Smoothness conditions on measures using Wallman spaces (Q1970275)

The following question is considered, using Wallman spaces: Let \(\mu \) be a nonnegative finite valued, nontrivial, finitely additive and bounded measure on the algebra generated by a lattice \(\mathcal{L}\) of subsets of an arbitrary nonempty set \(X,\) with \(X \in \) \(\mathcal{L}\). How do properties of \(\overline{\mu }\) (\(\overline{\mu }(W(A))=\mu (A)\)) imply properties (\(\sigma \)-smoothness on \(\mathcal{L}\), strongly \(\sigma \)-smoothness on \(\mathcal{L}\), countably additivity) of \(\mu\) ?