Why is Lebesgue so often spelled "Lebesque"?
Solution 1:
I am also a native French speaker (from France) and I confirm that Lebesgue is the correct spelling and that Lebesque is just a bad typo from this site.
The situation is not that bad, however. As the OP pointed out, there are 82 users who misspelled his name, but if you search for "Lebesgue" on this site you will find 8,587 entries. So the ratio is less than 1%. By the way, I remember having corrected this typo at least once and I invite you to help me until the number of such misspellings reaches Lebesgue measure $0$....
Solution 2:
I think this is a good question, and as the edit makes clear, the scope of it is certainly larger than stackexchange websites.
I have wondered about this myself over the years. Nate Eldredge's suggestions are as good as any I can come up with: in particular, yes, in older typefaces "Lebesgue" and "Lebesque" look very similar. Especially, slightly shoddy typesetting can lead to the bottom portions of these characters being removed, so I think it is likely that there is at least one classic text in which one cannot properly read the name. It would be interesting to find one.
I will mention what I think is one further ingredient: mathematics is decidedly less concerned with supplying source material (primary or otherwise) and in correctly associating ideas with names than most other academic fields. This is true to a degree which is simply remarkable once you notice it. For instance, a typical undergraduate mathematics textbook does not have a bibliography! If results are attributed to specific people, it is people like Newton, Leibniz and Cauchy, and this being done to leaven the text with human interest stories. (It is common enough to have a footnote giving a sentence or two of biographical information about someone like Cauchy. Which is nice, but calls into stark relief the absence of actual bibliographic information.)
The most common amount of effort taken in tracking down the reference in which Theorem X.Y was actually proved is zero. Rather, the association of a name to a result, when (too rarely) done in a mathematics textbook, is done entirely according to the author's whims. This turns the association of names with results into a game of whisper down the lane played by textbook authors and their readers.
I will give another example, closer to my own heart. For almost 20 years now I have been a fan of the Chevalley-Warning Theorem (and recently a bit more than a fan, but if there is any relevance to that, it is just that in this case I actually have read the primary literature). Now I have noticed over the years that about 5% of the mathematical world refers to this result as the "Chevalley-Waring Theorem". I don't think there can be any good reason -- no one calls it that after their attention is drawn to Ewald Warning's 1936 paper on the subject -- and yet I have seen it in published papers. Most recently I saw it in a review of my recent grant application.
Finally the PI has a proposal concerning 'combinatorial nullstellensatze and Chevalley-Waring'. I read this section, but have insufficient expertise in this subject area to judge the importance of this proposal. I see that for instance the PI has generalizations of Waring's theorem (saying that a system of polynomial equations over a finite field of low degree compared to the number of variables, either has no solutions or has quite a lot of solutions) to rings of integers modulo prime powers. This seems quite striking to a non-expert.
I think this is interesting glimpse of the psychology involved. Needless to say the grant application spoke of the "Chevalley-Warning Theorem". Probably the reviewer was, like many number theorists, much more familiar with the mathematician Edward Waring than the mathematician Ewald Warning, so substitutes one for the other without noticing. Or checking. In fact the reviewer includes quotation marks around what is by no means a direct quote (for that matter in my grant application it says Combinatorial Nullstellensätze, not combinatorial nullstellensatze, so a German language error has also been introduced inside a direct quote).
I don't mean to be too hard on the reviewer, who openly admitted lack of expertise in the area. They undoubtedly also had a lot of evaluative work to do in a short period of time, and getting the names of dead mathematicians right was not their priority. But that's my point: as a profession, getting the names of dead mathematicians right is typically our lowest priority. I think this is something to work on in the future, because it is so nicely enabled by modern technology. Time was you'd need to make a trip to a mathematical library to convince yourself that it's Chevalley-Warning, not Chevalley-Waring. Now, if you google Chevalley-Waring, it will show you instead results for Chevalley-Warning. Things are not as hard as they used to be.
P.S.: It is interesting to read the 48 MathSciNet citations to "Lebesque". In the reviews dating back to 1973, the mistake is always corrected in the title or remarked upon by the reviewer. Before then this seems mostly not to happen, and sometimes it is even the reviewer who introduces the mistake, as e.g. in MR0253894 from 1969. Here is an exception: the review of MR0179321 from 1963 is
Simples exercices sans difficulté et sans grand intérêt. (Le nom de Lebesgue est systématiquement estropié dans le titre et dans le texte.)
The second oldest use of "Lebesque" occurs in
Lebesque, Henri Sur la recherche des fonctions primitives. (French) Acta Math. 49 (1927), no. 3-4, 245–262.
However a look at the original paper shows that the author's surname is spelled LEBESGUE (capitals in the original).
The oldest use of "Lebesque" comes from the oldest paper I have yet seen on MathSciNet:
Lebesque, A.; Intégration d'un système d'équations linéaires du ne ordre. (French) J. Reine Angew. Math. 15 (1836), 185–190.
According to the first page of this paper, this author's name is indeed "Lebesque". Anyway, Lebesque is a real French surname.
Final Irony: Henri Lebesgue's father was a typesetter.
Solution 3:
I am a native French speaker, although not from France.
Lebesgue is definitely the correct spelling, and Lebesque sounds completely different; besgue sounds like beg in to beg, but the qu in que sounds like a k here.
I don’t know what you’re reading, but I have never seen it spelled like this in my life. I assume it comes from non-French speakers who remember the spelling wrong or mix up the letters, but as I have never seen the phenomenon myself, it’s hard to tell. (Maybe if you had an article or a text where it’s written? But then again....)
Hope that helps.
Solution 4:
Words ending in "que" are much more common than words ending in "gue". See Latin. :-)