INDUCED MEASURES ON WALLMAN SPACES

LetX be an abstract set and .t; a lattice of subsets ofX. To each lattice-regular measure we associate two induced measures and on suitable lattices of the Wallman space Is(L) and another measure IX' on the space I,(L). We will investigate the reflection of smoothness properties of IX onto t,

sequence {B,, } of sets of" with B,, , O then there exists a sequence {A,, } of sets of Z:l such that B,, CA,, andA,, , O. IfZ:l CL2 and IX EM(,:,)then the restriction of IX on .,(f_.t)will be denoted by v-Ix REMARK 2.1.We now list a 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 E IR(,), Vx .X ." is regular if and only if for any IX, I(.,) such that IX on " we have S(IX) S()." is T2 if and only if S() O or a singleton for any IX tE I(,).., is compact if and only if S(la) , for any Ix IR(,).
THEOREM 3.6.Let L be a complement generated and normal lattice of subsets of X.If t is strongly o-smooth on L, then Ix C M,(Z;).
Next, we generalize a result of Gardner [8].
Next, we have a restriction theorem, which although generally known, we prove for the reader's convenience.
We will briefly review the fundamental properties of this Wallman space associated with a regular lattice measure ix, and then associate with a regular lattice measure Ix, two measures t and on certain algebras in the Wallman space (see [3]).We then investigate how properties of ix reflect to those of t and !, and conversely, and 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 A(L), defined to be W(A {Ix EIR(L):IX(A 1 }.IfA,B CA(L) then 1) W(A UB)= W(A U W(B ).

4) W(A C W(B)
if ana only ira C B.

5)
W(A W(B if and only ira B.

6) W[A()] ., W()].
Let W() { W(L ),L z; }.Then W() is a compact lattice of IR(), and IR() with :W() as the topology of closed sets is a compact T1 space (the Wallman space) associated with the pair X,.It is a T2-space if and only if is normal.
One is now concerned with how further properties of IX reflect over to and respectively.The following are known to be true (see [1]) and we list them for the reader's convenience.TItEOREM 4.1.Let be a separating and disjunctive lattice.Let Ix MR(Z;), then the following statements are equivalent.
(K) 0 for all K C I()-X and K THEOREM 4.3.Let be a separating and disjunctive lattice.If IX MR() then the following statements are equivalent: ff {L,} L , and W(L C l(L X then W(L) 0.
THEOM 4.4.ff is a separating and disjunctive lattice of subse of X then, M(L) if and only vanishes on every closed subset of Is(L), contained in Es(L) X. THEOM 4.5.t be a separating and disjunctive lattice of subse ofX and the o statements aw equivalent: . (z).  .x is t*-measureabe and t*(X)- We now establish some eurther properties pertaining to the inuce measures t and t.First we snow TEOM 4.7.t e a separating an isjunctive lattice, an M() then is W() regular on (w())'.
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.Also this theorem enables us to improve corollary 3.12 namely: IfX is coutably paracompact and normal then each measure Ix tE Ms(2;) extends to a measure v Ms(Y) which is 2;-regular on 0. THEOREM 4.10.Suppose/; is a separating and disjunctive lattice.Let x if and only if {x} t Wo(L'.).
Therefore .6 must be replete.
The proof is a simple combination of the two previous theorems.

n.1 n-1
This theorem is somewhat more general than the previous corollary because we ask less from the lattice .6,however we get a set B o[W(L)] rather than a zero set z Z(xW(L)).EPL 4.15.We are going to apply corollary (4.13) to special cases of lattices. 1.
t X be a space and L 3 then X is 3-replete if and only if Vp X ] Z a zero set of such that p Z C -X. t X be a To countably paracompact space and L-Y then X is a-real compact if and only if p-X]Z a zero set of such thatpZC-X.ere is the Wallman compactification ofX. 3.
t X be a Tt space and L ( is normal and countably paracompact and ln() I()) then X is Borel-replete if and only if VpI(B)-X=I(B)-XZ a zero set of I(B) such that eZ CI(B)-X.
Let (CI) be the following condition: W(L) C I(L)-X ere exists a countable sequence {L} such that W(L) W(L) I(L)-X.so c c W(L,)c Z c 4(z,)-X so/; is Lindel6f if and only if for each compact K Mn(/;) there exists a zero set Z Z;((W(/;)) such that K CZ CIn(,)-X.
Let X be a T space and/; Z; then/; is Lindel6f if and only if for each compact K C I,(/;) X there exists a zero set Z such that K cZ c -X,Z e z(CWCz)). 2.
Let X be a 0-dim T space and/; C then/; is Lindel6f if and only if for each K C 0X -X there exists a zero setZ such thatZ Z;['W(/;)] andK CZ CX-X. 3.
X is a T space and/; B then B is Lindel6f if and only if for each compact K C I($) X there exists Z ;[:W($)] such thatK CZ CI($)-X.
Finally we give some further applications to measure-replete lattices.

2.
IfX is T1; , s then Ms(S) M(S) and M,'(S) M,(S) if and only if *(f) f) 0 for every F C I(B) -XF closed in I(B).IfX is a T1 space and L -9 then M,(.T) M(br) if and only if '(F) 0 for all F C wX-X; F closed in wX.

5.
IfX is T and , --2; thenZ is measure-compact if for each F C []X-X andF is closed in [d(, there exists a Baire set B of such that F CB C X-X. S. THE SPACE I(): DEFINITION 5.1.Let be a disjunctive lattice of subsets ofX.
PROOF.The proof of this Theorem is known.Let (C2) be the following condition: For each t H l(Z;) there exists at most one v THEOREM 5.5.LetL be a separating and disjunctive lattice of subsets ofX.Then (I(L),r.Wo(L)) is T2 if and only if (C2) holds. PROOF.
Restrict :k to t(Mg(W(r')).The restriction is unique because W(r') separates xW(L) and since t'(l,(L)) (I(L)) then .t. .tprojects onto l,(r') and is denoted by v. It' M(Wo()) and has a unique extension to M](xWo(r')) and of course v is that extension.
TItEOIM 5.11.Suppose is a separating, disjunctive and normal lattice of subsets of X, then the following statements are equivalent: 1.
us F C W(L 'i) but since W(L is compact en F C W(L ') W(L ') where L L and -1 -1

OM 4 .Let
. t L and L be o lattices of subsem of X such that L C L= and L separates Lz Ifv M(L) then v " on L'= and v on L'= where v PROOF.t v M(L=) then since L separates L M(L).Since L C L= then o(L) C

3 .
IfX is a 0-dim T space L C then Ms(C) M() and M(C) M(C) if and only if t(F) [*(F) 0 for F C [0X-XF closed in[oX.