Upper envelopes of inner premeasures
Let \({\mathfrak S}\) and \({\mathfrak T}\) be lattices of subsets of a nonempty set \(X\) both containing \(\emptyset\). Let, further, \(\beta:{\mathfrak S}\to [0, \infty[\) vanish at \(\emptyset\), be isotone and submodular (= strongly subadditive), and have the following regularity property: given \(A\in{\mathfrak S}\) and \(\varepsilon> 0\) there exists \(U\in{\mathfrak T}\) such that \(A\subset U\) and \(\beta(K)< \beta(A)+ \varepsilon\) whenever \(K\in{\mathfrak S}\) and \(K\subset U\). According to a classical theorem, in the special case where \({\mathfrak S}\) and \({\mathfrak T}\) are the families of compact and open subsets of a Hausdorff topological space \(X\), respectively, for every \(B\in{\mathfrak S}\) there exists a Radon premeasure \(\varphi:{\mathfrak S}\to [0,\infty[\) with \(\varphi\leq \beta\) and \(\varphi(B)= \beta(B)\). If, in addition, \(\sup\beta< \infty\), then there exists a Radon premeasure \(\psi:{\mathfrak S}\to [0,\infty[\) with \(\psi\leq\beta\) and \(\sup\psi= \sup\beta\). This theorem, originally formulated by G. Choquet (1953-54), has been studied by many authors, with the first complete proof given by F. Topsøe (1978). An abstract, non-topological generalization of the theorem was then found by W. Adamski (1977), who was also able to replace the single set \(B\) above by some systems of subsets of \(X\), e.g., increasing sequences of elements of \({\mathfrak S}\). Adamski's list of postulates on \({\mathfrak S}\), \({\mathfrak T}\) and \(\beta\) is, however, quite long and includes the stability of \({\mathfrak S}\) under countable intersections and the \(\sigma\)-continuity of \(\beta\) at \(\emptyset\). The present paper offers a simpler and (essentially) more general setting for the theorem under discussion. Namely, it merely assumes that each pair of disjoint elements of \({\mathfrak S}\) can be separated by a pair of disjoint elements of \({\mathfrak T}\) and for all \(U\in{\mathfrak T}\) and \(K\in{\mathfrak S}\) we have \(K\setminus U\in{\mathfrak S}\). As a representative result of the paper we quote the following (Specialization 3.8): If \({\mathfrak M}\subset{\mathfrak S}\) is totally ordered under inclusion, then there exists \(\varphi:{\mathfrak S}\to [0,\infty[\) with \(\varphi\leq \beta\) and \(\varphi|{\mathfrak M}= \beta|{\mathfrak M}\) which is an inner \(*\) premeasure, i.e., it extends to an additive \({\mathfrak S}\)-inner regular function on the algebra of subsets of \(X\) generated by \({\mathfrak S}\) taking values in \([0,\infty]\). The paper is written in the spirit of the author's book ``Measure and integration. An advanced course in basic procedures and applications'' (1997; Zbl 0887.28001) and uses its notation, terminology and results extensively. \{Reviewer's remarks: (1) Theorem 1.1 is due essentially to \textit{B. Fuchssteiner} [J. Funct. Anal. 16, 1-14 (1974; Zbl 0283.06007), Theorem 1]; for a generalization see \textit{P. Plappert} [Semigroup Forum 51, No. 3, 347-355 (1995; Zbl 0834.20060); Corollary 6]. (2) Example 1.4 is very similar to Example 5 in \textit{J. Kindler} [Math. Nachr. 134, 131-147 (1987; Zbl 0641.28001)]. (3) Example 3.4.1 should be made more specific to fit the assumptions of Proposition 3.3. (4) Well-ordered under inclusion systems of sets have already appeared in a similar context; see, e.g., \textit{A. El Kaabouchi} [Proc. Am. Math. Soc. 121, No. 3, 823-832 (1994; Zbl 0920.28003), Théorème 5]. (5) Reference [21] by the author has appeared [Note Mat. 17, 235-249 (1997)].
scientific article; zbMATH DE number 1448493
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