Measurable functions are of bounded variation on a set of dimension 1/2
نویسنده
چکیده
We show that for every Lebesgue measurable function f : [0, 1] → R there exists a compact set C of Hausdorff dimension 1/2 such that f is of bounded variation on C, and there exist compact sets Cα of Hausdorff dimension 1−α such that f is Hölder-α on Cα (0 < α < 1). These answer questions of M. 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.
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