On the extremal submeasures on a 3-set. (English) Zbl 0738.28003

Summary: Let \({\mathcal P}(\chi_ n)\) be an algebra of all subsets of an \(n\)-point set \(\chi_n\). A subadditive set function \(\varphi : \mathcal P(\chi_n)\to [0,\infty)\) is called a submeasure if it is increasing and \(\varphi(\emptyset)=0\). Moreover, if \(\varphi(\chi_n)=1\), \(\varphi\) is said to be normalized. A normalized submeasure \(\varphi\) is said to be extremal if \(\varphi\) is an extreme point of a convex set consisting of all normalized submeasures on the algebra \(\mathcal P(\chi_n)\). An open question dealing with finding an effective characterisation of extremal submeasures was posed by F. Topsœe [Math. Scand. 38, 159–166 (1976; Zbl 0319.28006)].
In the present note the author verifies Topsœ’s supposition that there are only twelve extremal submeasures on \(\mathcal P(\chi_3)\).


28A12 Contents, measures, outer measures, capacities
28A10 Real- or complex-valued set functions


Zbl 0319.28006