## Convexity of measures in certain convex cones in vector space $$\sigma$$- algebras.(English)Zbl 0568.46007

For $$0<\vartheta <1$$, -$$\infty \leq a\leq \infty$$ and $$0<s,t\leq \infty$$ set $$M_ a^{\vartheta}=(s,t)=(\vartheta s^ a+(1-\vartheta)t^ a)^{1/a}$$ if $$a\in R\setminus \{0\}$$, $$=\min (s,t)$$ if $$a=-\infty$$, $$=s^{\vartheta}t^{1-\vartheta}$$ if $$a=0$$, and $$=\max (s,t)$$ if $$a=\infty$$ $$(0^{\alpha}=\infty$$ if $$-\infty <\alpha <0)$$. For $$0\leq s,t\leq \infty$$ set $$M_ a^{\vartheta}(s,t)=0$$ if $$s=0$$ or $$t=0$$. Let E be a real locally convex space and C a closed convex cone in E (with vertex at 0); set $$<C>=\{K-C:E\supset K\quad compact\}.$$ A finite positive Radon measure $$\mu$$ on E is said to be :$$\alpha$$ :-concave in C (-$$\infty \leq \alpha \leq \infty)$$ if $$\mu (\vartheta A+(1-\vartheta)B)\geq M^{\vartheta}_{\alpha}(\mu (A),\mu (B))$$ for all $$A,B\in <C>$$ and $$\vartheta\in (0,1)$$. The author’s study of :$$\alpha$$ :-concave measures is motivated by their application in reliability theory, statistics, stochastic programming, convexity, absolute continuity of semi-norms etc. and by the possibility of deepening of the Brunn-Minkowski theory on vector spaces. For example, the author derives zero-one laws and integrability of some sublinear functions, and presents examples of stochastic processes with increasing path inducing :0:-concave measures in suitable convex cones.
Reviewer: J.Daneš

### MSC:

 46A55 Convex sets in topological linear spaces; Choquet theory 60F20 Zero-one laws 46G12 Measures and integration on abstract linear spaces
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