Abstract
Zero-one measure characterizations of lattice properties such as normality are extended to more general measures. For a given measure, we consider two associated outer measures and attempt to obtain the outer-measurable sets. We also seek necessary and sufficient conditions for the measure and outer measures to be equal on the lattice or its complement.
Highlights
AND NOTATION.We shall let/. denote a lattice of subsets of a set X and shall assume that the empty set and X are in/.. 4(/.) denotes the algebra generated by/
Zero-one measure characterizations of lattice properties such as normality are extended to more general measures
We seek necessary and sufficient conditions for the measure and outer measures to be equal on the lattice or its complement
Summary
M(L) denotes the set of all bounded and finitely additive measures defined on .A(L). Ms(L) will denote the set of all bounded and finitely additive measures which are a-smooth, and countably additive, on M(L). Is normal and complement generated /* Is(L implies/* 6, IRa(L) Suppose 1 c 2 where/‘1 separates 2" Let # MR(Z1) t, MR(Z2) and let extend the following are true: a) u is/.1-regular on/‘’2". The following theorem concerning supports is a generalization of a result in [5]. Ms(/.) provide a framework from which many of the remaining theorems of this section rely, with respect to results concerning It" and the It"-measurable sets. E e ff if and only if It’(E)= sup It(L) where
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