## 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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