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

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 I R ( L ) those elements of I ( L ) that are L -regular. It is known that I ( L ) = I R ( 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 I R ( σ , L ) those elements of I ( σ * , L ) that are L -regular. We then obtain necessary and sufficent conditions for I ( σ * , L ) = I R ( σ , L ) , and in particuliar ,we obtain conditions in terms of topologicial demands on associated Wallman spaces of the lattice.