Existence and integral representation of regular extensions of measures
Volume 87 / 2001
Colloquium Mathematicum 87 (2001), 235-243
MSC: Primary 28A12; Secondary 28C15, 46A55.
DOI: 10.4064/cm87-2-9
Abstract
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.