Author’s abstract: “The paper discusses the tension which occurred between the notions of set (with measure) and (trial-) sequence (or – to a certain degree – between nondenumerable and denumerable sets) when used in the foundations of probability theory around 1920. The main mathematical point was the logical need for measures in order to describe general nondiscrete distributions, which had been tentatively introduced before (1919) based on von Mises’s notion of the “Kollektiv” [R. von Mises, Math. Z. 5, 52–99 (1919; JFM 47.0483.01); correction ibid. 7, 323 (1920; JFM 47.0484.01)]. In the background there was a tension between the standpoints of pure mathematics and “real world probability” (in the words of J. L. Doob [Am. Math. Mon. 103, No. 7, 586–595 (1996; Zbl 0865.01011)]) at the time. The discussion and publication in English translation (in the Appendix) of two critical letters of November 1919 by the “pure” mathematician Felix Hausdorff to the engineer and applied mathematician Richard von Mises compose about one third of the paper. The article also investigates von Mises’s ill-conceived effort to adopt measures and his misinterpretation of an influential book of Constantin Carathéodory [C. Carathéodory, Vorlesungen über reelle Funktionen. Leipzig und Berlin: B. G. Teubner (1918; JFM 46.0376.12)]. A short and sketchy look at the subsequent development of the standpoints of the pure and the applied mathematician – here represented by Hausdorff and von Mises – in the probability theory of the 1920s and 1930s concludes the paper.”