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Symmetric submeasures on a finite set. (Sous-mesures symétriques sur un ensemble fini.) (French) Zbl 0915.28005

Azéma, Jacques (ed.) et al., Séminaire de probabilités XXXII. Berlin: Springer. Lect. Notes Math. 1686, 1-5 (1998).
Let \(E= \{0,1,\dots,n- 1\}\) and denote by \({\mathcal M}_n\) the set of all positive finite measures over \(E\). In the authors’ (nonstandard) terminology, a positive finite function \(J\) over \(E\) is a submeasure if there exists \(H\subset{\mathcal M}_n\) such that \(J(A)= \sup_{\mu\in H} \mu(A)\) for all \(A\subset E\). (The term “submeasure” has usually a wider meaning while those considered by the authors are called multiply or completely subadditive. Alternatively, in the case where \(H\) consists of probability measures, they are called upper probabilities.) Moreover, the submeasure \(J\) is called symmetric if \(J(A)\) depends only on the cardinality \(| A|\) of \(A\). Let \({\mathcal I}_n\), resp. \({\mathcal I}^{\text{sym}}_n\), stand for the set of all, resp. all symmetric, submeasures on \(E\). The authors show that for every \(J\in {\mathcal I}_n\), resp. \(J\in{\mathcal I}^{\text{sym}}_n\), one can choose \(H\) as above with \(| H|\leq 2^n- n-1\), resp. \(| H|\leq 2^{n-1}\), and that the latter estimate is best possible. They also give a description of the extreme points of the convex set of all \(J\in{\mathcal I}^{\text{sym}}_n\) with \(J(E)= 1\).
{Reference [3] of the paper under review has appeared in the meantime [J. B. Kadana and L. Wasserman, Ann. Stat. 24, No. 3, 1250-1264 (1996; Zbl 0862.60004)]}.
For the entire collection see [Zbl 0893.00035].

MSC:

28A12 Contents, measures, outer measures, capacities
52A20 Convex sets in \(n\) dimensions (including convex hypersurfaces)

Citations:

Zbl 0862.60004
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