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Functions equivalent to Borel measurable ones

Komisarski A., Michalewski H., Milewski P.

Bulletin of the Polish Academy of Sciences. Mathematics

2010

Vol. 58, no 1

55-64

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.

A non-ergodic phenomenon for some random dynamical system

Komisarski A.

Probability and Mathematical Statistics

1999

Vol. 19, Fasc. 2

407--412

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.