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Egoroff’s theorem and maximal run length. (English) Zbl 1170.28001
Summary: We construct a sequence of measurable functions converging at each point of the unit interval, but the set of points with any given rate of convergence has Hausdorff dimension one. This is used to show that a version of Egoroff’s theorem due to Taylor is best possible. The construction relies on an analysis of the maximal run length of ones in the dyadic expansion of real numbers. It is also proved that the exceptional set for a limit theorem of Renyi has Hausdorff dimension one.

MSC:
28A20 Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence
28A80 Fractals
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