On extremal extensions of regular contents and measures. (English) Zbl 0817.28002

Adamski, Wolfgang

Proc. Am. Math. Soc. 121, No. 4, 1159-1164 (1994).

Let \(\mathcal K\) be a lattice of subsets of a set \(X\), and denote by \(\alpha({\mathcal K})\) the algebra of subsets of \(X\) generated by \(\mathcal K\). Let, further, \(\lambda\) be a \({\mathcal K}\)-regular (\(={\mathcal K}\)-tight in another terminology) content on \(\alpha({\mathcal K})\), i.e., \(\lambda: \alpha({\mathcal K})\to [0,\infty)\) is additive and \(\lambda(A)= \sup\{\lambda(K): A\supset K\in {\mathcal K}\}\) for all \(A\in \alpha({\mathcal K})\), and let \(\mathcal L\) be another lattice of subsets of \(X\) with \({\mathcal K}\subset {\mathcal L}\). The author proves that \(\lambda\) extends to an \({\mathcal L}\)-regular content on \(\alpha({\mathcal L})\) which is an extreme point in the convex set of all such extensions (Theorem 2.3). From this he derives, under additional assumptions on \(\mathcal K\) and \(\mathcal L\), an analogous result on extreme extensions of \({\mathcal K}\)-regular measures (Theorem 2.4). Finally, several applications of the latter result in the case where \(\mathcal K\) and \(\mathcal L\) are of topological origin are given.
{Reviewer’s remarks: (1) In view of Plachky’s extremality criterion, Theorem 2.3 is a special case of a result of the reviewer [Colloq. Math. 51, 213-219 (1987; Zbl 0624.28006), Theorem 1]. The reviewer’s result is concerned with set functions with values in an Abelian complete Hausdorff topological group and involves a more general regularity assumption. (2) Similarly, all the topological applications of Theorem 2.4 discussed in the paper under review can be deduced, for group-valued measures, from Theorem 3 of the reviewer’s paper referred to above}.

Reviewer: Z.Lipecki (Wrocław)

MSC:

28A12	Contents, measures, outer measures, capacities
28C15	Set functions and measures on topological spaces (regularity of measures, etc.)

Keywords:

lattice of sets; regular content; extremal extension; regular measure

Citations:

Zbl 0624.28006

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