On unique extensions of positive additive set functions. II (Q1100576)

This is the second part of the author's paper [Arch. Math. 41, 71-79 (1983; Zbl 0505.28006)] and it contains improvements of the main results of that paper. Let G be an order complete Abelian lattice group, Let \({\mathfrak M}\) be an algebra of subsets of a set X, and let \(\mu\) : \({\mathfrak M}\to G\) be a quasi-measure (i.e., \(\mu\) is additive and nonnegative). We denote by \(\mu^*\) the outer quasi-measure on \(2^ X\) generated by \(\mu\). We say that \(\mu\) is \({\mathfrak U}\)-tight, where \({\mathfrak U}\subset 2^ X\), if, for every \(M\in {\mathfrak M}\), we have \[ \inf \{\mu (M\setminus \tilde M): \tilde M\subset U\subset M\quad and\quad (\tilde M,U)\in {\mathfrak M}\times {\mathfrak U}\}=0. \] We denote by \({\mathfrak R}\) the algebra of subsets of X generated by \({\mathfrak M}\cup {\mathfrak U}\) and by E(\(\mu)\) the set of all quasi-measures on \({\mathfrak R}\) with values in G extending \(\mu\). The main result of the paper asserts that the following two conditions are equivalent: (i) There exists a (unique) \(\rho\in E(\mu)\) with \(\rho (U)=\mu^*(U)\) for all \(U\in {\mathfrak U}.\) (ii) \(\sum^{n}_{i=1}\mu^*(U_ i)=\sum^{n}_{i=1}\mu^*(\cup_{1\leq i_ 1<...<i_ k\leq n}U_{i_ 1}\cap...\cap U_{i_ k})\) whenever \(U_ 1,...,U_ n\in {\mathfrak U}.\) Under these conditions, if \(\mu\) is \({\mathfrak U}\)-tight, then \(\rho\) is \({\mathfrak U}_{\ell}\)-tight, where \({\mathfrak U}_{\ell}\) is the lattice of sets generated by \({\mathfrak U}\). Also, a sufficient condition in order that \(\rho\) as above be \(\sigma\)-additive is given.