behaved map from the σ-smooth lattice regular measures on one set to
the σ− smooth lattice regular measures on a second set. After developing these
general theorems we specialize consideration to two valued latitce regular measures
and obtain in a new and consistent manner many important mapping and
subspace theorems on the preservation of different types of repleteness including
results of Dykes, Hager, Isiwata, Moran, Varadarajan, Gillman, Jerrison and
others.
