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

Borell, Christer

Math. Scand. 53, 125-144 (1983).

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

Keywords:

closed convex cone; finite positive Radon measure; :\(\alpha \) :-concave measures; Brunn-Minkowski theory; zero-one laws; integrability of some sublinear functions; stochastic processes