Abstract
The present paper is mainly concerned with establishing conditions which .assure that all lattice regular measures have additional smoothness properties or that simply all two-valued such measures have such properties and are therefore Dirac measures. These conditions are expressed in terms of the general Wallman space. The general results are then applied to specific topological lattices, yielding new conditions for measure compactness, Borel measure compactness, clopen measure repleteness, strong measure compactness, etc. In addition, smoothness properties in the general setting for lattice regular measures are related to the notion of support, and numerous applications are given.