## 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 $$\mathcal P(\chi_3)$$.

### MSC:

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

Zbl 0319.28006