# Kolmogorov's example of a measurable function not (generally) differentiable

In [1, page 7], the author says.

Kolmogorov showed that if the function $$f(x) = \sum_{n=1}^{\infty} \frac{\cos 3^n x}{3^n}$$ has a finite or infinite generalized derivative on a set of positive measure, then the function is nonmeasurable.

Where can I find a proof/explanation of this result (and/or other similar results) in English? The reference doesn't have to be to Kolmogorov's orignal paper; for example, a modern exposition would suffice (and might very well be better).

[1] A. N. Shiryaev, "Andrei Nikolaevich Kolmogorov", Theory of Probability and Applications, vol 34, no. 1, 1988.

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Note that I also posted this question on math.SE: math.stackexchange.com/questions/103000/… –  Quinn Culver Jan 27 '12 at 18:41
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## 2 Answers

A translation in english is in "Selected Works of A.N. Kolmogorov I" : "On the possibility of a general definition of derivative, integral and summation of divergent series" (page 33 and 34).

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Kolmogorov's original paper:

'Sur la possibilite de la definition generale de la derivee, de l'integrale et de la sommation des series divergentes', C.R. Acad. Sci. Paris 180(1925), 362-364.

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Thanks. I suppose that reference is in French, which I cannot read. Do you know if there's an English translation? –  Quinn Culver Jan 27 '12 at 15:38
Yes, see: V.M. Tikhomirov (Ed.): Selected works of A.N. Kolmogorov, Vol.1, Kluwer, pp.51-52 –  Ljubomir Cukic Jan 29 '12 at 8:22
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