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On the uniqueness of Lebesgue and Borel measures. (English) Zbl 0908.28003

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.

MSC:

28A12 Contents, measures, outer measures, capacities
28A05 Classes of sets (Borel fields, \(\sigma\)-rings, etc.), measurable sets, Suslin sets, analytic sets
28D05 Measure-preserving transformations
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