On a set in \({\mathbb{R}}^ n\) under coordinate transformations. (English) Zbl 0561.26009

Miller, Harry I.; Pal, Mukul

Čas. Pěst. Mat. 109, 225-235 (1984).

In this paper we prove several general theorems dealing with collections of transformations and their actions on subsets of \(R^ n\) of positive measure or on second category subsets of \(R^ n\) possessing the Baire property. As corollaries of our theorems we obtain results of H. Steinhaus, S. Piccard and S. Kurepa. In addition an example is given to show that a previous theorem of the first author cannot be improved. The following result is typical for the theorems in this paper. Theorem. Suppose A, \(A\subset {\mathbb{R}}^ n,\) is a set of positive Lebesgue measure and a is a point of density of A. If \(T_ 1,T_ 2,...,T_ p\) are p transformations satisfying \((\alpha)\quad T_ i:{\mathbb{R}}^ n\to {\mathbb{R}}^ n\) for each \(i=1,2,...,p;\) (\(\beta)\) \(T_ i\) is continuous at 0 and \(T_ i(0)=0\) for each \(i=1,2,...,p\); \((\beta)\quad (\gamma)\quad T_ i(a)=a\) for each \(i=1,2,...,p;\) (\(\delta)\) there exists an \(\bar R>0\) such that \(T_ i\) is a coordinate transformation on \(S(a,\bar R)\) for each \(i=1,2,...,p.\) Then there exists a ball K with center at the origin so that for every \(x\in K\) there are vectors \(a(x),a_ k^{k'}(x)\in A;\) \(k,k'\in \{1,2,...,p\}\) such that \(T_ k(a_ k^{k'}(x))=a(x)+T_{k'}(x);\quad k,k'\in \{1,2,...,p\}.\)

MSC:

26B10	Implicit function theorems, Jacobians, transformations with several variables
28A05	Classes of sets (Borel fields, \(\sigma\)-rings, etc.), measurable sets, Suslin sets, analytic sets

Keywords:

sets of transformations of \({\mathbb{R}}^ n\); Baire property; set of positive Lebesgue measure