A complete characterization of R-sets in the theory of differentiation of integrals

G. A. Karagulyan

Studia Mathematica, Tome 178 (2007), p. 17-32 / Harvested from The Polish Digital Mathematics Library

Résumé

Let $_{s}$ be the family of open rectangles in the plane ℝ² with a side of angle s to the x-axis. We say that a set S of directions is an R-set if there exists a function f ∈ L¹(ℝ²) such that the basis $_{s}$ differentiates the integral of f if s ∉ S, and $D ̅_{s} f (x) = l i m s u p_{d i a m (R) \to 0, x \in R \in_{s}} {| R |}^{- 1} \int_{R} f = \infty$ almost everywhere if s ∈ S. If the condition $D ̅_{s} f (x) = \infty$ holds on a set of positive measure (instead of a.e.) we say that S is a WR-set. It is proved that S is an R-set (resp. a WR-set) if and only if it is a $G_{δ}$ (resp. a $G_{δ σ}$ ).

Publié le : 2007-01-01

Zbl 1284.42056

EUDML-ID : urn:eudml:doc:284928

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G. A. Karagulyan. A complete characterization of R-sets in the theory of differentiation of integrals. Studia Mathematica, Tome 178 (2007) pp. 17-32. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm181-1-2/