MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
Note that I also posted this question on math.SE:… – Quinn Culver Jan 27 '12 at 18:41
up vote 2 down vote accepted

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).

share|cite|improve this answer

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.

share|cite|improve this answer
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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.