Existence and integral representation of regular extensions of measures

Werner Rinkewitz¹

¹ Mathematisches Institut Universität München Theresienstraße 39 D-80333 München, Germany

Colloquium Mathematicum, Tome 87 (2001) no. 2, pp. 235-243

Résumé

Let ${\cal L}$ be a $\delta $-lattice in a set $X$, and let $\nu $ be a measure on a sub-$\sigma $-algebra of $\sigma ({\cal L})$. It is shown that $\nu $ extends to an ${\cal L}$-regular measure on $\sigma ({\cal L})$ provided $\nu ^{\ast }| {\cal L}$ is $\sigma $-smooth at $\emptyset $ and $\nu ^{\ast }(L)=\mathop {\rm inf}\{ \nu ^{\ast }(U)\mid X\setminus U\in {\cal L},\ U\supset L\} $ for all $L\in {\cal L}$. Moreover, a Choquet type representation theorem is proved for the set of all such extensions.

Détail

DOI : 10.4064/cm87-2-9

Keywords: cal delta lattice set measure sub sigma algebra sigma cal shown extends cal regular measure sigma cal provided ast cal sigma smooth emptyset ast mathop inf ast mid setminus cal supset cal moreover choquet type representation theorem proved set extensions