Summary: We consider the uniqueness property for various invariant measures. Primarily, we discuss this property for the standard Lebesgue measure on the \(n\)-dimensional Euclidean space \(\mathbb{R}^n\) (sphere \(\mathbb{S}^n\)) and for the standard Borel measure on the same space (sphere), which is the restriction of the Lebesgue measure to the Borel \(\sigma\)-algebra of \(\mathbb{R}^n\) \((\mathbb{S}^n)\). The main goal of the paper is to show an application of the well-known theorems of Ulam and Ershov to the uniqueness property of Lebesgue and Borel measures.