Non-additive measure and integral. (English) Zbl 0826.28002

Denneberg, Dieter

Theory and Decision Library. Series B: Mathematical and Statistical Methods. 27. Dordrecht: Kluwer Academic Publishers. ix, 178 p. (1994).

Based on the theory of monotone set functions \(\mu\) the author presents the Choquet integral \(\int X d\mu\) for upper \(\mu\)-measurable functions \(X\) by means of the quantile functions associated with \(X\) and \(\mu\). In particular, properties of additivity and superadditivity of the Choquet integral are proved. In this connection the notion of comonotonicity and a characterization of this notion due to the author of this book plays an important role. Furthermore, versions of the classical monotone convergence theorem, the lemma of Fatou and the dominated convergence theorem are introduced. Highlights of this book are integral representations due to S. Schmeidler and G. Greco for linear functionals, from which classical representation theorems might be rederived.

Reviewer: D.Plachky (Münster)

MSC:

28-02	Research exposition (monographs, survey articles) pertaining to measure and integration
28A10	Real- or complex-valued set functions
28A12	Contents, measures, outer measures, capacities

Keywords:

monotone set functions; Choquet integral; additivity; superadditivity; comonotonicity; monotone convergence theorem; lemma of Fatou; dominated convergence theorem