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A negative result in differentiation theory. (English) Zbl 0531.28006

The author proves the following generalization of Saks’ rarity theorem: Let \({\mathcal B}\) be a homothecy invariant Busemann-Feller differentiation basis in \({\mathbb{R}}^ m\), \(H(r)=\cup \{R\in {\mathcal B}:\quad R\subset B_ 1,\quad | R\cap B_ r| /| R|>r\}\) where \(B_ r\) is the ball with radius r around 0, let \(| X|\) denote Lebesgue measure, \({\tilde \Phi}(u)=u^ m| H(1/u)|,\) and \(\Psi\) :(1,\(\infty)\to {\mathbb{R}}\) be increasing and convex. Then every f from the Orlicz class \(\Psi\) (L), except those in a set of the first category in \(\Psi\) (L), verifies that, for every rotation \(\gamma\) of \({\mathbb{R}}^ m\), the upper derivate of \(\int f\) relative to the basis \({\mathcal B}_{\gamma}\), obtained by rotating \({\mathcal B}\) through \(\gamma\), is \(+\infty\) almost everywhere.
Reviewer: Á.Császár

MSC:

28A15 Abstract differentiation theory, differentiation of set functions
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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