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Remarks on semiseparation of lattices. (English) Zbl 0733.28006
Let X be a non-empty set. For a lattice we refer to any lattice of subsets of X containing \(\emptyset\) and X. If \({\mathcal L}_ 1\) and \({\mathcal L}_ 2\) are two lattices such that \({\mathcal L}_ 1\subseteq {\mathcal L}_ 2,\) then \({\mathcal L}_ 1\) semiseparates \({\mathcal L}_ 2\) if \(L_ 1\in {\mathcal L}_ 1,\) \(L_ 2\in {\mathcal L}_ 2\) and \(L_ 1\cap L_ 2=\emptyset\) imply that there exists \(L\in {\mathcal L}_ 1\) such that \(L_ 2\subseteq L\) and \(L_ 1\cap L=\emptyset.\) Let \({\mathcal L}\) be a lattice. Then (1) \({\mathcal L}\) is a delta lattice if \({\mathcal L}\) is closed under countable intersections; (2) \({\mathcal L}\) is disjunctive if \(x\in X,\) \(L_ 1\in {\mathcal L}\) and \(x\not\in L_ 1\) imply that there exists \(L_ 1\in {\mathcal L}\) such that \(x\in L_ 2\) and \(L_ 1\cap L_ 2=\emptyset;\) (3) The symbol I(\({\mathcal L})\) denotes the set of all non-trivial \(\{0,1\}- \)valued finitely additive set functions defined on the algebra A(\({\mathcal L})\) generated by \({\mathcal L}\); (4) An element \(\mu \in I({\mathcal L})\) is \({\mathcal L}\)-regular if \(\mu (A)=\sup \{\mu (L):\;L\in {\mathcal L}\text{ and } L\subseteq A\}\) for all \(A\in A({\mathcal L});\) (5) The symbol \(I_ R({\mathcal L})\) denotes the set of all \({\mathcal L}\)-regular elements of I(\({\mathcal L})\); (6) The symbol \(I^{\sigma}_ R({\mathcal L})\) denotes the set of all countably additive elements of \(I_ R({\mathcal L})\); (7) An \({\mathcal L}\)- filter is a subset F of \({\mathcal L}\) satisfying the following conditions: \(i)\quad \emptyset \not\in {\mathcal F},\) ii) \({\mathcal F}\) is closed under finite intersections, and iii) If \(F\in {\mathcal F},\) \(L\in {\mathcal L}\) and \(F\subseteq L,\) then \(L\in {\mathcal F};\) (8) \({\mathcal L}\) is an I-lattice if every \({\mathcal L}\)-filter with the countable intersection property is contained in an \({\mathcal L}\)-ultrafilter with the countable intersection property. It is easy to verify that the set \(\{\{\mu \in I_ R({\mathcal L}):\;\mu (L)=1:\;L\in {\mathcal L}\}\) is a base for the closed sets of a topology on \(I_ R({\mathcal L})\), called the Wallman topology on \(I_ R({\mathcal L}).\)
Now the main results of the paper under review can be stated as follows:
Theorem A. Let \({\mathcal L}_ 1\) and \({\mathcal L}_ 2\) be two lattices such that \({\mathcal L}_ 1\subseteq {\mathcal L}_ 2\), let \(\mu \in I_ R({\mathcal L}_ 1)\) and, for every \(E\subseteq X,\) let \(\mu '(E)=\inf \{\mu (X\setminus L_ 1):\;L_ 1\in {\mathcal L}_ 1\text{ and } E\subseteq X\setminus L_ 1\}\) and \({\tilde \mu}(E)=\inf \{\mu (L_ 1):\;L_ 1\in {\mathcal L}_ 1\text{ and } E\subseteq L_ 1\}.\) Then \(\mu '| {\mathcal L}_ 2={\tilde \mu}| {\mathcal L}_ 2\) if and only if \({\mathcal L}_ 1\) semiseparates \({\mathcal L}_ 2.\)
Theorem B. Let \({\mathcal L}_ 1\) and \({\mathcal L}_ 2\) be two lattices such that \({\mathcal L}_ 1\subseteq {\mathcal L}_ 2\), \({\mathcal L}_ 1\) is delta and an I-lattice, \({\mathcal L}_ 2\) is disjunctive and, for every \(\mu \in I_ R({\mathcal L}_ 2),\) the restriction \(\mu | A({\mathcal L}_ 1)\) is \({\mathcal L}_ 1\)-regular. If the map \(\chi (\mu)=\mu | A({\mathcal L}_ 1)\) is closed with respect to the corresponding Wallman topologies, then \({\mathcal L}_ 1\) semiseparates \({\mathcal L}_ 2\).
MSC:
28C15 Set functions and measures on topological spaces (regularity of measures, etc.)
28A60 Measures on Boolean rings, measure algebras
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