Lebesgue published his celebrated integral in 1901-1902. Poincaré passed away in 1912, at full mathematical power.

Of course, Lebesgue and Poincaré knew each other, they even met on several occasions and shared a common close friend, Émile Borel.

However, it seems Lebesgue never wrote to Poincaré and, according to Lettres d’Henri Lebesgue à Émile Borel, note 321, p. 370

… la seule information, de seconde main, que nous avons sur l’intérêt de Poincaré pour la « nouvelle analyse » de Borel, Baire et Lebesgue

the only second-hand information we have on Poincaré's interest in the "new analysis" of Borel, Baire and Lebesgue

is this, Lebesgue to Borel, 1904, p. 84:

J’ai appris que Poincaré trouve mon livre bien ; je ne sais pas jusqu’à quel point cela est exact, mais j’en ai été tout de même très flatté ; je ne croyais pas que Poincaré sût mon existence.

I learned that Poincaré finds my book good; I do not know to what extent that is accurate, but I nevertheless was very flattered; I did not believe that Poincaré knew of my existence.

See also note 197, p. 359

Nous ne connaissons aucune réaction de Poincaré aux travaux de Borel, Baire et Lebesgue.

We do not know any reaction of Poincaré to the works of Borel, Baire and Lebesgue.

To my mind this situation is totally unexpected, almost incredible: the Lebesgue integral and measure theory are major mathematical achievements but Poincaré, the ultimate mathematical authority at this time, does not say anything??? What does it mean?

So, please, are you aware of any explicit or implicit statement by Poincaré on the Lebesgue integral or measure theory?

If you are not, how would you interpret Poincaré’s silence?

Pure disinterest? Why? Discomfort? Why? Something else?

This question is somewhat opinion-based, but

The true method of forecasting the future of mathematics is the study of its history and current state.

according to Poincaré and his silence is a complete historical mystery, at least to me.

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    $\begingroup$ Well, at that time Poincaré was working on many subjects different from integration theory (relativity, analysis situs, dynamical systems, math foundations), so maybe he did not have the time to write extensively about Lebesgue's new theory. $\endgroup$ – Francesco Polizzi Nov 5 '19 at 6:33
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    $\begingroup$ @FrancescoPolizzi Thanks. Yes, Poincaré was quite busy :) but not a single statement between 1904 and 1912 at least about such important works is really hard to understand, precisely because when Poincaré is not happy (e.g. Cantor set theory), he's not reluctant to tell it. Moreover, see e.g. p. 260, Lebesgue to Borel, 1910: "I met Poincaré only to talk about Drach.". It seems like Lebesgue even never discussed his works privately with Poincaré! Same for Borel, apparently. Crazy story. $\endgroup$ – Fabrice Pautot Nov 5 '19 at 8:47
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    $\begingroup$ @ Polizzi "first non-Lebesgue-misurable set" do you mean miserable or measurable ? :) $\endgroup$ – meh Nov 5 '19 at 21:45
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    $\begingroup$ @aginensky misurabile :) $\endgroup$ – R W Nov 5 '19 at 21:49
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    $\begingroup$ It seems to me that we can only speculate to answer the question. I, on the contrary, think that there's nothing unusual in what you describe. Even nowadays not all the big names working in a field talk actively to each other. $\endgroup$ – user347489 Nov 7 '19 at 1:51

It has nothing to do with the conflict with Borel which developed later, and one can find a pretty explicit answer in the aforementioned letters of Lebesgue to Borel.

(These letters were first published in 1991 in Cahiers du séminaire d’histoire des mathématiques; selected letters with updated commentaries were also published later by Bru and Dugac in an extremely interesting separate book.)

In letter CL (May 30, 1910) Lebesgue clearly states:

Poincaré m'ignore; ce que j'ai fait ne s'écrit pas en formules.

Poincaré ignores me, [because] what I have done can not be written in formulas.

EDIT In interpreting this statement of Lebesgue I trust the authority of Bru and Dugac who in "Les lendemains de l'intégrale" accompany this passage with a footnote (missing in the 1991 publication) stating that

Dans [the 1908 ICM address] Poincaré ne semble pas considérer l'intégrale de Lebesgue comme faisant partie de "l'avenir des mathématiques", puisqu'il ne mentionne pas du tout la théorie des fonctions de variable réelle de Borel, Baire et Lebesgue.

In [the 1908 ICM address] Poincaré does not seem to consider the Lebesgue integral as a part of the "future of mathematics", as he does not mention at all the theory of functions of a real variable of Borel, Baire and Lebesgue.

I would rather interpret the meaning of "formulas" in the words of Lebesgue in a more straighforward and naive way. It seems to me that he was referring to the opposition which was more recently so vividly revoked by Arnold in the form of "mathematics as an experimental science" vs "destructive bourbakism".

By the way, it is interesting to mention that the first applications of the Lebesgue theory were - may be surprisingly - not to analysis, but to probability (and the departure point of Borel's Remarques sur certaines questions de probabilité, 1905 is clearly and explicitly the first edition of Poincaré's "Calcul des probabilités"). Poincaré had taught probability for 10 years and remained active in this area (let me just mention "Le hasard" that appeared first in 1907 and then was included as a chapter in "Science et méthode", 1908 and the second revised edition of "Calcul des probabilités", 1912), and still he makes no mention of Lebesgue's theory. This issue has been addressed, and there are excellent articles by Pier (Henri Poincaré croyait-il au calcul des probabilités?, 1996), Cartier (Le Calcul des Probabilités de Poincaré, 2006, the English version is a bit more detailed) and Mazliak (Poincaré et le hasard, 2012 or the English version). To sum them up,

[Poincaré's] seemingly limited taste for new mathematical techniques, in particular measure theory and Lebesgue’s integration, though they could have provided decisive tools to tackle numerous problems (Mazliak)

is explained by his approach of

a physicist and not of a mathematican (Cartier)

to these problems.

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    $\begingroup$ Very nice answer. Are you French RW? I've already read Poincaré, Borel, Cartier and Mazliak (who's doing a great job) but not Pier and Bru (who's doing a great job too and I should read his book). In fact, by a probabilistic reasoning (probability theory belonging to mathematical physics according to Poincaré), I guessed that Poincaré the physicist and philosopher of science could not welcome the Lebesgue integral and measure theory, and I was not surprised at all to check a posteriori that he never said anything about them, probably in order to preserve his friendship with Borel... $\endgroup$ – Fabrice Pautot Nov 6 '19 at 21:54
  • $\begingroup$ Therefore, I will answer my own question. Hope you will appreciate... $\endgroup$ – Fabrice Pautot Nov 6 '19 at 21:56
  • $\begingroup$ Your input is CRUCIAL. I'd like to discuss with you more extensively if ever possible please. May I recap the situation like this: for some reasons to be understood, Poincaré could ESPECIALLY not welcome the Lebesgue integral because he was also a physicist and, in particular (he used to conceive le calcul des probabilités as a branch of mathematical physics), a probabilist... and a philosopher of science. But since Kolmogorov Grundbegriffe (and even Borel 1905 paper as you pointed it out), probability theory, for mathematicians, physicists and statisticians, relies on the Lebesgue integral. $\endgroup$ – Fabrice Pautot Nov 9 '19 at 0:20
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    $\begingroup$ Concerning the statement that "since Kolmogorov ... probability theory ... relies on the Lebesgue integral". Alas, this is quite far from the reality. Most probabilists don't know measure theory, are afraid of using it, and, in particular, avoid dealing with probability measures (!). There are several very instructive articles by Doob about it written in the 90s (for instance, Probability vs. Measure which he concludes by saying that "adherents of each aspect [of probability] are human and therefore scorn adherents of the other"). I could give more recent examples as well... $\endgroup$ – R W Nov 9 '19 at 1:54
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    $\begingroup$ Yes - I agree - just wanted to emphasize that the reality is somewhat different from what it is supposed to be :) Have a look at Doob's articles - they are quite entertaining. $\endgroup$ – R W Nov 9 '19 at 2:03

The main applications of Lebesgue integral to concrete problems of analysis found before Poincare's death are the Riesz-Fischer theorem (1907) and Fatou's work (1906). All this is somewhat remote from the main interests of Poincare. Applications of measure theory to mechanics (ergodic theory) were found later, after his death.

You cannot expect even the greatest mathematician to react quickly to ALL important discoveries.

EDIT. Recently I read an old survey of harmonic analysis written by N. Wiener, "Historical background of harmonic analysis". I cite:

"...The notion of "almost all" has became an accepted part of the equipment of every physicist. It was under the influence of ideas belonging to this domain that Poincare at the end of the last century developed the philosophy in questions of the theory of probability which marked the first really great progress in that theory since the days of Laplace."

"The ideas of statistical randomness and phenomena of zero probability were current among the physicists and mathematicians in Paris around 1900, and it was in a medium heavily ionized by these ideas that Borel and Lebesgue solved the mathematical problem of measure."

I discussed with colleagues what does he exactly mean by "first really great progress since the days of Laplace", and we decided that this was his famous "return theorem". And certainly this was very long before the measure theoretic foundations of probability "were laid down by Kolmogorov".

  • $\begingroup$ Of course, that's a possibility even if i) 8 years (1904-1912) is a very long time for Poincaré ii) a non-reactive Poincaré to a major next-door, French discovery is simply not Poincaré iii) if Poincaré the mathematician might not have reacted "quickly", there are good reasons to think that Poincaré the physicist might have reacted immediately and very strongly (e.g. Borel never understood Poincaré about the physical continuum vs the mathematical one). My own guess is that Poincaré reacted, perhaps very strongly, in 1904, but that he decided to keep his reaction for himself. $\endgroup$ – Fabrice Pautot Nov 6 '19 at 8:13

Poincaré studied with Hermite, who famously in a letter 1893 to Stieltjes wrote „I turn with terror and horror from this lamentable scourge of continuous functions with no derivative.“ Poincaré himself is often quoted „Heretofore when a new function was invented it was for some practical end; today they are invented expressly to put at fault the reasoning of our fathers; and one will never get more from them than that.“ Of course these quotes are older than the Lebesgue integral, yet they may explain why integration of pathological functions was not considered to be important by Poincaré and other French mathematicians.

  • $\begingroup$ The quote you attribute to Poincare is actually due to Andre Bloch. This is what he wrote in connection with Fatou's example, which is known nowadays as the "Fatou-Bieberbach example". $\endgroup$ – Alexandre Eremenko Nov 8 '19 at 12:50
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    $\begingroup$ @AlexandreEremenko No, Thiku is right, this is Poincaré's. Original quote: "Autrefois, quand on inventait une fonction nouvelle, c'était en vue de quelque but pratique ; aujourd'hui, on les invente tout exprès pour mettre en défaut les raisonnements de nos pères, et on n'en tirera jamais que cela.". Science et méthode, bottom of page 132 jubilotheque.upmc.fr/fonds-physchim/PC_000305_001/… $\endgroup$ – Fabrice Pautot Nov 8 '19 at 13:56
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    $\begingroup$ @Fabrice Pautot: Thank you very much. I clearly remember seeing this in Bloch. Probably Bloch CITES Poincare. $\endgroup$ – Alexandre Eremenko Nov 8 '19 at 14:06
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    $\begingroup$ @AlexandreEremenko My pleasure. $\endgroup$ – Fabrice Pautot Nov 8 '19 at 14:09
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    $\begingroup$ Many Fourier series yield functions which are Lebegue-integrable but not Riemann-integrable. The Riesz-Fischer theorem $L^2(\left[-\pi,\pi\right])=l^2({\mathbf N})$ wouldn‘t be true with Riemann‘s notion of integral. $\endgroup$ – ThiKu Nov 8 '19 at 23:35

The Lebesgue-Borel conflict may provide a hint why Poincaré was, like Borel, not impressed by Lebesgue's contribution:

As we hinted earlier, all was not well between Borel and Lebesgue and their long-standing friendship deteriorated until it finally collapsed, at Lebesgue's instigation, in 1917. The evidence we have is provided by letters preserved at the Institute Poincaré, which Lebesgue wrote to Borel starting in 1901. The reasons, both psychological and scientific, are complex. To begin with, Borel, along with such luminaries as Kronecker and Poincaré, was a constructivist, so he rejected Lebesgue's generalisation of his measure concept as having no meaning since it was non-constructive.

—G. T. Q. Hoare and N. J. Lord, 'Intégrale, longueur, aire' the Centenary of the Lebesgue Integral, The Mathematical Gazette Vol. 86, No. 505 (2002) pp. 3–27, doi:10.2307/3621569.

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    $\begingroup$ Thanks Carlo, I'm aware of the deterioration of the friendship between Borel and Lebesgue. But according to Lebesgue quotation above, Poincaré was rather happy with his works at least in 1904! Unfortunately, we don't know how Lebesgue learned that "Poincaré finds my book good." Another main issue for understanding this strange story is that the letters from Borel to Lebesgue are lost, unfortunately. $\endgroup$ – Fabrice Pautot Nov 5 '19 at 9:34
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    $\begingroup$ Are there any evidences that Poincaré was aware of Vitali's counterexample? $\endgroup$ – Francesco Polizzi Nov 5 '19 at 9:40
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    $\begingroup$ @FrancescoPolizzi I'd like to know! $\endgroup$ – Fabrice Pautot Nov 5 '19 at 14:38

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