Let X and Y be two Polish spaces. Functions ƒ, g : X → Y are called equivalent if there exists a bijection φ from X onto itself such that g º φ = ƒ. Using a theorem of J. Saint Raymond we characterize functions equivalent to Borel measurable ones. This characterization answers a question asked by M. Morayne and C. Ryll-Nardzewski.
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In [2] Jajte formulated the following question: Let h0 (x) and h1 (x) be homeomorphisms of the interval [0, 1] onto itself. Is it true that for any x € [0, 1] and almost any t ϵ (0, 1) there exists a limit of a sequence [formula] for n → ∞, where t = (0, t1 t2…)2 is a binary representation of t, i.e. t = Σi ≥ 1 ti 2-i and ti ϵ {0, 1}? The answer is negative. We describe the set of condensation points of the sequence in some special cases.
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