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 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.

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.

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.