On regular and sigma-smooth two valued measures and lattice generated topologies

Abstract

Let X be an abstract set and L a lattice of subsets of X. I(L) denotes the non-trivial zero one valued finitely additive measures on A(L), the algebra generated by L, and IR(L) those elements of I(L) that are L-regular. It is known that I(L)=IR(L) if and only if L is an algebra. We first give several new proofs of this fact and a number of characterizations of this in topologicial terms.Next we consider, I(σ*,L) the elements of I(L) that are σ-smooth on L, and IR(σ,L) those elements of I(σ*,L) that are L-regular. We then obtain necessary and sufficent conditions for I(σ*,L)=IR(σ,L), and in particuliar ,we obtain conditions in terms of topologicial demands on associated Wallman spaces of the lattice.