Outer measures associated with lattice measures and their application (Q1902923)

Summary: Consider a set \(X\) and a lattice \({\mathfrak L}\) of subsets of \(X\) such that \(\emptyset, X \in {\mathfrak L}\). \(M ({\mathfrak L})\) denotes those bounded finitely additive measures on \({\mathfrak A} ({\mathfrak L})\) (the algebra generated by \({\mathfrak L})\) which are studied, and \(I({\mathfrak L})\) denotes those elements of \(M ({\mathfrak L})\) which are 0-1 valued. Associated with a \(\mu \in M ({\mathfrak L})\) or a \(\mu \in M_\sigma ({\mathfrak L})\) (the elements of \(M ({\mathfrak L})\) which are \(\sigma\)-smooth on \({\mathfrak L})\) are outer measures \(\mu'\) and \(\mu''\). In terms of these outer measures various regularity properties of \(\mu\) can be introduced, and the interplay between regularity, smoothness, and measurability is investigated for both the 0-1 valued case and the more general case. Certain results for the special case carry over readily to the more general case or with at most a regularity assumption on \(\mu'\) or \(\mu''\), while others do not. Also, in the special case of 0-1 valued measures more refined notions of regularity can be introduced which have no immediate analogues in the general case.