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Nondifferentiable functions, Haar null sets and Wiener measure. (English) Zbl 0981.26004

Summary: Recently, B. R. Hunt [Proc. Am. Math. Soc. 122, No. 3, 711-717 (1994; Zbl 0861.26003)] proved that the set \(S\) of continuous functions \(f\in C[0,1]\) which have a finite derivative at a point \(x\in (0,1)\) is Haar null in Christensen’s sense. Let \(\mu\) be the Wiener measure on \(C[0,1]\). We show that a natural very slight modification of the well-known simple proof of A. Dvoretzky, P. Erdős and S. Kakutani [Proc. 4th Berkeley Symp. Math. Stat. Probab. 2, 103-116 (1961; Zbl 0111.15002)] that \(\mu(S)= 0\) gives that \(\mu(S+ f)= 0\) for each \(f\in C[0,1]\), which gives Hunt’s result. A related conjecture concerning Jarník points is formulated.

MSC:

26A27 Nondifferentiability (nondifferentiable functions, points of nondifferentiability), discontinuous derivatives
46G12 Measures and integration on abstract linear spaces
60J65 Brownian motion
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