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Approximation and differential properties of measurable sets. (English. Russian original) Zbl 0556.28002
Math. USSR, Sb. 49, 401-418 (1984); translation from Mat. Sb., Nov. Ser. 121(163), No. 3, 403-422 (1983).
If E is a Lebesgue measurable set and t is an arbitrary real number, let \(E_ t=\{x:x-t\in E\},\) and let the function \(\omega (\cdot,E):(0,+\infty)\to {\mathbb{R}},\) the modulus of E, be defined by \(\omega (h,E)=\sup \{\rho (E,E_ t):0<t\leq h\},\) where \(\rho (E,E_ t)=| E\Delta E_ t|,\) the Lebesgue measure of the symmetric difference of E and \(E_ t\). Further, if n be an arbitrary natural number, let \(\sigma_ n(E)=\inf \{\rho (E,I_ n):I_ n\) is a set consisting of not more than n segments\(\}\). The authors first study the connection between \(\omega\) (\(\cdot,E)\) and \(\sigma_ n(E)\). For example, they show that for the condition \(\omega (h,E)=O(h^{\alpha})\), \(0<\alpha <1\), it is both necessary and sufficient that \(\sigma_ n(E)=O(n^{-\alpha /1-\alpha})\). The early results on the approximation of measurable sets then are used to study local properties of measurable sets that are connected with the following notion of density: A real number x is a point of \(\lambda\)-density \((\lambda >0)\) of the measurable set E if \(| [x-h,x+h]\setminus E| /2h=o(h^{\lambda}),\) as \(h\to 0^+\); x is a point of \(\lambda\)-dispersion of E if it is a point of \(\lambda\)-density of the complement of E. The authors show, for example, that if \(\int_{0^+}[\omega (t,E)/t^{1+\alpha}]dt<+\infty,\) for some \(\alpha\) in (0,1), then, for an arbitrary positive \(\lambda\leq \alpha /1- \alpha\), save for the possible exception of a set of \((\lambda +1)(1- \alpha)\)-Hausdorff measure zero, each real number is either a point of \(\lambda\)-density or a point of \(\lambda\)-dispersion of E. In particular, almost every point of E is a point of (\(\alpha\) /1-\(\alpha)\)-density. Finally, the authors show that the convergence of the above integral is essential for the given conclusion. In particular, they construct a closed set E such that \(\omega (E,h)=O(h^{\alpha}\log^{-1}(1/h))\), no point of E is a point of dispersion and, for each \(\lambda\) in (0,\(\alpha\) /1-\(\alpha)\), \(\{\) \(x\in E:x\) is not a point of \(\lambda\)- density of \(E\}\) has positive \((\lambda +1)(1-\alpha)\)-Hausdorff measure.
Reviewer: R.E.Zink
MSC:
28A05 Classes of sets (Borel fields, \(\sigma\)-rings, etc.), measurable sets, Suslin sets, analytic sets
28A75 Length, area, volume, other geometric measure theory
28A15 Abstract differentiation theory, differentiation of set functions
41A25 Rate of convergence, degree of approximation
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