Subject: Re: [HM] Presque partout
From: Udai Venedem (venedem@wanadoo.fr)
Date: Sun Jun 25 2000 - 16:08:27 EDT
Dear Historians and other Measure Theorists,
some times ago (Sat June 3rd, 2000), I had asked about the phrase "almost
everywhere", as introduced by Lebesgue:
> 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.
and Dave L. Renfro <dlrenfro@gateway.net>, on Sun 25 June 2000 gives very
interesting data, but wondered:
> 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'"?
The answer is no. Indeed, in his long note in half of the page 179, Lebesgue
gives an account on the controversy he had with Denjoy about the PHRASE
"presque partout". He gives this (wrong, as we know now by Dave) precision:
*****************
Nous conviendrons de dire qu'une proprie/t/e a lieu *presque partout*
(*-* in italic) dans un intervalle (a, b), ou sur un ensemble E, si les
points de (a, b) ou de E en lesquels elle n'a pas lieu ou bien n'existent
pas, ou bien forment un ensemble de mesure nulle.
Cette locution, introduite dans la premie\re e/dition de ce Livre, a
e/t/e ge/ne/ralement adopte/e.
*****************
where *locution* = phrase
then, Lebesgue goes on the critics by Denjoy. Of course, this does not mean
that Lebesgue did not give the IDEA of this in the first edition of
his book (1904). On the contrary, at the end of it (p. 125-129), he is very
clear about properties found to be true everywhere except for points forming
a set of measure zero.
Still, Dave writes:
> At this place in his book Hawkins is discussing a paper of Lebesgue's
> that appeared in 1907.
Could it be possible to know which is this paper, so we could conclude on
the question of the first appearance of the phrase?
BTW, for the second edition (1928) of Lebesgue's book, see at
http://perso.wanadoo.fr/alta.mathematica/tour3.html#459
Udai Venedem
venedem@wanadoo.fr
welcomes you at
http://perso.wanadoo.fr/alta.mathematica/
a site devoted to collector's books of mathematics
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