Extensions of measures invariant under countable groups of transformations

Abstract:

We consider countably additive, nonnegative, extended real-valued measures vanishing on singletons. Given a group $G$ of bijections of a set $X$ and a $G$-invariant measure $m$ on $X$ we ask whether there exists a proper $G$-invariant extension of $m$. We prove, among others, that if $\mathbb {Q}$ is the group of rational translations of the reals, then there is no maximal $\mathbb {Q}$-invariant extension of the Lebesgue measure on $\mathbb {R}$. On the other hand, if ${2^\omega }$ is real-valued measurable, then there exists a maximal $\sigma$-finite $\mathbb {Q}$-invariant measure defined on a proper $\sigma$-algebra of subsets of $\mathbb {R}$.