MEASURES ON COALLOCATION AND NORMAL LATTICES

Let ℒ1 and ℒ2 be lattices of subsets of a nonempty set X. Suppose ℒ2 coallocates ℒ1 and ℒ1 is a subset of ℒ2. We show that any ℒ1-regular finitely additive measure on the algebra generated by ℒ1 can be uniquely extended to an ℒ2-regular measure on the algebra generated by ℒ2. The case when ℒ1 is not necessary contained in ℒ2, as well as the measure enlargement problem are considered. Furthermore, some discussions on normal lattices and separation of lattices are also given.


INTRODUCTION
Let X be an arbitrary set and land 2are lattices of subsets of X.
If ,c , and if coallocates ,, then any ,-regular finitely additive measure on the algebra generated by , can be uniquely extended to an -regular measure on the algebra generated by 2. This situation has been investigated by J. Camacho in [2].
We extend his results in several directions in this paper.We will consider the case where lis not necessary contained in (see Theorem 3.1) and show that under suitable conditions any peMR() (see below for definitions) gives rise to a v6 MR().We will also J.K. CHAN consider besides measure extension problems, measure enlargement problems (see e.g.Theorem 3.3) and will finally apply these results to the case of a single lattice , thereby extending results of M.
We begin by giving some standard lattice and measure theoretic background in Section 2.
Our notation and terminology is consistent with [1,4,6,7,9].In Section 3, we consider the general coallocation theorem and a variety of consequences of it.Section 4 is devoted to a more detailed discussion of normal lattices and to separation of lattices.This work extends to some extent that of G. Eid [3].

BACKGROUND AND TERMINOLOGY
In this section, we summarize some lattice and measure theoretic notions and notations.This is all fairly standard and as previously mentioned is consistent with standard references.

Definition 2.1
Let X be a nonempty set and e(X) is the power set of X.A lattice is a collection of subsets of X, which is closed under finite unions and finite intersections, and , X e .Let ' -= L' Le where L' denotes the complement of L.
' is a lattice if is.
DeHnHon 2.2 Let , I and 2be any lattices of subsets of X.
Usually, we simply refer # to as a measure on a lattice .to mean that # is a finitely additive measure defined on the algebra A(.).A f.a.measure # defined on the algebra A() is (i Or, equivalently, ,(A) inf (') 'DA, {.6 ).

MEASURES ON COALLOCATION LATTICES
In this section we extend some of the work of [$] and [2] on the unique extendability of a measure # e MR(,) to a measure e MR(2) where ,and are lattices of subsets of X. we note that it is not always necessary to assume that c nor that X belongs to the lattices in order for the main results of the =oallocatlon theorem to hold (see Theorem 3.1).We first define two functions which form an inner-outer measure pair.Definition
t Le S. To prove that L is '-measurable, we have to show, by (6), that VA62, let P,Q6 (.'.Q6) s.t.We conclude, from (6), that eve element of is -measurable.

( 4
and Lc A, then by monotonicity of . )Is] Let L,I and A, s.t.L,c A'