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

G. A. Karagulyan (2007)

Studia Mathematica

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 ) 0 , x R s | R | - 1 R f = almost everywhere if s ∈ S. If the condition D ̅ s f ( x ) = 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 ).

Density covers

A. M. Bruckner, Charles L. Thorne (1973)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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