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](http://www.numdam.org/article/CSHM_1991__12__1_0.pdf), 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.



A historical mystery : Poincaré’s silence on Lebesgue integral and measure theory?
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](https://mathscinet.ams.org/mathscinet-getitem?mr=1110360) 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](https://mathscinet.ams.org/mathscinet-getitem?mr=2406268).)


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](http://www.numdam.org/article/BSMF_1905__33__123_1.pdf), 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?](http://www.numdam.org/article/PHSC_1996__1_4_69_0.pdf), 1996), Cartier ([Le Calcul des Probabilités de Poincaré](http://preprints.ihes.fr/2006/M/M-06-47.pdf), 2006, the [English version](https://mathscinet.ams.org/mathscinet-getitem?mr=2647630) is a bit more detailed) and Mazliak ([Poincaré et le hasard](http://www.bourbaphy.fr/mazliakfr.pdf), 2012 or the [English version](https://mathscinet.ams.org/mathscinet-getitem?mr=3329414)). 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.


