Subject: Re: [HM] Presque partout
From: Dave L. Renfro (dlrenfro@gateway.net)
Date: Mon Jun 26 2000 - 03:56:57 EDT
Udai Venedem <venedem@wanadoo.fr>
[Historia-Matematica Sun, 25 Jun 2000 22:08:27 +0200]
<http://forum.swarthmore.edu/epigone/historia_matematica/tinraxlox>
wrote (in part, including a quote from my previous post):
>> 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?
See reference [11] below. However, assuming Hawkins is correct
when he wrote that the phrase 'presque partout' does not appear in
[11], nothing in [11] would help in determining date of first
appearance of the phrase 'presque partout'. This is why I didn't
bother to give reference [11] in my previous post.
De la Vallee-Poussin wrote a very influential paper, [7], which
might give some information as to when Lebesgue introduced the
term 'presque partout'. Unfortunately, I only have a copy of one
page (page 457, where the Lebesgue density theorem is discussed) of
this paper. Nonetheless, the term 'presque partout' is used several
times on page 457. This paper can be down-loaded (I think) at
<http://www.jstor.org/journals/00029947.html>,
if you're somewhere where you would have the appropriate
access permission (which I'm not).
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].
Probably the easiest way to pin down the specific paper(s) where
Lebesgue began using the term 'presque partout' is to simply
spend a couple of hours looking though the papers in his
collected works (which any research library should have), paying
attention to their submission dates.
I also have two papers from 1912 in which other people use the
phrase 'presque partout'. 'Presque partout' appears throughout
Montel [15] and 'presque partout' is actually part of the title
of Young [16].
At the JFM site
<http://www.emis.de/MATH/JFM/JFM.html>,
a title-contains-the-expression "presque partout" search generated
7 hits, although Young [16] WASN'T among them (this site is
still under construction). The earliest date for these papers was
1916 (2 papers by Sierpinski have 'presque partout' in their title
in 1916), which is not early enough to be of much significance here.
I found a long footnote on page 766 of Denjoy [6] that gives
another statement of his feelings about the phrase 'presque partout'.
The following quote from page 187 of Medvedev [14] might be of
interest:
<< In what follows, the original numbers [2], [3], and [6]
have been replaced with my numbers [8], [9], and [10]. >>
The principle of negligibility of various point sets in
considering one problem or another in analysis or the theory
of functions came into use at a very early stage. Originally,
only finite point sets were regarded as negligible, but later,
about at the time of Cantor, also various kinds of infinite
sets (reducible, countable, etc.). Towards the end of the
nineteenth century, Baire in considering certain problems
in the theory of functions was led to neglect sets of the
first category. Lebesgue, starting with the note [8] and
his dissertation [9], began to make frequent use of the
principle of negligibility of sets of measure zero. He gave
an explicit formulation of this principle in 1903 ([10], 1229),
basing it on an argument due to Baire, and then, as we have
already said, applied it systematically to problems of
differentiation, integration, convergence, summability, etc.,
in a word, to the most diverse problems of the metric theory
of functions. It was just, through the work of Lebesgue, that
the principle of negligibility of sets of measure zero became
one of the essential methods of the theory of functions of a
real variable.
Finally, Letta [13] might be worth a look, if you can read
Italian. Here's what MR 96g:01026 (Thomas Archibald) says:
The author surveys studies of Riemann's integrability condition
in the period up to the early 1890's. In particular, he argues
that Riemann's work was instrumental in focussing attention on
the notion of a negligible set, which is of importance in the
development of measure theory. The survey includes work on
so-called discontinuous functions due to Hankel and Darboux,
integration criteria of Ascoli and Volterra, examples due to
H. J. S. Smith and Volterra, and some erroneous conceptions
of Harnack and P. du Bois-Reymond. In particular, he disagrees
with Thomas Hawkins' 1973 statement to the effect that Harnack
was the first to correct a criterion due to Hankel, noting
that Harnack offered no proof and further that both Ascoli
and Volterra had stronger results earlier.
[6] Arnaud Denjoy, "Sur les nombres derives", C. R. Acad. Sci.
Paris 160 (1915), 763-766.
[7] C. de la Vallee-Poussin, "Sur l'integrale de Lebesgue",
Trans. Amer. Math. Soc. 16 (1915), 435-501.
[8] Henri Lebesgue, "Sur une generalisation de l'integrale
definie, C. R. Acad. Sci. Paris 132 (1901), 1025-1028.
[9] Henri Lebesgue, "Integrale, longueur, aire", Annali di
Mathematica Pura ed Applicata (3) 7 (1902), 231-359.
[10] Henri Lebesgue, "Sur une propriete des fonctions", C. R. Acad.
Sci. Paris 137 (1903), 1228-1230.
[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.
[12] Henri Lebesgue, "Sur l'integration des fonctions discontinues",
Annales Scientifiques de l'Ecole Normale Superieure (3) 27
(1910), 361-450.
[13] Giorgio Letta, "Riemann integrability conditions and their
influence on the origin of the concept of measure" (Italian),
Rend. Accad. Naz. Sci. XL Mem. Mat. (5) 18 (1994), 143-169.
[14] Fyodor A. Medvedev, "The work of Henri Lebesgue in the theory
of functions of a real variable" (translated by J. Friberg),
Russian Math. Surveys 30:4 (1975), 179-191.
[15] Paul Montel, "Sur l'existence des des derivees", C. R. Acad.
Sci. Paris 155 (1912), 1478-1480.
[16] William H. Young, "Sur les series de Fourier convergentes
presque partout", C. R. Acad. Paris 155 (1912), 1480-1482.
Dave L. Renfro
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