Lévy to Jessen

Paris, 38 Avenue Théophile Gautier (16^o), 27 September 1934.

Dear Sir,

Thank you for your friendly letter. As soon as I have finished writing up some results that I have had already for several months, I will not fail to look into those works of yours that you have pointed out to me. Unfortunately I am a little discouraged by the difficulty of researching ζ(s); I have not obtained any important result on this question (1). I was happier with functions of infinitely many variables.

Professor Bohr spoke to me about your work, and what he said interested me very much. To tell you the truth, your theory of measure and that of H. Steinhaus have been familiar to me for a long time, perhaps since 1920 (2). For me they are elementary concepts that I specify as I need them. But in the use you have made of them, you have gone far beyond what I knew. I informed M. Denjoy of your communication to the Oslo Congress (3).

He has just re-discovered the theory of measure (Note of June 6, 1933 to the Académie des Sciences) (4) and I have cited it in an article which I have just finished drafting and which will appear in the Bulletin des Sciences Mathématiques (5). It is a pity that I do not know Danish and cannot read the more elaborate report that M. Bohr has given me (6).

I am presenting a note to the Academy summarising my new memoir (7). It supplements, and corrects the summary I gave to the Société Mathématique de France on May 23, 1934, and which I have sent to M. Bohr and to M. Lublin (8). I also point out that, in my paper in Studia Mathematica, theorems XI and XII are true, not for , but for

, being, for each n, a suitably given constant which can be the term of a semi-convergent series

x_n

∑

x_n−a_n

( )

∑

(9). I noticed this error only while returning from Denmark, so that it is not corrected on the copies I left at the Copenhagen Institute.

I would like to ask you to remember me to all your colleagues I saw in Copenhagen, M.M. Norlund, Bohr, Steffensen, Petersen, Bonnessen, Mollerup,… (10) but they are too

many and I cannot really ask. I have excellent memories of the days I spent in Copenhagen, and regret not having met you there. But I hope to see you one day in Paris.

In the meantime trust in my devoted feelings.

P. Levy 2. Lévy to Jessen.

Paris, 4 April 1935.

Dear Sir

I think of interesting you by sending the proofs of a memoir which soon will appear and which is related to your own work. It was written last summer and sent to the editor of the Bulletin des Sciences Mathématiques before you sent me your article from Acta Mathematica. I could only indicate one of the common points in a note added afterwards (1).

For a few days now I have been occupied in studying your article more completely. I have to give an account of it in the presence of M. Hadamard and I see that the common points between your ideas and mine are even more numerous than I had thought (2).

My lemma I is fundamentally the same as the theorem in your § 14 (Representation of function as limit of an integral). Only I establish it directly (3). Your “important lemma” of § 11 is then a special case of mine; and your theorem of § 13 is obtained readily enough from my lemma I.

Also I think that you are unaware of a lecture I gave to M. Hadamard’s seminar in January 1924; it appeared in the Revue de Métaphysique et de Morale, and I reproduced it in my Calcul des Probabilités (p. 325-345) (4). Re-reading it recently, I found that, as well as an error on page 330 (l. 12 to 19), which M. Steinhaus pointed out to me (5), there is something unfortunate on p. 332 (l. 5), (6). In spite of this, I introduced at that time the ideas that M.

Steinhaus and you have developed and clarified, without you suspecting that some were already in my article of 1924—and even in a course I gave in 1919 (7). What I call a partition corresponds to what you call “construction of nets” (8). I indicate it for abstract sets, and then (p. 334, l. 6 to 13) I indicate the means of realising it for the cube in an infinite number of dimensions (9); it is what you have done. As for your “transferring principle”, I was not very explicit in the article, but when I wrote p. 332 (towards the bottom) that one can carry out the image of the partition on a segment of right-hand side, it was of this principle that I was thinking (10). Indeed a partition is not simply an unspecified subdivision of the relevant set, but a subdivision where each cell has a weight and which leads to one definition of the probability (or, if you prefer, of measure). I recognise that I should have been more explicit.

Perhaps I was at the seminar; after 11 years I am no longer sure (11). I am sure that I knew this result which seemed so obvious to me that it was enough to indicate it by a word. On the other hand, is only very recently, in particular after reading the paper by Steinhaus (Studia t.

II) and your communication to the Oslo congress of 1929 (12), that I saw how the principle could be used for denumerable probabilities.

In any case it clearly follows from my article that the principles of the theory of measure in any set are those of M. Lebesgue.

Of course the very brief indications in my article are not always sufficient, and your very complete study remained necessary. Besides it taught me many things that I did not know (in particular § 3; before reading it I had not thought that it of interest to specify whether one considered open or closed intervals; and I had never studied the representation of the measure in Qω by the symbol of an integral of infinite order (13); and I do not speak of the application to Fourier series which I have only just begun to study).

Excuse this slightly long letter. Like Steinhaus, who, however, knew my 1924 article and seemed in no doubt what it contained, I think it cannot always be very clear, and that it is of interest if I indicate to you explicitly the points which are connected to your work.

Believe, dear Sir, with my most cordial feelings.

P. Levy Please recall me to the memory of M. Harald Bohr.

Note attached: (14)

We apply lemma I, using E to denote the inequality ^h

2^p ≤ f x

( )

Applying this result for h=0,1,...,2^p−1, one sees that the previous inequality holds, except in the case of probability <ε_p, for all n>N_p.

Put p=1,2,... ; ε_p = ε

2^p. We see that, except in the case of probability

<

∑

As ²

2^p +ε_p is arbitrarily small, we have the theorem from § 13 of M. Jessen.

4/4/35 P. Lévy