Subject: Re: [HM] Presque partout
From: Dave L. Renfro (dlrenfro@gateway.net)
Date: Sun Jun 25 2000 - 07:22:51 EDT
Udai Venedem <venedem@wanadoo.fr>
[Historia-Matematica Sat, 03 Jun 2000 21:47:45 +0200]
< http://forum.swarthmore.edu/epigone/historia_matematica/tinraxlox >
wrote
> Dear HMrs,
> 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. Could it be that Lebesgue
> is wrong, and that the (small) controversy he has with Denjoy
> about this expression (God knows why, Denjoy did not find it
> proper), was aroused somewhere else in the interval 1904-1928?
> Or did not I look carefully enough?
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. I wonder if
perhaps Lebesgue had meant to comment on "the IDEA of having full
measure", rather than "the PHRASE 'almost everywhere'"?
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.
I glanced at some of Denjoy's, Lebesgue's, and Young's work that
I have to see if I could find any mention of when Lebesgue began
using the term 'presque partout', but I wasn't able to find out
anything more. However, while doing this I came across the
following two comments by Denjoy that might be of interest.
Top of page 5 from Denjoy [2]:
L'expression << presque partout >> n'a pas eu mon
approbation. J'en ai propose une autre: << sur une
plenitude >> (et auparavant: << sur une epaisseur
pleine >>). Connaitre une fonction << presque partout >>
n'avance pas plus que de ne la connaitre << presque nulle
part >> si l'on ne s'est pas assure, par les proprietes
speciales, generalement topologiques ou << descriptives >>
de cette fonction, qu'il est indifferent pour le but a
atteindre d'ignorer ses valeurs sur un ensemble de mesure
nulle. Du point de vue topologique, c'est-a-dire pour les
caracteres invariants par une transformation topologique
[y = f(x) continue et croissante, dans le cas d'une
variable], un ensemble de mesure nulle peut etre un
<< presque partout >>, son complementaire, pleine
espaisseur metrique, etant topologiquement un << presque
nulle part >>.
In a footnote at the bottom of page 107 from Denjoy [1]:
M. Lebesgue suppose (Integration, p. 179 en Note) que je
voulais donner deux significations speciales a la locution
<< presque partout >>. Tout au contraire, je ne veux lui
en donner aucune; parce que ces mots ont trop de sens
dans le vocabulaire formant le fonds courant du langage
pour que des intuitions illusoires ne soient provoquees,
et des intuitions justes contrariees, si l'on pretend
specialiser ce sens.
[1] Arnaud Denjoy, LECONS SUR LE CALCUL DES COEFFICIENTS D'UNE
SERIE TRIGONOMETRIQUE, Gauthier-Villars, 1941-1949,
xiv + 714 pages.
[2] Arnaud Denjoy, Lucienne Felix, and Paul Montel, "Henri
Lebesgue: Le savant, le professeur, l'homme", Enseignement
Math. (2) 3 (1957), 1-18.
[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.
Dave L. Renfro
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