ON MEASURE REPLETENESS AND SUPPORT FOR LATTICE REGULAR MEASURES

. The present paper is mainly concerned with establishing conditions which .assure that all lattice regular measures have additional smoothness properties or that simply all two-valued such measures have such properties and are therefore Dirac measures. These conditions are expressed in terms of the general Wallman space. The general results are then applied to specific topological lattices, yielding new conditions for measure compactness, Borel measure compactness, clopen measure replete-ness, strong measure compactness, etc. In addition, smoothness properties in the general setting for lattice regular measures are related to the notion of support, and numerous applications are given

of closed regular Borel measures in just T topological spaces expressible in terms of mX-X, where mX is the Wallman compactification of X and also to clopen regular Borel measures in o-dimensional T spaces expressible in terms of BoX-X, where BoX is the Banaschewski compactification of X.
In the first part of this paper we utilize the framework of the previously mentioned paper and obtain new results for lattice repleteness, measure repleteness and strongly measure repleteness. We then apply these results to specific topological lattices and obtain new conditions for measure compactness, Borel measure compactness, and clopen measure repleteness and similar facts for strongly measure compactness, strongly Borel measure compactness, and strongly clopen measure repleteness. (See, in particular, Theorems 2.4, 2.6 and their consequences and associated examples.) It is advantageous to be able to characterize various repleteness properties in terms of support of certain measures. We pursue this in general in the second part of the paper. We cite here just one of the more important results (see Theorem 3.3): If L is separating and disjunctive, then L is measure replete iff the support of every o-smooth, L-regular measure (which is not the zero measure) is nonempty. This result has many applications. Thus, in this part of the paper, we concentrate on various aspects of support of a measure.

TERMINOLOGY AND NOTATION.
I. Most of the terminology used in the present paper goes back to Wallman [10] and Alexandroff [l], [2]. Some of the more recent terminology appears inNoebeling [7] and Frolik [6], as well as in [5], [8]. For the reader's convenience, in this part we will collect some of the special terminology which is used throughout the paper.
Consider any set X and any lattice of subsets of X, L. The algebra of subsets of X generated by L is denoted by A{LI. The o-algebra of subsets of X generated by L is dnoted by o{LI. Next, consider any algebra of subsets of X, A. A measure on A is defined to be a function, g, from A to R, such that g is bounded and finitely additive. The set whose general element is a measure on A{L is denoted by M(L). For the general element of M(L), , the support of is defined to be n{e e L/Ig l(e) Igl(X)} and is denoted by S(). An element of M(L), g, is said to be L-regular iff for every element of A{LI, E for every positive number, e there exists an element of L L such that L E and Ig(E)-g(L)l < e. ment of M(6) which is 6-tight is denoted by M(t,6). The set whose general element is an element of (6), , such that (A([)) {0,i}, that is, the set of 0-I measures is denoted by I([).
6 is said to be replete iff whenever an element of 1 (6) belongs to IR(o,6), then S() # . 6 is said to be prime complete iff whenever an element of I(6), belongs to I(o*,6), then S() # . 6 is said to be measure replete iff MR(o,[) MR (r,6). [ is said to be strongly measure replete iff MR(o,6) MR(t,6).
Next, consider any topological space X and denote its collection of closed sets by its collection of open sets by 0, its collection of clopen sets by C, and its collection of zero sets by Z. In case X is T3, X is said to be realcompact iff Z is replete. X is said to be s-complete iff F is replete. X is said to be Ncompact iff C is replete. Moreover, X is said to be measure compact iff Z is measure replete. X is said to be Borel measure compact iff F is measure replete.
X is said to be clopen measure replete iff C is measure replete. NOTE. Since every element of M (6) is expressible as the difference of nonnegarive elements of M(6), without loss of generality, we shall work with nonnegative elements of M (6).
II. Among the principal tools utilized in the present work are three measures induced by the general element of M(6), denoted by (these measures are denoted by , , and ') and certain criteria for o-smoothness, r-smoothness, or tightness, which are expressed in terms of , , or '. (See [5].) For the reader's convenience, in this part we collect thedefinitions of0, , and ' and we summarize (in the form of a theorem) the principal facts pertaining to the criteria mentioned above.
Preliminaries. Consider any set X and any lattice of subsets of X, 6, such that 6 is separating and disjunctive. It is known that the topological space < IR (6), tW(6) > is compact and TI; it is T 2 iff 6 is normal. (See e.g., [4] and [9]). Consider the function which is such that the domain of # is X and for every element of X, x, (x) Then is a < t6, tW(6) > -homeomorphlsm.
x For this reason, (X) is topologically identifiable with X. Moreover, (X) is dense in IR (6). Consequently IR (6) is a compactificatlon of X. In case (X) is identified with X, X is said to be embedded in IR (6) Next, consider any element of M(L), and the function ' which is such that the domain of ' is A(Wo(L)) and for every element of A(Wo(L)), ( THEOREM 1.1. Consider any set X and any lattice of subsets of X, L, such that t is.(separating) and disjunctive. For every element of MR(L), u: MR(,L) iff *(X) (IR(L)). 3. ' MR(T,wo(L)) iff *(IR(o,L)) (IR(L)). 4. If L is also separating and normal, or T 2, then p MR(t,L) iff *(X) (IR(L)) and X is *-measurable.
We not for example, that the statement of part I, " e MR(o,L) iff *(X) (IR(L))" is equivalent to "p e MR(o,L) iff ,(IR(L) X) 0" or to " e MR(o,L) iff for every sequence in L, < L i >, if < gi is decreasing and W(Li) IR(L) -g then [W(Li)) 0". Similarly, equivalent statements are obtainable for the other parts.
(For more details refer to [I].) 3. NECESSARY AND SUFFICIENT CONDITIONS.
In this section we work with an arbitrary set X and an arbitrary lattice of subsets of X, L such that L is separating and disjunctive and we give necessary and sufficient conditions for L to be a) LindelSf, b) replete, c) measure replete, d) strongly measure replete. a) LindelSf property. (2). Consider any topological space X such that X is T and 0-dimensional and let L C. Then, by Corollary 2.3, X is N-compact iff whenever E BoX X, then there exists a zero-set of BoX KO, such that E K 0 BoX X. Observation. Note for every element of IR(L), IR(,[) iff ' IR(, W ILl). Next for the general element of IR(o,W ([)), ' note S(') n{W (L)/L and ' (W (L)) 1}. Consider any element of L, L such that ' (Wo(L)) I. Then, by the definition of ', (L) I. Consequently We will obtain a necessary and sufficient condition for W (L) to be measure replete.
Preliminaries: Consider the set whose general element is an element of MR([), such that ' MR(,W([)). This set is denoted by R([). (See [5], P. 1517.) According to [5], Theorem 3.2,part   (3). Consider any topological space X such that X is T and 0-dimensional and let C. Then, by Corollary 2.6, X is clopen measure replete iff X is Ncompact and R(C) MR(o,C). Examples. (I). Consider any topological space X such that X is T31/2 and let L Z. If for every closed subset of BX, K, K BX X implies there exists a Balre set of BX, B such that K B BX X, then, by Corollary 2.7, X is measure compact.
(2). Consider any topological space X such that X is T and let t If for every closed subset of X, K K X X implies there exists a Balre set of X, B such that K B X X, then, by Corollary 2.7, X s Borel measure compact.
Examples. (I). Consider any topological space X such that X is T31/2 and let [ Z. If X is a Baire set of 8X, then, by Theorem 2.6, X is strongly measure compact.
(2). Consider any topological space X such that X is T31/2 and normal or simply T 2 and let i F. If X is a Baire set of X, then, by Theorem 2.6, X is strongly Borel measure compact.
(3). Consider any topological space X such that X is T and O-dimensional and let L C. If X s a Bare set of BoX, then, by Theorem 2.6, X is strongly clopen measure replete.

REPLETENESS PROPERTIES.
It is advantageous to be able to characterize various repleteness properties in terms of support of certain measures. In this section we pursue this matter in general Consider any set X and any lattice of subsets of X, [ such that L is separating and disjunctive. ) Repleteness and support.
Preliminaries. Consider the set whose general element is an elemen of MR (1)   The purpose of the following example is to show that the condition "there exists an element of MR(o,/), such that S(9) # " is not sufficient for i to be measure replete.
Example. Assume L is not compact. Then IR(/) X # . e) Consider any element of IR(/) X, and any element of X, x. Then, consider B + x and denote it by v. Since E IR(/) and x IR(L) (because L is We will give a necessary and sufficient condition for measure repleteness in terms of support. MR(,L). Consequently L is measure replete. Remark. This theorem generalizes [8], Theorem 2.2, where it is assumed that L is Examples. (I). Consider any topological space X such that X is T31/2 and let [ Z. Then, by Theorem 3.3,(or by [8], Theorem 2.2), X is measure compact iff for every element of MR(o,7) {0} , S() # . (2). Consider any topological space X such that X is T and let [ F.
Then, by Theorem 3.3,(or by [8], Theorem  Example. Consider any topological space X such that X is regular and countably compact and let L F Then, by Observation  Example. Consider any topological space X such that X is T31/2 and let [ Z. Then, by Observation   This result generalizes the following well-known result: Consider any topological space X such that X is T31/2 and let [ 2. Then M(o,2) MR(o,Z); expressed otherwise: every Baire measure is regular.
Remark. Note the condition "[ is separating and disjunctive" was not needed in the proof.
Finally, we will consider the question of when the support of a measure is LindelSf. there exists a sequence of subsets of G, < G >, such that G uG and for every n, n n G is discrete. Consider any such G and any such < G >. Note to show there exists n n a subset of A, A* such that S() u{O e, A*} and A* is countable, it suffices to show that for every n, G s countable. To do this, proceed as follows: