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Vitali systems in \(R^n\) with irregular sets. (English) Zbl 0893.28003

Let \(\Phi^n\) denote the family of Lebesgue measurable functions, \(\varphi:(0,+\infty)\to(0,+\infty)\), for which \(\varphi(r)\leq(2r)^n\), \(\forall r>0\), let \(\mathcal K\) be a class of compact subsets of \(\mathbb{R}^n\), and let the closed ball with center \(x\) and radius \(r\) be denoted by \(B[x,r]\). Let \[ {\mathcal B}_r(x)={\mathcal B}^n_r(x;\varphi,{\mathcal K})=\{K\in{\mathcal K}:K\subseteq B[x,r],| K|\geq\varphi(r)\}, \] where \(| K|\) represents the outer Lebesgue measure of \(K\) and let \({\mathcal B}^n[\varphi,{\mathcal K}]=\{{\mathcal B}_r(x):x\in\mathbb{R}^n\), \(r>0\}\). An associated Vitali system, \({\mathcal V}^n[\varphi,{\mathcal K}]\), is a set of pairs \((A,{\mathcal S})\) such that \(A\subseteq\mathbb{R}^n\) and \(\mathcal S\) is a family of closed subsets of \(\mathbb{R}^n\) for which \(\forall x\in A\) \(\exists r_0\) \(\forall r\leq r_0:{\mathcal S}\cap{\mathcal B}^n[\varphi,{\mathcal K}]\neq\emptyset\). A Vitali system \({\mathcal V}={\mathcal V}^n[\varphi,{\mathcal K}]\) has the packing property if, for every pair \((A,{\mathcal S})\in{\mathcal V}\) there is a countable subset, \({\mathcal S}_0\) of \({\mathcal S}\), whose elements are pairwise disjoint, such that \(| A\setminus\cup\{S:S\in{\mathcal S}_0\}|=0\). For positive constants \(\eta\) and \(\kappa\) let \({\mathcal K}^n_{\eta,\kappa}\) be the class of compact sets, \(K\), such that for every closed ball, \(B\), either \(| B\cap K|\geq\kappa| B|\), or there is an open ball, \(B_1\subset B\setminus K\), such that \(\text{diam }B_1\geq\eta\text{ diam }B\). The principal result of the article is the following proposition, which generalizes a result demonstrated by the authors in one of their earlier papers [Math. Ann. 230, 183-193 (1977; Zbl 0364.28003)].
Theorem 1. Let \(\eta\) and \(\kappa\) be positive constants for which \(2\eta+\root{n}\of{\kappa}\leq 1\) and let \(\varphi\in\Phi^n\). Then \({\mathcal V}^n[\varphi,{\mathcal K}^n_{\eta,\kappa}]\) has the packing property iff \[ \int^{r_0}_0\varphi(r)r^{-n-1}dr=+\infty,\quad\forall r_0>0.\tag{*} \] The authors observe that if \(\mathcal K\) be the class of all convex, compact subsets of \(\mathbb{R}^n\), and if (*) be satisfied, then \({\mathcal V}^n[\varphi,{\mathcal K}]\) will have the packing property, since the condition \(2\eta+\root{n}\of{\kappa}\leq 1\) is both necessary and sufficient for \(K^n_{\eta,\kappa}\) to contain the convex, compact sets. It also follows from Theorem 1 that, if (*) be satisfied and if \((A,{\mathcal S})\) satisfies the condition \[ \forall x\in A\quad\exists r_0,\quad\eta,\kappa\quad\forall r\leq r_0:{\mathcal S}\cap{\mathcal B}^n_r(x;\varphi,{\mathcal K}^n_{\eta,\kappa})\neq\emptyset, \] then there is a countable subset \({\mathcal S}_0\) of \(\mathcal S\), whose elements are pairwise disjoint, such that \(| A\setminus\cup\{S:S\in{\mathcal S}_0\}|=0\). Finally, from Theorem 1 and the earlier-established result to which reference has been made above, the authors note that if \({\mathcal V}^n[\varphi,{\mathcal K}^n_{\eta,\kappa}]\) has the packing property then every \({\mathcal V}^n[\varphi,{\mathcal K}^n_{\eta,\kappa}]\)-meager set is a null set.

MSC:

28A15 Abstract differentiation theory, differentiation of set functions
28A12 Contents, measures, outer measures, capacities
52A20 Convex sets in \(n\) dimensions (including convex hypersurfaces)

Citations:

Zbl 0364.28003
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