Extensions of isometrically invariant measures on Euclidean spaces

Abstract:

We consider countably additive, nonnegative, extended real-valued measures that vanish on singletons. Given a group $G$ of isometries of ${{\mathbf {R}}^n}$ and a $G$-invariant $\sigma$-finite measure on ${{\mathbf {R}}^n}$ we study the problem of determining whether it has a proper $G$-invariant extension. We prove that it does, provided that the set of all points with uncountable $G$-orbits is not of measure zero. We also characterize those groups $G$ for which every $G$-invariant $\sigma$-finite measure on ${{\mathbf {R}}^n}$ has a proper $G$-invariant extension.