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 The following quote is from a letter of Lebesgue to Borel, in February 17, 1904, where he talks about an unpublished publication of him, (see p. 52 in Lettres d’Henri Lebesgue à Émile Borel, Cahiers du séminaire d’histoire des mathématiques, tome 12 (1991), p. 1 -506),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

Cantor existe-t-il?

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    $\begingroup$ Might fit better here: hsm.stackexchange.com $\endgroup$ – Stopple Oct 16 at 22:05
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    $\begingroup$ Question on HSM: hsm.stackexchange.com/questions/12349/… $\endgroup$ – LSpice Oct 16 at 23:11
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    $\begingroup$ The obvious thing is to check Joseph Dauben's biography of Cantor, and look for Lebesgue in the index -- the GoogleBooks excerpts discuss Lebesgue's continuation of Cantor's ideas, but not Cantor's response, so it may be that Cantor had nothing to say on the topic. $\endgroup$ – Matt F. Oct 16 at 23:55
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    $\begingroup$ I just read an1983 article (in German) Von Riemann zu Lesbegue - Zur Entwicklung der Integrationstheorie from Riemann to Lesbegue - on the history of the theory of integration. In there no reaction of Cantor is mentioned; it is however interesting to read that contemporary mathematicians were not enthusiastic about Lesbegue's theory of integration if not even hostile. That may explain why no reaction of Cantor is known. $\endgroup$ – Manfred Weis Oct 17 at 14:59
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    $\begingroup$ Why the close votes? $\endgroup$ – Timothy Chow Oct 17 at 15:04

The following quote is from Joseph Dauben (in his article Georg Cantor: The Personal Matrix of His Mathematics, Isis, Vol. 69, No. 4, Dec., 1978, pp. 534-550).

When he suddenly suffered his first breakdown in May 1884, Cantor had just returned from an apparently successful, quite enjoyable trip to Paris. He had met a number of French mathematicians, including Hermite, Picard, and Appell, and was delighted to report to Gosta Mittag-Leffler that he had liked Poincare very much and was happy to see that the Frenchman understood transfinite set theory and its applications in functional analysis.

It appears that Cantor did not keep up with the work of the later generation of French analysts. Recall that Lebesgue wrote his dissertation under the supervision of Borel, and Borel was born in 1871, so Borel was only 13 years old at the time of Cantor's visit. One key reason for this might be that after his first breakdown (1884), Cantor's range of interests were widely expanded to many other domains, as indicated by the following excerpt from Dauben's aforementioned paper:

He began to emphasize other interests. The amount of time he devoted to various literary and historical problems steadily increased, and he read the history and documents of the Elizabethans with great attentiveness in hopes of proving that Francis Bacon was the true author of Shakespeare's plays. As time progressed, he also began to intensify his study of the Scriptures and of the church fathers, and he developed new interests in Freemasonry, Rosicrucianism, and Theosophy.

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    $\begingroup$ Cantor attended the third ICM in Heidelberg, 1904, where Kőnig presented a disproof of the continuum hypothesis. Borel was also an invited speaker. I wonder if Cantor heard of his work on measure theory, at least. $\endgroup$ – Conifold Oct 20 at 5:16
  • $\begingroup$ I'm not sure that the conclusion is true. In a letter that Lebesgue sent to Borel in January, 1904, he says among other things 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 ' i dee d' une ou deux personnes." (see p. 52 in Lettres d’Henri Lebesgue à Émile Borel, Cahiers du séminaire d’histoire des mathématiques, tome 12 (1991), p. 1 -506). It shows at least there are some Correspondence between these people. $\endgroup$ – XIE Oct 20 at 16:38
  • $\begingroup$ @Ali It might be good if you added your answer over at hsm.stackexchange.com/questions/12349/… $\endgroup$ – David Roberts Oct 21 at 5:29

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