OUTER MEASURE ANALYSIS OF TOPOLOGICAL LA’r’I’ICE PROPERTIES

ABSTRACT. Let X be a set and E a lattice of subsets ofX such that 0, X E. 4(E) is the algebra generated by 12, M(12) the set of nontrivial, finite, normegative, finitely additive measures on 4(12) and I() those elements of M(E) which just assume the values zero and one Various subsets of M(E) and I(E) are included which display smoothness and regularity properties. We consider several outer measures associated with dements of M(E) and relate their behavior to smoothness and regularity conditions as well as to various lattice topological properties In addition, their measurable sets are fully investigated. In the case of two lattices 121, E2 with 121 c 129., we present consequences of separation properties between the pair of lattices in terms of these outer measures, and further demonstrate the extension of smoothness conditions on 1 to 2 KEY WORDS AND PIIRASES: Lattice, measure, associated outer measure, regular outer measure, weakly regular, vaguely regular, almost countably compact, complement generated, countabl