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Certain transformations \(T_\omega \) and Lebesgue measurable sets of positive measure. (English) Zbl 0953.28002
Let \(\Omega \) be a metric space, \(\omega _n \rightarrow \omega _0\) in \(\Omega \) and \(\alpha _n \rightarrow \alpha _0 \not = 0\) in \(\mathbb R\). Let \(T_{\omega _n}\) be transformations of \(\mathbb R^N\) into \(\mathbb R^N\). Conditions are studied in order to guarantee that for every set \(A\) of positive Lebesgue measure in \(\mathbb R^N\), the set \[ \frac 1{\alpha _0} A \cap T_{\alpha_1}\Big (\frac {1}{\alpha _{n_1}}A\Big) \cap T_{\alpha _2} \Big (\frac {1}{\alpha _{n_2}}A\Big)\cap \ldots \cap T_{\alpha _p} \Big (\frac 1{\alpha _{n_p}} A\Big) \] is a set of positive Lebesgue measure for all large enough \(n_i\).
Reviewer: V.Zizler (Praha)
MSC:
28A05 Classes of sets (Borel fields, \(\sigma\)-rings, etc.), measurable sets, Suslin sets, analytic sets
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References:
[1] T. Neubrunn, T. Šalát: Distance sets, ratio sets and certain transformations of sets of real numbers. Čas. Pěst. Mat. 94 (1969), 381-393. · Zbl 0188.11501
[2] M. Pal: On certain transformations of Sets in \(R_N\). Acta F.R.N. Univ. Comen. Mathematica XXIX, 1974, pp. 43-58. · Zbl 0291.28014
[3] K.C. Ray: On two theorems of S. Kurepa. Glasnik Mat-Fiz. 19 (1964), 207-210. · Zbl 0138.03604
[4] N.G. Saha, K.C. Ray: On sets under certain transformations in \(R_N\). Publ. Inst. Math. 22(36) (1977), 237-244. · Zbl 0375.28012
[5] R.L. Wheeden, A. Zygmund: Measure and Integration. Marcel Dekker, Inc., New York, 1977, pp. 45. · Zbl 0362.26004
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