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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 $D ̅_{s} f (x) = l i m s u p_{d i a m (R) \to 0, x \in R \in_{s}} {| R |}^{- 1} \int_{R} f = \infty$ 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 $G_{δ σ}$ ).

A Decomposition Theorem for Additive Set-Functions, with Applications to Pettis Integrals and Ergodic Means.

Michel Talagrand, David H. Fremlin (1979)

Mathematische Zeitschrift

A derivative to match the v-integral.

S.G. Wayment, L. Hatta (1972)

Journal für die reine und angewandte Mathematik

A general result on the equivalence between derivation of integrals and weak inequalities for the Hardy-Littlewood maximal operator

Ireneo Alonso (1977)

Studia Mathematica

We prove an extension of a result by Peres and Solomyak on almost sure absolute continuity in a class of symmetric Bernoulli convolutions.

Some notions of limit weaker than the topological one are studied.

Displaying 1 – 16 of 16

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

A Decomposition Theorem for Additive Set-Functions, with Applications to Pettis Integrals and Ergodic Means.

A derivative to match the v-integral.

A general result on the equivalence between derivation of integrals and weak inequalities for the Hardy-Littlewood maximal operator

A generalization of Rademacher's theorem on complete differential

A generalization of the Lagrange mean value theorem to the case of vector-valued mappings.

A martingale proof of the theorem by Jessen, Marcinkiewicz and Zygmund on strong differentiation of integrals

A negative result in differentiation theory

A note on angular semigroups.

A Precise Vitali Theorem for Lebesgue Measure.

A Radon-Nikodym theorem for the v-integral.

Adjoint differential inclusions in necessary conditions for the minimal trajectories of differential inclusions

Almost sure absolute continuity of Bernoulli convolutions

An analytic study on the self-similar fractals: Differentiation of integrals.

An inequality for the Hardy-Littlewood maximal operator with respect to a product of differentiation bases

Approximative limit and related notions

Currently displaying 1 – 16 of 16