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.