OUTER MEASURES AND ASSOCIATED LATTICE PROPERTIES

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. IEY WORDS AND PHRASES. Measure, lattice, outer-measure, measurable, normal, regular, almost countably compact, separates, semiseparates, support. 1991 AMS SUBJECT CLASSIFICATION CODES. 28C15, 28A12.

In general, if u is an arbitrary outer measure on the power set of x, it is very difficult to give a description of the u-measurable sets, or even to give nontrivial classes of sets which are u- measurable.In the case of #, a measure, and u it* the induced outer measure, then, of course, classes of sets which are u-measurable are well-known.This is also the case in metric spaces with u a metric outer measure.Here, we consider # E M() or Ma( and two associated "outer" measures u t," and u " and attempt to obtain the t-measurable sets.A full description can be given in case # e l(Z) or Ia( (see Section 4), and we attempt to extend some of these results to the more general situation.We also seek necessary and sufficient conditions for various of the ,, t,', t," to be equal on or ', the complementary lattice, under varying conditions on the measure and on .
Finally, in Section 5, we give lattice separating conditions between pairs of lattices 1 and/'2 in terms of/f, #" or some other "outer" measure.
We begin by giving a brief review of the basic lattice and measure theoretic terminology and notation which will be used throughout the paper.This terminology will be consistent with standard usage (see e.g.[1], [6], [7], [8], [9], [11]).

BACKGROUND 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/.. If/. is closed under countable intersections then /. is said to be a &lattice./. is said to be normal if whenever A, B /. such that A N B , there exist C, D 6 L such that A C C/, B C D" and C" o D'= $.L is regular if for each z 6, X and A 6, such that z A, there exist B,C 6, L with z 6, B',A C C" and B'NC" .L is complement generated if for all +0o L6-L, L= A L is complement generated if for all L6, L,L f') A',, Ao 6, L. L is countably |'-1 +Oo par&compact f whenever {A0} is a decreasing sequence of lattice sets in which A }, there exists a decreasing sequence of L" sets {B,'} such that A C B i" for all and f] B,"---$'.If L and L 2 are lattices of subsets of X and L C L 2, then 1 separates/ if whenever A,B there exist C, D L such that A c C, B C D, and C D O; 1 semisepar&tes/-2 if whenever A 6 L and B 6,/'2 such that A r B O, there exists (7 6, L such that B C C and A (7 0. If/-1 separates L 2 then L is normal if and only if L 2 is normal.
M(L) denotes the set of all bounded and finitely additive measures defined on .A(L).Without loss of generality, we assume that these measures are non-negative.A measure/* is a-smooth on L if L 6,/. and L $ implies a(Li) o.Ms(L) will denote the set of all bounded and finitely additive measures which are a-smooth, and hence countably additive, on M(L).If for all A 6, t(L), ,(A)= sup /*(L), where L C A, /,6-L, then /* is said to be /--regtdar.MR(L denotes the subset of M(L) consisting of all L-regular measures, and MRs(L) that subset of MR(L) consisting of a-smooth, L- regular measures, i.e., MRs(L)= MR(L)Ma(L ).Ma(L denotes those measures in M(L) which are a-smooth on L. I() denotes the subset of M(L) containing precisely the 0-1 nontrivial measures; similarly, lit(L), is(/-), iita(L) and la( denote those subsets of MR( MS( MRs(L) and Ma(L respectively, which are in I(L).We note that there is a one-to-one correspondence between prime filters on/-and measures in I(L), and between L-ultrafilters and measures in lit(L).Furthermore, a prime filter on L has the countable intersection property (i.e., the intersection of any countable number of prime filter set is nonempty) if and only if the corresponding measure is in la().If #6-M(L), S(#) denotes the support of /*, i.e., S(/*)= L such that L6,L and /,(L) /*(X).If /*,v 6. M(L) we will write /* < v(L), or /* < v on L, whenever /*(L) < v(L) for all L 6, L. One can show (c.f.[10]) that if , 6, M(/-) then there exists a v 6, MR(L) such that/* < v() and/*(X) v(x); if/-is normal and/* 6, I(L), then v 6, IR(L and u is unique.
PROOF.Let A , A" , A" # , let # MR('), and let 3g {B IA" c B for some i}.Now {A'i} can be enlarged to an/." ultrafilter.Therefore, there exists u IR(') such that t,(A'i)= for all i.Since is a.c.c., t, la(/').Thus has the countable intersection property.Suppose Li , and suppose #(Li)> e > 0 for all i.Since # MR('), there exist A'ic Li, A , and A" such that i(A'i)> el2 for all i.Now A" # for any and Li .Therefore : does not have the countable intersection property, a contradiction.THEOREM 3.4.Suppose 1 c 2 where/'1 separates 2" Let # MR(Z1) t, MR(Z2) and let extend .Then the following are true: a) u is/.1-regular on/''2" b) If t,1 MR(Z2) and Ul extends then t, Ul.

=1
The sets Is(/.and Ms(/.) provide a framework from which many of the remaining theorems of this section rely, particularly with respect to results concerning It" and the It"-measurable sets.

REMARK.
If It e M(/.) then E e " iff It'(A') > It'(A'n E) + It'(A'n E') for all A" LEMMA 4.1.Let It e M(/.) and let E C X. Then E e ff if and only if It'(E)= sup It(L) where Lc E and Le PROOF.Suppose It'(E) sup It(L).Let > 0 be given.Then there exists L e/. such that L C E and It'(E)-it(L)</2.Similarly, by definition of It', there exists fiDE such that Be and It(B')-it'(E)< /2.Therefore, It(B')-it(L)< and LC EC B'.Let A'e L'.Now It'(A'nE)<it(A'fB')= Conversely, suppose E e Then It'(if) It(X)-It'(E).Also, if > 0, there exists L e L such that L'D E" and It(L')-It'(E') < e.Therefore, It'(E)-it(L) < .
ii) Let >0 be given and let L t.There exist Lil LC U_IL and g'(L)+ > Ett(L'i).Since/. is 6, if A'= ,=lLi then A .Since /. is normal, there exist B,C /" such that L C B'C C A'.
The following theorem shows that set inclusion of #'-measurable sets is preserved under inequalities with respect to the lattice/-.THEOREM 4.9.Let 0,, M(/-),# < on/-and #(X) u(X).Then *Y c Y PROOF.E r implies #(E) sup #(L) < sup ,(L) < '(E), L C E, L /-.But since v < # on/-" and hence v'(E) < #'(E).We next note some extensions of some results which are known for zero-one measures that require the notion of a regular outer measure.We begin by defining this concept and list some consequences.
Let u be a finitely subadditive outer measure.Then , is regular if for every G C X, there exists E *_rt, such that G c E and u(G) u(E).
The following properties are noted for completeness: i) Let , be a regular outer measure.If Ell E C X, then t,(limEi)= lirn t,(Ei).
REMARK.Clearly, i) is not true if , is a finitely subadditive outer measure.For example, let g l(/-)-Ia(/-).Then there exist Li/-such that L t/I and P(Li)= for all i.Therefore, '(x) (x) but t'(L i) 0 for all i.We now show that the converse of Theorem 4.6 is valid when g" is regular.THEOREM 4.10.Let g Ma(/-).If t;= t" on/-" and if g" is regular then Ms(/-).
We end this section with some further consequences of regular outer measures which are stated without proof in the following theorem.THEOREM 4.11.Let t Ma(/-) and let " be regular.Then the following hold: a) ,c/" b) If g" on/-then Ma(/').