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Measurable functions are of bounded variation on a set of dimension ½
Author(s) -
Máthé András
Publication year - 2013
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms/bds106
Subject(s) - mathematics , hausdorff dimension , bounded variation , bounded function , packing dimension , hausdorff measure , lebesgue integration , hausdorff space , dimension function , effective dimension , compact space , dimension (graph theory) , lebesgue measure , minkowski–bouligand dimension , limit set , measurable function , limit (mathematics) , measure (data warehouse) , outer measure , discrete mathematics , pure mathematics , mathematical analysis , fractal dimension , fractal , database , computer science
We show that for every Lebesgue measurable function f :[0, 1] → ℝ there exists a compact set C of Hausdorff dimension ½ such that f is of bounded variation on C , and there exist compact sets C α of Hausdorff dimension 1 − α such that f is Hölder‐ α on C α (0 < α < 1). These answer questions of Elekes, which were open even for continuous functions f . Our proof goes by defining a discrete Hausdorff pre‐measure on ℤ, solving the corresponding discrete problems, and then finding suitable limit theorems.