The uniqueness of Lebesgue and Borel measures

Abstract

This section is devoted to the uniqueness property of various invariant measures. Primarily, we discuss this property for the standard Lebesgue measure on a finite-dimensional Euclidean space (sphere) E and for the standard Borel measure on the same space (sphere), being the restriction of the Lebesgue measure to the Borel σ-algebra of E. Our main tool here is the measure extension theorem due to Ershov (see Section 9 of the book). In particular, applying this theorem and some simple facts from the general theory of a Haar measure on a locally compact topological group, we present natural sufficient conditions for the uniqueness of the standard Borel measure on E. One problem closely related to this result is also posed which seems to be interesting from the point of view of the theory of invariant measures.