A historical mystery : Poincaré’s silence on Lebesgue integral and measure theory? 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.
 A: 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.
A: 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".
A: 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. 
A: 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.
