In the archived version of Kolmogorov's Foundations of the Theory of Probability, at the very end of the book, p. $84,$ few books have been listed, one being E. Hopf's Ergodentheorie, where it is mentioned

Measure theoretic viewpoints are preferred over topological ones throughout because, as the author says, ergodic theory is statistics and statistics is measure theory.

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I have done preliminary googling but so far I didn't come across any helpful site or source that mentions his quote, the time (at least earlier than $1937,$ the publication year of the said book) he asserted that or the context.

So, the question is: when and where did E. Hopf assert the concerned statement?

  • $\begingroup$ Why do you think Hopf said it before 1937? The edition of Kolmogorov's Foundations where it is mentioned is from 1950, and Kolmogorov apparently refers to what is written in Hopf's book. $\endgroup$
    – Conifold
    Oct 6, 2022 at 4:49
  • $\begingroup$ @Conifold, by $1937, $ I am referring to the publication of Hopf's Ergodentheorie as written beneath the name in the snapshot I provided. $\endgroup$ Oct 6, 2022 at 5:39
  • $\begingroup$ The text seems to be a review of the book in which the reviewer quotes from Hopf's book so I assume that is where it is to be found. $\endgroup$
    – mdewey
    Oct 6, 2022 at 9:58
  • $\begingroup$ Okay @mdewey. I don't have any copy of this. So, if you are aware of it, you can post this as an answer. I would accept that. $\endgroup$ Oct 6, 2022 at 10:12

1 Answer 1


The blurb shown in the question quotes from a review of Eberhard Hopf's book Ergodentheorie, Berlin: Springer 1937, by Béla Szőkefalvi-Nagy in Acta Scientiarum Mathematicarum (Szeged), Vol. 9, No. 3, 1938-40, pp. 197-198 (volume online). The relevant sentence is on page 197 and reads as follows:

Maßtheoretische Gesichtspunkte nehmen in der ganzen Darstellung den Vorrang vor den topologischen ein, da, wie der Verfasser betont, Ergodentheorie Statistik ist und Statistik Maßtheorie.

The language pretty clearly indicates that the reviewer is quoting Hopf from the book under review, but I do not have access to the book to verify that. I am making this a community wiki answer to allow easy addition of this information at a later stage.

  • $\begingroup$ I have been able to find Hopf's book soft copy. Since it is not in English, I would need some time. Thanks for the answer, njuffa. I will let you know. $\endgroup$ Oct 6, 2022 at 11:05
  • $\begingroup$ Update: using google translate, I have been able to trace down the line "Statistik ist Maßtheorie" in Vorwort (forward). But the other parts of the quote is not present, at least along with it. $\endgroup$ Oct 6, 2022 at 11:22
  • $\begingroup$ I google translated ergodic theory is statistics into Ergodentheorie ist Statistik and searched it in the copy but didn't find any match. $\endgroup$ Oct 6, 2022 at 11:34

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