Subject: Re: [HM] Presque partout
From: Dave L. Renfro (dlrenfro@gateway.net)
Date: Wed Jul 26 2000 - 19:49:56 EDT
In the thread "[HM] Presque partout" at
http://forum.swarthmore.edu/epigone/historia_matematica/tinraxlox
Udai Venedem wrote (in part) on June 3, 2000:
> in the second edition (1928) of his LEC,ONS SUR L'INTE/GRATION
> ET LA RECHERCHE DES FONCTIONS PRIMITIVES, Lebesgue by a note
> on page 179 pretend that the locution "presque partout"
> (to mean "except on a zero-measured set") was introduced in
> the first edition of his book (1904), where I do not find it.
On June 25, 2000 [I had been out of town the previous 5 weeks.]
and in this same thread I responded (in part) with:
> At the top of page 138 of [3] Hawkins writes: "The phrase 'almost
> everywhere'--analogous to Harnack's 'in general'--was later
> introduced by Lebesgue to signify that a condition holds for all
> points except those forming a set of measure zero." At this place
> in his book Hawkins is discussing a paper of Lebesgue's that
> appeared in 1907. Thus, it would appear that the phrase 'presque
> partout' did not appear until after 1907, contrary to what
> Lebesgue wrote in the 1928 edition of his book.
> In the middle of page 114 of [5] Medvedev writes: "The term 'almost
> everywhere' can be found in a 1909 paper of Lebesgue [14, p. 43]."
> The paper in question is Lebesgue [4] (which I don't happen to
> have a copy of). Thus, it appears that an upper bound for the
> date of first appearance for the phrase 'almost everywhere' is
> 1909.
> [3] Thomas Hawkins, LEBESGUE'S THEORY OF INTEGRATION: ITS ORGINS
> AND DEVELOPMENT, Chelsea Publishing Company, 1975.
>
> [4] Henri Lebesgue, "Sur les integrales singulieres", Ann. Fac.
> Sci. l'Univ. Toulouse (3) 1 (1909), 25-117.
>
> [5] Fyodor A. Medvedev, SCENES FROM THE HISTORY OF REAL
> FUNCTIONS, (English translation by Roger Cooke), Birkhauser,
> 1991.
Finally, on June 26, 2000 (same thread) I gave the specific
paper that Hawkins [3, p. 138] refers to . . .
> [11] Henri Lebesgue, "Sur la recherche des fonctions primitives
> par l'integration", Atti della R. Accademia dei Lincei.
> Transunti (5) 16(1) (1907), 283-290.
and wrote (in part):
> In my previous post I mentioned that Medvedev's book says
> the phrase 'presque partout' appears in a certain 1909 paper
> by Lebesgue. Although I don't have a copy of this 1909 paper,
> I did manage to locate in my collection a 1910 paper by Lebesgue
> that uses this phrase. The phrase 'presque partout' appears in
> lines 1 and 2 of section 34 on page 407 of [12].
> [12] Henri Lebesgue, "Sur l'integration des fonctions
> discontinues", Annales Scientifiques de l'Ecole Normale
> Superieure (3) 27 (1910), 361-450.
I was at a library recently that had Lebesgue's 5 volume
collected works and I found 'presque partout' in two papers
([17] and [18] below) by Lebesgue that were written before [11],
the paper that Hawkins appears to be saying predates Lebesgue's
first usage of 'presque partout'.
[17] Henri Lebesgue, "Sur les fonctions de/rive/es", Atti della
R. Accademia dei Lincei. Rendiconti (5) 15(2) (1906), 3-8.
[JFM 37 (p. 312)]
[18] Henri Lebesgue, "Encore une observation sur les fonctions
de/rive/es", Atti della R. Accademia dei Lincei. Rendiconti
(5) 16(1) (1907), 92-100.
[JFM 38 (p. 426)]
Footnote (2) on page 8 of [17] is:
Vitali, Sulle funzioni ad integrale nullo (Rend. del
Circolo Mat. di Palermo, XX, 1905). Puisque j'ai
l'occasion de citer cette Note, sans en contester
l'inte/re^t, je ferai remarquer que le the/ore\me qui
en fait l'objet: une fonction dont l'inte/grale
indefinie est nulle ne diffe\re de ze/ro qu'aux
points d'un ensemble de mesure nulle, est une
conse/quence imme/diate du fait qu'une telle fonction
est, presque partout, la de/rive/e de son inte/grale
inde/finie.
The 2'nd paragraph on page 99 of [18] begins with:
Passons a\ l'autre critique: je sais que le nombre
de/rive/ $\lambda$ de l'inte/grale infinie d'une
fonction $\varphi$ positive est, presque partout,
au moins e/gal a\ $\varphi$.
The phrase 'presque partout' occurs again in the 2'nd
paragraph on page 99 of [18]. However, in neither [17]
nor in [18] did I see an explicit definition for 'presque
partout', although I suppose one could say that its meaning
is implicit in the quote I gave from [17].
Dave L. Renfro
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