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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)
28A05 Classes of sets (Borel fields, \(\sigma\)-rings, etc.), measurable sets, Suslin sets, analytic sets
Full Text: DOI EuDML
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