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 The following quote is from a letter of Lebesgue to Borel, in February 17, 1904, where he talks about an unpublished publication of him, (see p. 52 in Lettres d’Henri Lebesgue à Émile Borel, Cahiers du séminaire d’histoire des mathématiques, tome 12 (1991), p. 1 -506),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

 A: 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.

