SPECIAL MEASURES AND REPLETENESS

Let X be an abstract set and ℒ a lattice of subsets of X . To each lattice-regular measure μ , 
we associate two induced measures μ ⌢ and μ ˜ on suitable lattices of the Wallman space I R ( ℒ ) and another 
measure μ ′ on the space I R σ ( ℒ ) . We will investigate the reflection of smoothness properties of μ onto μ ⌢ , μ ˜ 
and μ ′ ; and try to set some new criterion for repleteness and measure repleteness.

X-U L ', where L'o, E A (,) the algebra generated by o(,) the o-algebra generated by ,.
,) the Lattice of countable intersections of sets of ,.
:(,) the Lattice of arbitrary intersection of sets of p(,) the smallest class containing , and closed under countable unions and intersections.

Measure Terminology
Let, be a lattice of subsets ofX.M(L) will denote the set of finite valued bounded finitely additive measures on A(,).Clearly since any measure in M(,) can be written as a difference of two non-negative measures there is no loss of generality in assuming that the measures are non-negative, and we will assume so throughout this paper.

1-
A measure ia EM(L) is said to be o-smooth on, if forL,.EL andL.

4-
A measure t E M(,) is said to be L-regular if for any A E.,q(,) bt(A sup la(L LCA If , is a lattice of subsets of X, then we will denote by" MR(L,)the set of L-regular measures of Mo(') the set of o-smooth measures on , of M(,) M(L)= the set of o-smooth measures on M,(,) the set of regular measures of M(,) M,(,) the set of z-smooth measures on , of MR(L') M(,) the set of tight measures on , of MR(t.).

Clearly
DEFINITION 2.3.IfA EA(L) then lax is the measure concentrated atx EX.
{ifx EA lax(A)= if x qA I(,) is the subset of M(,) which consists of non-trivial zero-one measures which are finitely additive on (c).IR(,) the set of ,-regular measures of 1(12,) Io(,) the set of o-smooth measures on , of I(,) I"(L)= the set of o-smooth measures on (,) of I(,)  It(L)-the set of x-smooth measures on L of I(,) I,(L) the set of ,-regular measures of 1(,) I,(L) the set of ,-regular measures of I,(L) DEFINITION 2.4: If la M(L) then we define the support of IX to be: s(,)-c /L )-(X).
Consequently if Ix 1(,) DEFINITION 2.$.lf, is a Lattice of subsets ofX, we say that , is replete if for any Ix I(,) then S(Ix) ,, O. DEFINITION 2.6.Let , be a lattice of subsets of X.We say that , is measure replete if S(ta) ;, for all Ix M(, ), Ix # O.

Separation Terminology
Let , and -2 be two Lattices of subsets of X. DEFINITION 2.7., separates ,2 if for A2,B2 '2 and A,.f'IB,.O then there exists At,Bt C. Ll such thatA2 CAI,B2 C.BI andAt f"lB O. REMARK 2.1.We now list few known facts found in [1] which will enable us to characterize some previously defined properties in a measure theoretic fashion.

1.
, is disjunctive if and only if Ixx tE IR(L), lx ft.X.

3.
, is T2 if and only if S(Ix)= O or a singleton for any Ix l(,)., is compact if and only if S(IX) O for any IX I,(,).

THE INDUCED MEASURF
If , is a disjunctive lattice of subsets of an abstract set X then there is a Wailman space associated with it.
We will briefly review the fundamental properties of this Wallman space, and then associate with a regular lattice measure two measures and on certain algebras in the Wallman space (see [1]).We then investigate how properties of Ix reflect to those of and and conversely, then give a variety of applications of these results.Let X be an abstract set and , a disjunctive lattice of subsets of X such that O and X are in,.For anyA in.l(,), define W(A)-{IxIs(,):IX(A)-1}. IfA,B ,(,)then 1) Let W(,) {W(L ),L , }.Then W(,) is a compact lattice of I(,), and I(,) with xW(,) as the topology of closed sets is a compact Tt space (the Wallman space) associated with the pair X,,.It is a T2-space if and only if, is normal.For Ix .M(,) we define I on.,(W(,))by" W(A )) IX(A whereA (,).Then t M(W(,)), and Mn(W(,)) if and only if ta M,().
Finally, since "r,W(,) and W(,) are compact lattices, and W(,) separates xW(,), then has a unique extension to .MR('tW(,)).We note that by compactness and l[t are in M(W(,)) and M('tW(,)) respectively, where they are certainly x-smooth and of course o-smooth.can be extended to o(W(L)) where it is 6W(,)-regular; while can be extended to o('r,(W(,))), the Borei sets oflR(.r_,),and is :W(,)-regular on it.One is now concerned with how further properties of I,t reflect over to and respectively.The following are known to be true (see [1 ]) and we list them for the reader's convenience.
THEOREM 3.1.Let , be a separating and disjunctive lattice of subsets of X, and let la C_M(,).
IfL is also normal (or T) then (L)if and only ifX is O*-measurable and *(X)-(ls(L)).
We now give some further results related to the induced measures and .
THEOREM 3.2.t L be a separating and disjunctive lattice, and Ms(L) then is W(L) regular on (W())'.
) where L GL and i.e. that !" is regular on "W(f_,).On the other hand since W(L) is then F-W(L)and [ W(L)]-inf W(L))-inf(W(L)) where F C W(L), L .erefore fi" on W(L).
THEOREM 3.4.t L and : be o lattices of subse of X such that L C : and L separates L2 If v M(L) then v " on L'2 and v on '2 where v ]z,.
therefore, v" s '.Now on v" =s ".Suppose : : such that v(L:) < '(L:) then since v M(2),v(L2) infv('2),L C' and 2 then L and by separation t, t such that L Ll, C 't '2 and therefore v(L2) inf L) where Le C L inf v(') where L C ' < *(L2) Ve >0 t such that LeCL and (Lt)-e <v(L:) <(L) but since LCL then '(L:)p(L) < v(L) + e which is a contradiction to our assumption.erefore v " on and thus v .on ' This theorem is a generalization of the previous one in which we used the compactness of W() to have a regular restriction of the measure.Next consider the space I() and the induced measure '.

4)
Wo(A C Wo(B) if and only ifA C B 5) [wo()]w()] The proof is the same as for W() by simply using the properties of W() and the fact that Wo(A W(A )l() and Wo(B W(B )I().
RE 3.1.It is not difficult to show that Wo()] Wo()].Also, for each M() we define D' on [W'o()] as follows: IX'[W,,(A )] p.(A whereA tEA(L) IX' is defined and the map IX Ix' from M(L) to M(W,,(L)) is onto.In addition, it can readily be checked that, THEOREM 3.5.Let , be disjunctive then 1) IX IEM(L) if and only if IX' IEM[W,,(L)] 2) IX lE M(L) if and only if IX' lE M[Wo(L)] 3) IX IEMo(L) if and only if IX' IEM,,[Wo(L)] 4) IX lE M(,) if and only if ix' lE M"[Wo(,)] 5) IX lE M(L) if and only if It' lE MWo(,)] THEOREM 3.6.Let L be a separating and disjunctive lattice of subsets of X, and let Ix lE M(L).
If L is also normal or T then IX' IEM(W,,(L)) if and only if I(L) is *-measurable and i'(I()) (I.()).
We note some consequences.
COROLLARY 3.1.If L is a separating, disjunctive and replete lattice of subsets of X, then IX' lE [M(L)] implies la lE M(,).
PROOF.Since L is replete thenX-I,(L)then from the previous theorem we have lR(L)) (t'(l(L)) (t'(X) i.e.IX lE M(L) from theorem 3.1.
PROOF.Let IX lEI(L) then since Wo(L is replete IX' IE[Wo(L)] then by hypothesis IX lEI(L) therefore I(,) I(,) or is replete.If we combine the two corollaries we get the following: THEOREM 3.7.Let , be separating and disjunctive.Then L" is replete if and only if THEOREM 3.8.Let L be a separating, disjunctive, normal and replete lattice.Then IX' lE M[W,,(L)] if and only if IX lE M(L). PROOF. 1.
Let IX' lE M[Wo(,)] then since is replete then X I(,) and X is t'-measurable and then by theorem 3.1 we get that tx lE M(L). 2.

SPECIAL MEASURES AND REPLETENESS
In this section we define a purely finitely additive measure (p.f. a.), a purely o-additive measure (p.o. a.) and a purely z-additive measure (p.x. a.) and for each type we give a characterization theorem.Then we will define strong o-additive measures (s.o. a.) and (s.x. a.) measures and give for each a characterization theorem.Finally we will investigate relationships among these measures under repleteness.
LEMMA 4.1.Let L be a lattice of subsets of X and Ix -MR(,). 1.
Consider on o[W(L)]; we saw in earlier work that I] is 6(W()) regular on o[W()].Let H C In(L) such that '(H) a 0 then :lp countably additive on o[xW(L)] and x W(,) regular such that 0 p < and p'(H) p(ln(L)) a O.
LetH C In(L such that '(H) a 0 then :lp countably additive on xW(L) regular on oxW(L)] such that 0 p and p'(H) p(l,(L)) a DEFINITION 4.1. 1.
Let Ix :_Mn(L); we say that Ix is p. f. a. if for y Mo(L) and O y Ix on q(L) then y O.

2.
Let Ix MR(L); we say that Ix is p. o. a. if for y E Mo(), y z-smooth on L and 0 y Ix then y 0. THEOREM 4.1.Let L be a separating and disjunctive lattice and Ix (E Mn(L) then: 1.
If we further assume that L is 6 and o(L) p(L) then the converses are true.
PROOF.The proof will be given only for part (1) and is similar for the second one. 1.
Suppose Ix is purely finitely additive.If '(X) * 0 then from previous Lemma 4.1 there exists p _ MR[W(L)] M[W(L)] such that 0 p t and p*(X) p(ln(L)) a,' then p y and y M(,) so from the definition of purely finitely additive which is a contradiction because '/[ln(L)] a 0 and therefore I*(X) 0.
Let t _ MR(L), we say that is o.f. a. if for ), such that 0 "/ I,t on A(L) and y' C MWo(L)] then 2.
t M(L), we y that is s.o. a. if for y such that 0ysB on g(L) and ?' MWo(L)], ?' z-smooth on Wo(L) then y 0.
THEOREM 4.2.t L be a disjunctive lattice of subse ofX.t B C M(L) then: 1.
Suppose is strong o additive but '(I(L)) a 0 then from iemma (4.1) ]p countably additive on W(L)] and W(L) regular such that 0 s p s and p'(l(L)) l(L)) a from previous lemma 4.2 p where y' M(Wo(L)) then and since is s.o. a. then 0 which is a contradiction to the fact that I()) (I,()) , 0 and hence *(I(L)) O.
o. a. is p. o. a.

2.
Suppose is p. f. a. and y' M[Wo(L)];O y then y M(L) and 0 y y 0 by purely finitely additive.erefore is s.f. a.
PROPOSION 4.3.If L is replete then is s.o. a. if and only if is p. o. a.
PROOF.L replete = X Ij(L) I,() then L Wo(L) and so y C M(L) and :-smooth on y' CM(W,,(L))and x-smooth or Wo therefore the definitions are equivalent.
We saw in proposition 4.3 that if is replete then p. o. a. s. o. a.
We say that IX is p. x a. if for y Mo(L), 0 Ix, and yL-tight then y 0.

2.
We say that IX is s.:. a. if for y' C M[Wo(L)], 0 y Vt on .,q(L)and y' is Wo(L)-tight then y 0.
If we further assume that ., is 6 and o(/,) p() then the converses are true.
PROOF.We will prove only the second proposition and the proof of the first is similar.