Smoothness Conditions on Measures Using Wallman Spaces

In this paper, X denotes an arbitrary nonempty set, a lattice of subsets of X with ∅, X∈ , A( ) is the algebra generated by and M( ) is the set of nontrivial, finite, and finitely additive measures on A( ), and MR( ) is the set of elements of M( ) which are -regular. It is well known that any μ ∈M( ) induces a finitely additive measure μ̄ on an associated Wallman space. Whenever μ ∈MR( ), μ̄ is countably additive. We consider the general problem of given μ ∈ M( ), how do properties of μ̄ imply smoothness properties of μ? For instance, what conditions on μ̄ are necessary and sufficient for μ to be σ -smooth on , or strongly σ -smooth on , or countably additive? We consider in discussing these questions either of two associated Wallman spaces.