OUTER MEASURES AND WEAK REGULARITY OF MEASURES

This paper investigates smoothness properties of probability measures on lattices which imply regularity, and then considers weaker versions of regularity; in particular, weakly regular, vaguely regular, and slightly regular. They are derived from commonly used outer measures, and we analyze them mainly for the case of I(ℒ) or for those elements of I(ℒ) with added smoothness conditions.


DEFINITIONS AND NOTATIONS
Let X 1)(, an al),,tt,(-t ,,(,t.L(,t 1)e a lattice of slbsets of X.We assume thr(mgh()ut that 0 an(l X ar(' in .If .4C X, then we will denote the complement of A by .4(i.e. A' X A).If is a latti('(' ()f t)sets ()f X. then '={L']L } is the lattice ()f .Lattice Terminology (2.1) DEFINITION: Let /2 be a lattice of subsets of X. V:e say that: 1-is a 5-1atticc if it is closed under countable intersections; 5() is the lattice of countable intersections of sets of L.
2-is complement generated if L implies L L', where L,, /. 3-is countably paracompact if, for every sequence {L,} in such that L, J. 0, there exists a sequence {,,} in such that L,, C ,; and ', 0. 4is disjunctive if and only if x X, L , and x L imply there exists A, B such that x A, LCB, andAOB=0.
5-is compact if and only if X U Lc,', L , implies there exists a finite number of L', that cover X. 6-is countably compact if and only if X U L',, L, implies there exists a finite number of the L', that cover X. 7-/2 is normal if and only if A, B and A B 0 imply there exists (7, D such that ACC',BCD',andC'OD' =0.

MEASURE TERMINOLOGY
Let /; be a lattice of subsets of X. M(.) will denote the set of finite-valued, bounded, finitely additive measures on A().We may clearly assume throughout that all measures are non-negative.
2i measure tt M()is said to be (,-smooth on A()if A,, E A() and A, , qt imply #(A,,) 0. 3-A measure # M() is said to be -regular if, for any A A(), #(A) sup{#(L): L C A,L e }.
(2.3) NOTATIONS: If is a lattice of subsets of X, then we will denote by: 2I(/2) the set of a-smooth measures on/2 of M(L;) M'(L;) the set of a-smooth measures on A(.) of M() MR(f.) the set of/;-regular measures of M() M(/) the set of/;-regular measures of M() (2.4) DEFINITIONS: 1-If A A(), then #,(A)={1 if x A, and 0 if x A} is the measure concentrated at .rX.
(2.7) REMARKS: 1-I() is in one-to-one correspondence with the set of all primefilters.
2-L,() is in one-to-one correspondence with primefilters which have the countable intersection property.
3-IR() is in one-to-one correspondence with the set of allultrafilters.SEPARATION TERMINOLOGY (2.8) DEFINITIONS: Let : and be two lattices of subsets of X.

2-/2 separates
if A:, B .a nd A C B: q) imply there exists A1, B ( : such that A: C A, B2 C B1, and A B 0.
3-Let 1;1 C .:i s 1-countably bounded if, for any sequence {B,} of sets of 2 with B, $ 0, there exists a sequence {A,} of sets of 1 such that B, C A, and A, $0.

3.OUTER MEASURES
In this section ve consider p E M(), and associate with it certain "outer measures" #' and p".In general, they differ fl'om the customary induced "outer meazures" p an(l p*.\\'e seek to investigate the interplay of these outer measures on the lattice L; and, conversely, the effect of L; on them.We will consider mainly the case where p I(L:), and fir this reason we will usually restrict discussion of #" to the case where E I() since, otherwise, p" O.
(3.1) DEFINITIONS: Let # M() such that # > 0 and let E be a subset of X.
(3.2) DEFINITION: u is said to be a regular outer measure (or regular finitely- subadditive outer lneasure) if u is an outer measure (finitely subadditive) and if, for A, E C X, there exists E S, (where S, denotes the u-measurable sets) such that A C E and v(A) v(E).
PROOF: We will just prove 2 and 5.
, (L u(L)= .(X)-,(Lv(X)-up{u(L).L c L',L } 4. WEAKER NOTIONS OF REGULARITY Previously we have considered some properties related to p E MR() or E IR(Z).We now want to consider weaker notions of regularity, and see when they might coincide with regularity; and, in general, to investigate their properties and interplay with the underlying lattice.
(4.1) DEFINITIONS: Let L Z, where is a lattice of subsets of X. 1-A measure p M() is said to be weakly regular if t,(L') sup{'(L) L c L',L }.
Then J(Z) C Iv(Z) C Iw(Z) f I(Z).