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On transformations of sets in \(R^n\). (English) Zbl 0475.28002

Summary: Motivated by translation and other concrete transformations of measurable subsets of the real line, T. Neubrunn and T. Šalát [Čas. Pěst. Mat. 94, 381–393 (1969; Zbl 0188.11501)J considered arbitrary families of transformations \((T_\omega)_{\omega\in\Omega}\), where \(\Omega\) is a metric space, satisfying three conditions. Various authors have extended the work of Neubrunn and Šalát, for example: M. Pal [Acta Fac. Rer. Natur. Univ. Comenian, Math. 29, 43–53 (1974; Zbl 0291.28014)], N. G. Saha and K. C. Ray [Publ. Inst. Math. Beograd, N. Ser. 22(36), 237–244 (1977; Zbl 0375.28012)]. H. I. Miller [Publ. Inst. Math. Beograd, N. Ser. 27(41), 175–178 (1980)].
In this paper, motivated by similarity transformation in \(R^n\), the author considers arbitrary families \((T_\omega)_{\omega\in\Omega}\) of transformations of the measurable subsets in \(R^n\), satisfying three conditions. These conditions are shown to be strictly less restrictive than the conditions of Neubrunn and Šalát and in addition several of their theorems are shown to hold in this more general setting.
In the second part of this paper two theorems are proved. They are the Baire property analogues of results of T. K. Khan and M. Pal [Glas. Mat., III. Ser. 16(36), 29–37 (1981; Zbl 0471.28001)], and A. Mazumdar [Glas. Mat., III. Ser. 16(36), 25-27 (1981; Zbl 0478.28001)], respectively.
Reviewer: Harry I. Miller

MSC:

28A05 Classes of sets (Borel fields, \(\sigma\)-rings, etc.), measurable sets, Suslin sets, analytic sets
54C50 Topology of special sets defined by functions
28D05 Measure-preserving transformations
28A75 Length, area, volume, other geometric measure theory