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Regular measures and normal lattices. (English) Zbl 0810.28004

The purpose of the paper under review is to investigate the set of all real bounded finitely additive measures on the Boolean algebra generated by a normal lattice of sets. More precisely: Let \(X\) be a non-empty set, le \(\mathcal L\) be a lattice of subsets of \(X\) containing \(\emptyset\) and \(X\), let \(A({\mathcal L})\) be the Boolean subalgebra of \(2^ X\) generated by \(\mathcal L\) and let \(M({\mathcal L})= \{\mu: A({\mathcal L})\to \mathbb{R}\mid \mu\geq 0\) and \(\mu\) is finitely additive}. Write \(M_ R({\mathcal L})= \{\mu\in M({\mathcal L})\): \(\mu\) is \({\mathcal L}\)-regular}, \(M_ \sigma{\mathcal L}= \{\mu\in M({\mathcal L})\): if \(L_ n\in {\mathcal L}\) and \(L_ n\downarrow \emptyset\), then \(\mu(L_ n)\to 0\}\) and \(M^ \sigma({\mathcal L})= \{\mu\in M({\mathcal L})\): \(\mu\) is countably additive}. We say that the lattice \(\mathcal L\) is normal if, for any \(L_ 1,L_ 2\in {\mathcal L}\) with \(L_ 1\cap L_ 2= \emptyset\), there exist \(L_ 3,L_ 4\in {\mathcal L}\) such that \(L_ 1\subseteq L^ c_ 3\), \(L_ 2\subseteq L^ c_ 4\) and \(L^ c_ 3\cap L^ c_ 4= \emptyset\), where \(^ c\) denotes the usual set complementation. If \({\mathcal L}_ 1\) and \({\mathcal L}_ 2\) are two lattices of subsets of \(X\) with \({\mathcal L}_ 1\subseteq {\mathcal L}_ 2\), we say that \({\mathcal L}_ 1\) separates \({\mathcal L}_ 2\) if \(A,B\in {\mathcal L}_ 2\) and \(A\cap B= \emptyset\), then there exist \(C,D\in {\mathcal L}_ 1\) such that \(A\subseteq C\), \(B\subseteq D\) and \(C\cap D= \emptyset\).
Now the main results of the paper can be stated as follows:
Theorem A. Suppose \({\mathcal L}\) is normal and let \(\mu\in M_ \sigma({\mathcal L})\), \(\nu\in M_ R({\mathcal L})\) with \(\mu\leq \nu\) on \(\mathcal L\) and \(\mu(X)= \nu(X)\). Then \(\nu\in M_ \sigma({\mathcal L}')\) where \({\mathcal L}'= \{L^ c: L\in {\mathcal L}\}\).
Theorem B. Suppose \(\mathcal L\) is normal and let \(\mu\in M({\mathcal L})\), \(\nu_ 1,\nu_ 2\in M_ R({\mathcal L})\) with \(\mu\leq \nu_ 1\), \(\nu_ 2\) on \(\mathcal L\) and \(\mu(X)= \nu_ 1(X)= \nu_ 2(X)\). Then \(\nu_ 1= \nu_ 2\).
Theorem C. Let \({\mathcal L}_ 1\subseteq {\mathcal L}_ 2\) and suppose that \({\mathcal L}_ 1\) separates \({\mathcal L}_ 2\). Let \(\mu\in M_ R({\mathcal L_ 1})\) and consider the extension \(\nu\in M_ R({\mathcal L}_ 2)\). Then: a) \(\nu\) is \({\mathcal L}_ 1\)-regular on \({\mathcal L}_ 2'\). b) If \(\mu\in M_ R({\mathcal L}_ 1)\) then \(\nu\in M_ \sigma({\mathcal L}_ 2')\), c) \(\nu\) is unique.

MSC:

28A60 Measures on Boolean rings, measure algebras
28C15 Set functions and measures on topological spaces (regularity of measures, etc.)
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