Infinite variation and derivatives in \(R^m\)
Let \({\mathcal C}\) be a family of subsets in \(\mathbb R^m\), then: (a) it is fine on a set \(E\) if \(\forall x\in E\), and \(\forall r>0\), \(\exists C\in {\mathcal C}\) with \(x\in C\subset B(x,r)\); if \(E= \mathbb R^m\) we just say that \({\mathcal C}\) is fine; (b) it is a Vitali family if for every sub-family \({\mathcal C} ^*\) fine in \(E\) \(\exists\) disjoint \(C_i\), \(i=1,2,\ldots\), sets from \({\mathcal C} ^*\) such that \(| E\setminus \cup_i C_i| = 0\); (c) the upper \({\mathcal C}\)-derivative at \(x\) of the real-valued function \(G\) defined on \({\mathcal C}\) is \( \overline D_{\mathcal C} G(x)= \limsup_{r\to 0}\{{G(C)\over | C| }\); \(C\in{\mathcal C}\), \(x\in C\subset B(x,r)\}\); (d) the \({\mathcal C}\)-variation generated by \(G\) for \(E\) a Borel set, \(V_{\mathcal C}(G,E) =\lim_{\delta\to 0}\sup\sum_i| G(C_i)| \), where the sup is over all disjoint families \(\{C_i\); \(i=1,2,\ldots\}\) in \({\mathcal C}\) with \(C_i\cap E\neq\emptyset\), and \(d(C_i)< \delta\), \(i=1,2,\ldots\) . Now it is known that the upper derivative is infinite on sets of positive measure, and so the variation is also infinite on such sets. However, the behaviour of the variation on sets of measure zero is not known. The authors show that if \({\mathcal C}\) is a Vitali family the previous property is shared by many measures that are singular with respect to Lebesgue measure. More precisely, their main result is: if \({\mathcal C}\) is a fine Vitali family of measurable sets, \(G\) a finite-valued set function defined on \({\mathcal C}\) such that \(| \{x\); \(\overline D_{\mathcal C}G(x)= \infty\}| >0\) then for every increasing function \(\omega\) with \(\lim_{r\to 0} \omega(r)/r^m = \infty\), \(\exists \) a non-zero Radon measure \(\mu\) with the properties: (i) \(\mu\) is supported by a compact set \(N\) with \(| N| = 0\); (ii) \( \mu(B(x,r))\leq \omega(r)\), \(\forall x, r\); (iii) \(V_{\mathcal C}(G, E)= \infty\) if \(\mu_{e}(E)>0\).
scientific article; zbMATH DE number 1215752