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Derivates of interval functions. (English) Zbl 0734.26003

Mem. Am. Math. Soc. 452, 96 p. (1991).
As the author notes the intention of this memoir is to introduce a new perspective on certain applications of the Vitali covering theorem. The main notions are a covering relation \(\beta\) and variational measures defined as follows. A covering relation \(\beta\) on a set E of real numbers is a collection of pairs \((I,x)\) where I is a closed interval and \(x\in (I\cap E),\) while a packing is a covering relation \(\pi\) with the property that for distinct pairs \((I_ 1,x_ 1)\) and \((I_ 2,x_ 2)\) belonging to \(\pi\) the intervals \(I_ 1\) and \(I_ 2\) do not overlap. A covering relation \(\beta\) is said to be full at a point x provided that there exists a \(\delta >0\) so that \(([y,z],x)\in \beta\) for every \(y<x<z\) with \(0<z-y<\delta.\) Such a relation is said to be full covering relation on a set E if it is full at each point of E. The other essential notion is that of a fine covering relation on a set E which coincides with the more familiar notion of a Vitali covering. For an interval point function h which assigns a real value to each \((I,x)\in \beta\) the variation of h relative to \(\beta\) is defined as \[ Var(h,\beta)=\{\sum_{(I,x)\in \pi}| h(I,x)|; \pi \beta,\;\pi \text{ a packing}\}. \] The full and the fine variational measures \(h^*\) and \(h_*\) for a set E are the \[ h^*(E)=\inf \{Var(h,\beta);\;\beta \text{ is a full covering relation on } E\} \] and \[ h_*(E)=\inf \{Var(h,\beta);\;\beta \text{ is a fine covering relation on } E\}. \] These become metric outer measures and if \(h((I,x))=| I|^ s\) for some s with \(0<s<1\) the measure \(h_*\) is the s-dimensional Hausdorff measure and \(h^*\) is a kind of s-dimensional packing measure. The author then investigates the relations of these metric outer measures to all of the general theory of derivation and integration on the real line. Especially, he gives a new perspective on old results of Rogers and Taylor from 1961 concerned with determining the nature of functions that are absolutely continuous with respect to the Hausdorff measures. This now includes a parallel characterization of the functions that are absolutely continuous with respect to some s-dimensional packing measure of Tricot.
Remark. The packing measure \(\mu^ s\) considered on page 65, Theorem 6.5 is not the usual s-dimensional packing measure obtained by packings with closed centered balls since a full covering \(\beta\) allows to have packings \(\pi \subset \beta\) such that for \((I,x)\in \pi\) the point x may be near as possible to the endpoints of I.

MSC:

26A21 Classification of real functions; Baire classification of sets and functions
26A24 Differentiation (real functions of one variable): general theory, generalized derivatives, mean value theorems
28A78 Hausdorff and packing measures
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