On the uniqueness property for products of symmetric invariant probability measures.

Kharazishvili, A.

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We construct a transformation T:[0,1] → [0,1] having the following properties: 1) (T,|·|) is completely mixing, where |·| is Lebesgue measure, 2) for every f∈ L¹ with ∫fdx = 1 and φ ∈ C[0,1] we have $\int φ (T^{n} x) f (x) d x \to \int φ d μ$ , where μ is the cylinder measure on the standard Cantor set, 3) if φ ∈ C[0,1] then $n^{- 1} \sum_{i = 0}^{n - 1} φ (T^{i} x) \to \int φ d μ$ for Lebesgue-a.e. x.