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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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