Regular measures and normal lattices
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.
scientific article; zbMATH DE number 650811