Abstract

Let be an arbitrary nonempty set and a lattice of subsets of such that , . () denotes the algebra generated by , and () denotes those nonnegative, finite, finitely additive measures on (). In addition, () denotes the subset of () which consists of the nontrivial zero-one valued measures. The paper gives detailed analysis of products of lattices, their associated Wallman spaces, and products of a variety of measures.

1. Introduction

It is well known that given two measurable spaces and measures on them, we can obtain the product measurable space and the product measure on that space. The purpose of this paper is to give detailed analysis of product lattices and their associated Wallman spaces and to investigate how certain lattice properties carry over to the product lattices. In addition, we proceed from a measure theoretic point of view. We note that some of the material presented here has been developed from a filter approach by Kost, but the measure approach lends to a generalization of measures and to an easier treatment of topological style lattice properties.

2. Background and Notations

In this section we introduce the notation and terminology that will be used throughout the paper. All is fairly standard, and we include it for the reader’s convenience.

Let be an arbitrary nonempty set and a lattice of subsets of such that , . A lattice is a partially ordered set any two elements () of which have both and .

denotes the algebra generated by ; is the algebra generated by ; is the lattice of all countable intersections of sets from ; is the lattice of arbitrary intersections of sets from ; is the smallest class closed under countable intersections and unions which contains .

2.1. Lattice Terminology

The lattice is called:-lattice if is closed under countable intersections; complement generated if implies , , (where prime denotes the complement); disjunctive if for and such that there exists with and ; separating (or ) if and implies there exists such that , ; if for and there exist such that , , and ; normal if for any with there exist with , , and ; compact if for any collection of sets of with , there exists a finite subcollection with empty intersection; countably compact if for any countable collection of sets of with , there exists a finite subcollection with empty intersection.

2.2. Measure Terminology

denotes those nonnegative, finite, finitely additive measures on .

A measure is called: -smooth on if for all sequences of sets of with , ; -smooth on if for all sequences of sets of with , , that is, countably additive.

-regular if for any ,

We denote by the set of -regular measures of ; the set of -smooth measures on , of ; the set of -smooth measures on of ; the set of -regular measures of .

In addition, , , , , are the subsets of the corresponding ’s which consist of the nontrivial zero-one valued measures.

Finally, let be abstract sets and a lattice of subsets of and a lattice of subsets of . Let and .

The product measure is defined by

2.3. Lattice-Measure Correspondence

The support of is .

In case then the support is .

With this notation and in light of the above correspondences, we now note:For any , there exists such that on (i.e., for all ). For any , there exists such that on . is compact if and only if for every . is countably compact if and only if . is normal if and only if for each , there exists a unique such that on . is regular if and only if whenever and on , then . is replete if and only if for any , . is prime-complete if and only if for any , . Finally, if is the measure concentrated at , then , for all if and only if is disjunctive.

For further results and related matters see [1–3].

2.4. The General Wallman Space and Wallman Topology

The Wallman topology in is obtained by taking all as a base for the closed sets in and then is called the general Wallman space associated with and . Assuming is disjunctive, is a lattice in , isomorphic to under the map , . is replete and a base for the closed sets , all arbitrary intersections of sets of .

If , then and the following statements are true:

The Induced Measure
Let and consider the induced measure , defined by The map is a bijection between and .

3. The Case of Finite Product of Lattices

3.1. Notations

Let , be abstract sets and a lattice of subsets of and a lattice of subsets of . We denote: (1), (2), the lattice generated by . We have the following:(3), (4), (5), (6), (7).

3.2. Results

Theorem 3.1 (the finite product of lattices/regular measures). Let , be abstract sets and let , be lattices of subsets of and , respectively. Then .

Proof. For , we have , disjoint union and , .
Let and and consider defined on .
If , then for some .
Then μ, and since and are zero-one valued measures, () and . By the regularity of and there exist , with and , with .
Therefore and .
If we let , then

Conversely, let and define on by , . Since is a zero-one measure on , it follows that is a zero-one measure on , that is, .

Suppose ; there exists such that and . Then and which shows that . Similarly take on defined by , .

Then, as before is regular on .

Finally for any and any we have which shows that , and therefore.

Theorem 3.2 (the product of lattices/-smooth regular measures). Let , be abstract sets and let , be lattices of subsets of and , respectively. Then .

Proof. Let and . Hence for with we have and for with we have , .
Consider the sequence of sets from . As in Theorem 3.1  , disjoint union and , .
Suppose that , that is, for all . Therefore or or both:

Conversely, let and define μ₁ on by

If is a sequence of sets with and , then , and since it follows that .

Therefore .

Similarly, defining on by , we get .

Hence .

Theorem 3.3 (product of supports of measures). Let , be abstract sets and let , be lattices of subsets of   and , respectively. The following statements are true:(a)if then ;(b)if   and are compact lattices then is compact.

Proof. We have(a), But and ,(b), since, being compact.

Theorem 3.4 (product of Wallman spaces/Wallman topologies). Consider the spaces with the Wallman topologies  , .
It is known that the topological spaces are compact and . Then the topological space is also compact and .

Proof. Since are compact topological spaces,
We have
Therefore so that , and then is compact.

To show that is a -space, let and suppose . Since with and with we get and . There exist and with Therefore , , , which implies , ; , .

Theorem 3.5 (product of normal lattices). Let , be abstract sets and let , be normal lattices of subsets of and , respectively. Then is a normal lattice of subsets of .

Proof. Let and such that on .
Then, since , and , we obtain , on , .
normal lattices ; therefore , that is, .

3.3. Examples

4. The General Case of Product of Lattices

Let be a collection of abstract sets ( an arbitrary index set) and let be the lattice of subsets of for all .

We denote