© Electronic Publishing House SMOOTHNESS CONDITIONS ON MEASURES USING WALLMAN SPACES

Abstract. In this paper, X denotes an arbitrary nonempty set, a lattice of subsets of X with∅, X∈, A() is the algebra generated by andM() 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 µ ̄ on an associated Wallman space. Whenever µ ∈MR(), µ ̄ is countably additive. We consider the general problem of given µ ∈ M(), how do properties of µ ̄ imply smoothness properties of µ? For instance, what conditions on µ ̄ are necessary and suffi-cient 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