Abstract 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 R n (sphere S n ) and for the standard Borel measure on the same space (sphere), which is the restriction of the Lebesgue measure to the Borel σ –algebra of R n ( 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.
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