Infinite variation and derivatives in \(R^m\)
type: Item
instance of: scholarly article
title: Infinite variation and derivatives in \(R^m\)
zbMATH Open document ID: 0927.28001
author: Benedetto Bongiorno
author: Luisa Di Piazza
author: David Preiss
published in: Journal of Mathematical Analysis and Applications
publication date: 1999-11-28T00:00:00Z
review text: 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 \(\mu_{e}(E)>0\).
Mathematics Subject Classification ID: 28A15
Mathematics Subject Classification ID: 26B30
zbMATH DE Number: 1215752
zbMATH Keywords: charges
zbMATH Keywords: variation
zbMATH Keywords: absolute continuity
zbMATH Keywords: Vitali family
zbMATH Keywords: upper derivative
zbMATH Keywords: gage integral
zbMATH Keywords: Radon measure
reviewed by: P. S. Bullen
MaRDI profile type: Publication
full work available at URL: jmaa.1998.5982
OpenAlex ID: W1979678727
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DOI: 10.1006/JMAA.1998.5982
description: scientific article; zbMATH DE number 1215752