Georg Cantor died in 1919, more than ten years after appearance of the Lebesgue theory of measure and integration at the beginning of the twentieth century. Lebesgue theory has a deep connection with Cantor's theory of sets, for instance one of first Lebesgue's contributions after his thesis was about Fourier series, which is one of motivations of Cantor in developing theory of sets. It seems interesting to know about any (possible) reaction of Cantor to the measure and integration theory of Lebesgue.
Added in Edit: It seems that there are at-least some correspondence. The following quote is from a letter of Lebesgue to Borel, in February 17, 1904, where he talks about an unpublished publication of him, (see p. 52 in Lettres d’Henri Lebesgue à Émile Borel, Cahiers du séminaire d’histoire des mathématiques, tome 12 (1991), p. 1 -506), Lebesgue writes that
Vous pouvez envoyer a Fatou et G. Cantor et vous savez qu' il m' en restera tres probablement pendant quel que temps si vous avez l' idee d' une ou deux personnes.
I do not claim that this prove anything (I even don't know whether Cantor received such document). Two days later, in another letter (ibid, p. 54), Lebesgue writes