Abstract
These ten letters are mainly about two erroneous theorems published by Joseph Doob in 1938. Both concerned probability measures on abstract spaces. The first concerned the construction of measures on infinite-dimensional product spaces from their finite-dimensional margins (the Daniell-Kolmogorov construction). The second concerned the existence of regular conditional probabilities. Ten years after the theorems were published and used by Doob’s students, counterexamples were independently discovered by Jean Dieudonné and by Erik Sparre Andersen in collaboration with Børge Jessen. The correspondence begins when Jessen writes to Doob about the counterexample he and Sparre Andersen had discovered. It reveals that Doob had not encountered Jessen’s theorem until then, and it suggests that Doob’s return to martingales, which he had left aside after his initial work on them in 1940, was inspired by his learning about Jessen’s work on the topic.
Introduction and footnotes translated from the French by John Aldrich, University of Southampton e-mail: john.aldrich@soton.ac.uk
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Notes
- 1.
We follow Jessen and other contemporaries in referring to “Sparre Andersen” as if this were his last name. Other sources (the yearbooks of the Danish Academy of Sciences, for example) give his name is given as “Andersen, E. S.”, and we follow this practice in our list of references.
- 2.
See the chapter by Bernard Locker in the present volume.
- 3.
This refers to the first theorem in Doob’s 1938 article [13]. The article in which Sparre Andersen and Jessen questioned it, [3], was submitted on 5 October 1945 (before postal service between America and Denmark was restored) and printed on 1 April 1946. Sparre Andersen and Jessen wrote (p. 22, Sect. 24, note 1):
An analogous theorem on arbitrary measures in product sets has been given by Doob, but his proof seems incomplete (it is not seen how the sets on p. 92 are chosen). The proof by Sparre Andersen of a more general theorem is incomplete...
Sparre Andersen had published his more general erroneous theorem in 1944 [2].
It is possible that Doob objected to the 1946 criticism, and that Jessen wanted to be apologize, but we have found no proof of this. Doob himself was not sparing in writing notes of this kind (see e.g. the notes on pp. 91 and 135 of [13]), and Doeblin (who was at least as tough as Doob) complained to him about it; see Doeblin’s correspondence with Doob in [11]. In any case, the letter’s main point is the counterexample discovered in the spring of 1948 by Sparre Andersen and Jessen, which left in no doubt the irreparable inaccuracy of Doob’s theorem.
- 4.
Here we omit a passage that reproduces word for word the description of the counterexample as it appears in the Sparre Andersen’s and Jessen’s 1948 article [4, Sect. 4]. The article was submitted on 28 June 1948, after Doob’s reply.
- 5.
The “first theorem” in question here and in the following letter is Theorem 1.1 in Doob’s 1938 article [13, p. 90], which extends to a general abstract framework the Daniell-Kolmogorov theorem for the real case or for the abstract case for product probabilities (the case of independent variables). The “following theorem” (or the “second theorem” lower down), is Theorem 3.1 on page 96, which asserts the existence of abstract disintegrations (or the existence of regular conditional probabilities), and this is equally erroneous. Jessen’s counterexample also works for this theorem, which Doob already knows is false from an example of Dieudonné’s that Halmos had communicated to him (see the letter in Sec. 5 below). But until Jessen’s first letter arrived, Doob still thought that his Daniell theorem was correct.
- 6.
See Kakatuni’s [24, I], which has a very simple treatment of the independent case. S. Kakutani (1911–2004) was at the Princeton Institute for Advanced Study between 1940 and 1942, and it was there that he learned Doob’s erroneous general theorem.
- 7.
The formulation Doob proposes here is a version of the very general result, without topological assumptions, that was published in 1949 by C. Ionescu Tulcea. Neveu [29, V-1] has a presentation of this theorem, which is adequate for the general theory of Markov chains, and of which Jessen said in the following letter that he had become convinced “in the course of my attempts to prove this theorem [the theorem of Daniell-Doob]”.
- 8.
The search for the minimal topological assumptions ensuring the validity of Doob’s disintegration theorem produced an abundance of literature in the 50s and later. See the very numerous references in [8, Vol. II, p. 462].
- 9.
From this last sentence it can safely be concluded that by 17 May 1948 Doob had read Sparre Andersen and Jessen’s 1946 article [3] and undoubtedly also Jessen’s articles from the 30s, and had understood that they contained a satisfactory version of his first theory of 1940 [14], which he was not yet calling the theory of martingales.
- 10.
Such a note is indeed in the final paragraph of Sect. 3 of Sparre Andersen and Jessen’s [4]. Yet as we will see, the project of a joint Doob-Jessen article did not materialize (see the letter in 11 below). This consolation prize was obviously not much motivation for Doob, and Jessen rather quickly realized that it held no interest to him, except as a way of expressing his sympathy for a colleague he had put into difficulty.
In any event, Doob preferred to work alone at home and published very little with others. In his interesting conversation with Snell [31], he says:
I corresponded with many mathematicians but never had detailed interplay with any but Kai Lai Chung and P.- A. Meyer in probability and Brelot in potential theory. My instincts were to work alone and even to collect enough books and reprints so that I could do all my work at home.
- 11.
William Feller (1906–1970) fled Nazi Germany in 1933 for Denmark and Sweden, before emigrating in the United States in 1939. See [7] for Feller’s links with Danish mathematicians.
- 12.
Paul Halmos (1916–2006) was Doob’s first doctoral student in Urbana, and he defended his Ph.D. thesis in 1938. His 1941 article [23] was based on Doob’s theorem of disintegration; it is referred to below in the correspondence with Dieudonné. See also Halmos’s autobiography [22] and Burkholder and Protter’s obituary of Doob [9].
Warren Ambrose (1914–1996) completed his Ph.D. under Doob’s direction in 1939. He used Doob’s theorem in [1].
For Kakutani, see footnote 6 above. He used Doob’s theorem in [24, II].
The brilliant careers of these three mathematicians do not seem to have especially suffered from this unfortunate mistake.
- 13.
Jessen’s proof, probably dating back to 1934–1935, was published in Danish in 1939 and in English in 1946.
- 14.
Reference [28].
- 15.
- 16.
- 17.
Jessen reconsiders this point in his 1948 article with Sparre Andersen [5, Sect. 1]. He explains there that he was unaware of Doob’s 1940 work [14] in 1946, and that while he had preferred to adopt the viewpoint of set functions, this had been only for convenience of exposition, the results being just as valid for point functions. The 1948 article makes this explicit, making the two theorems perfectly symmetrical so that they include Doob’s 1940 results. The 1948 article was received by the journal on 16 August 1948, and published on 23 October. It is discussed in the letter in Sect. 9 below.
- 18.
Doob and Feller met for the first time at the meeting of the American Mathematical Society at Darmouth in 1940 [31]. He was, according to Doob “the first mathematical probabilist I ever met.” Doob and Feller had very different visions of mathematics. To be convinced, simply compare Doob’s 1953 book and Feller’s 1950 book [17, 21]. Doob did not like calculations and looked for the most general possible results and concepts. Feller loved only formulas and the rare and precious flowers that appear only after complicated calculations and particularly meticulous investigations. That did not prevent them from getting together to try to convince American mathematicians that the theory of probability was a branch of mathematics like any other, contrary to general belief. See also [15, 18, 19].
- 19.
Reference [4].
- 20.
This is a reference to the Nancy Conference of June 1947 (see Jessen’s letter to Paul Lévy, dated 13 September 1948, in the collection of correspondence between the two of them in the present volume). Dieudonné was in Brazil at the time.
- 21.
- 22.
This letter and the following one are dated from Chicago. These are incomplete drafts and were doubtless altered at the time of sending. From these letters, it seems that the Jessen-Doob meeting took place in Urbana in the first fortnight of May 1949, without much scientific result, and that the two men met again at the end of June 1949, after Jessen had given his course in Chicago.
- 23.
This refers no doubt to Doob’s oldest son Steve. Burkholder and Protter [9] describe Doob’s life in Urbana.
- 24.
Gene Autry (1907–1998), “the singing cowboy”, was a famous American actor and singer.
- 25.
Turkey Run is a national park in Indiana.
Gerhard Hochschild was born in Berlin in 1915. An important algebraist, he was a student of Chevalley at Princeton. In 1949 he was professor at Urbana, before being appointed to Berkeley.
The draft ends in a badly written, crossed out sentence: “Please give my kind regards to the Cairns, Landen?” Stewart S. Cairns (1904–1982), a student of Marston Morse at Harvard, was mathematics professor at Urbana from 1948 to 1972.
- 26.
Alberto Calderón (1920–1998), a mathematician of Argentinian origin, was one of the most important analysts of the 20th century. Discovered by A. Zygmund at a conference in Buenos Aires in 1949, he followed Zygmund to the USA, where he spent his whole career. In Chicago, where he was appointed in 1959, he and Zygmund established an important school of analysis. See [10] for a glimpse of his work and his influence.
- 27.
Leopoldo Nachbin (1922–1993), a brilliant Brazilian mathematician, was a student of Dieudonné and Weil when they taught in São Paulo. He was a professor in Rio de Janeiro and then in Rochester. In 1949 he was a Guggenheim foundation fellow. See [6].
- 28.
Here there is a fragment of a crossed out phrase: “If we decide to drop the matter I...”
- 29.
Thus ended the collaboration of Doob and Jessen.
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Acknowledgements
We are very grateful to Christian Berg, who obtained these letters for us from the Jessen archives at the Institute for Mathematical Sciences of the University of Copenhagen.
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Bru, B., Eid, S. (2022). Counterexamples to Abstract Probability: Ten Letters by Jessen, Doob and Dieudonné. In: Mazliak, L., Shafer, G. (eds) The Splendors and Miseries of Martingales. Trends in the History of Science. Birkhäuser, Cham. https://doi.org/10.1007/978-3-031-05988-9_16
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