The Lebesgue differentiation theorem revisited

We prove a general version of the Lebesgue differentiation theorem where the averages are taken on a family of sets that may not shrink nicely to any point. These families of sets involve the unit ball and its dilated by negative integers of an expansive linear map. We also give a characterization o...

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Bibliographic Details
Published inExpositiones mathematicae Vol. 37; no. 3; pp. 322 - 332
Main Authors Dubon, E., San Antolín, A.
Format Journal Article
LanguageEnglish
Published Elsevier GmbH 01.09.2019
Subjects
[formula omitted]-approximate continuity
[formula omitted]-density point
Expansive linear maps
Lebesgue differentiation theorem
Lebesgue measurable functions
Lebesgue differentiation theorem
A-approximate continuity
Expansive linear maps
Lebesgue measurable functions
A-density point
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Summary:We prove a general version of the Lebesgue differentiation theorem where the averages are taken on a family of sets that may not shrink nicely to any point. These families of sets involve the unit ball and its dilated by negative integers of an expansive linear map. We also give a characterization of the Lebesgue measurable functions on Rn in terms of approximate continuity associated to an expansive linear map.
ISSN:0723-0869
1878-0792
DOI:10.1016/j.exmath.2019.02.001