Emile Borel's Epoch-making Theorem
Kai Lai Chung
In July, 1999, a celebrating colloque was held at BorelXs birth village
Saint-Afrique, Aveyron. On this occasion Professor Bernard Bru wrote a
historic appreciation including hAe recent disovery of Borel's miraculous
equation for the renowned St-Petersburg Paradox, published in his little
book in the Que Sais-je series. By coincidence I had been preLaring a
volume on RANDOM TIME in which that paradox and other paradoxes furnish
vivid illustrations. My commentary and critique has been added tc Bru's
It was Borel, around the turn of the last century, namely c. 190C, who
conceived and initiated the notion of a countably additive measure that
extends the common human notion of "length" of a segment, already formalize
in Euclid's Elements. A first absolutely fundamental step is to "see" that
if such a segment is divided up into a number of sub-segments, the length
of the former is equal to the sum of the lengths of the subsegments. Is
this not totally obvious? In a Comptes Rendu note in there French Academie
des Sciences (Paris), dated 1698 to 1902, that I once read buy no longer
recall the exact datum, Borel said: "This appears obvious; AND YET I WILL
GIVE A PROOF!" Fis proof has been reproduced perhaps in hundreds of books
and I have taught it several dozens of times in courses of analysis. It
is based on a famous theorem known as Heine-Borel Theorem that I first read
in Hardy's Pure l.athematics in 1956 (in the library of Tsinghua University)
Vy curiosity made me dig up Heine's article, in which he used the argument
to prove a certain uniform continuity of a function. But it was Borel who
perceived the topological significance of the result which should properly
be named BorelXs Covering Theorem (as is so named in some more recent text
bock). Let us state it here: if a compact interval Cab] is a subset of
a collection of open intervals, then it'i's already a subset of a selected
finite collection (of the given collection. This sophisticated result is
then applied to the "obvious"
Borelks Theorem. Let [a,b) be a finite interval in ( -oo, m) and
(1) [a,b) = E [ajbj)
where E denotes a/union of disjoint members,/countable Then we have
(2) b-a = Z (b.-a.).
In order to ap ly Borel's Cover Theorem, we must first close [ab)
at b, and enlarge the intervals [aj.b) that a tiny little so as to include b.
After such diddling, we apply BorelXs Cover Theorem to reduce the possibly
uncountably infinite collection to a finite collection that cpvers the [ab].
Now the, what? Is it obvious now that (2) is true provided we add an E>O
on the right side owing to the slight enlarging of the intervals there? As
) is arbitrarily tiny, we infer that (2) becomes true when the "=" is relaxed
into "<". Now "it is easy to see" (Laplace) without any trickery atht the
reverse inequality with ">" replacing "=" in (2) is true, BECAUSE it is true
when the infinite sum over j is cut down to any finite sum, et cetra.
I have sketched the usual proof above in order to bring out the point
that even after the reduction t( a finite cover, we still must prove the re-
salting inequality. To do it correctly there is no way but to use mathemati-
cal induction on the total finite number of the covering intervals. In class
room instruction this was frequently left as an exercise and probably not car-
ried out by the students "except a happy few". It is detailed in some serious
textbooks (reader: check yours!)
Borel's idea of reduction to finite is seminal: nowadays the very con-
cept of compactness is defined to be the possibility of reducing any open
covering to a selected finite covering. 7hen I got to Princeton in 1945,
the term "bicompact" was still in use (and enunciated by the chairman Solomon
The moral is: the novice should ponder the possibilities concealed in
the notation employed in the formula (1). To illustrate let
faxxhx a = 2-+22 +...+2-k b +2-k-
S a2k-l .. b2k-1 2k-1
a k = l+a2kl k=l,2,...,n,... (ad infinitum)
[abb) = [0,2)
Then (1) holds. Half of this construction may be satmit Zeno of Elea (c.
490 B.C. -430 B.C. Clearly the two infinite credited to
sequences may be redoubled again and again, but is it so clear that after
an infinite sequences of infinite sequences, we can continue with "and so
on and so fcrth"? This is the theory of "ordinal numbers" created by Cantor.
At the time of Borel, mathematicians like Baire made great use of it. The
elements of the theory are expounded in classic treatises such as Hausdorff's
Set Theory, and is also discussed in more Ilementary textbooks such as Natan-
son'a Even if the reader has not studied it before, he will be able to get
the idea which is extremely intuitive in fact "transparent". It has been
shown to two "probabilists", one European and the other American, as well as
a logician. None of these three mathematicians consulted have seen such a
direct proof of Borel's theorem. For this reason it is published here as a
sup element to the article commemorating Borel. The reader is welcome to send
any comments directly to the author at the Department of Mathematics, Stanford
University, Stanford, Ca. U.S.A. 94505. No E-mail.
Bernard Bru l .jV fC VIA (5
Laboratoire de Statistique Mddicale
University R. Descartes (Paris V)
45 rue des Saints-Peres
75270 Paris cedex 06
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March 26. 1995
Dear Professor Bru;
Please excuse me for not replying to your most.kind letter of Jan. 31.
I have.been engaged in drafting a long expose of the Newton-Coulomb-Green-
Gauss-Dirichlet-Einetein connexions of Brownian Movement Process, and spent
an enoremus time retracing certain historical points, without.teo mZu*suc-
cess.. This taught me how hard it is to discover the "facts" in history, even
in the most celebratedand constantly repeated situations. For instance,
who was it that realized that the distribution of the Brownian motion is
normal? Einatain certainly gave his "proof" if 19o5 before he heard of the
Brown phenomenon. But in his original paper (available in a paperback, S5)
he made a crucial assumption (the existence of a time interval t, neither
too long nor toe short!) in order to obtain the heat equation for the normal
-density p(t,x5. More than one Ameriean authors cited Einstein's, sentence
without obviously trying to understand what he really meant. .... In Levy's
? 1948 book, he called the process Bachelier-Wieners and iove followed hia.
I took out the large volume of Ann. Eeele Norm. 1900 in order to see whether
Bachelielr had in mind the Br6wn case'which must be known'to him via Perrin,
if not Einst in (1900<1905), but after spending an hour on that wonderfully
written article as well as his book "Speeulation" (which you mentioned in an
earlier letters I thinkn$hich I have), I did not find any "hint" (Loeveta
word in Vol. II of his book) of the application to botany? Bachelier was
intent on the stock markets PIN. Yet of course his models emphasizing conti-
CA1 nuous timek no discrete approximation, was clearly adaptable to the movements
-a~tof the plant pollens observed by Robert Brown. However, he put down the nor-
mal density without "proving" it as Einstein did. Perhaps he gave the proof
a im his Calcul des probabilities If you have the latter ready could you see
I am now sdrry that for some reason I have three of Bachelier's son raphs
AW1V'but not his Calcul, which I cited in my doctorate thesis (an abstract was
sent to you I recall) 1947, and to whish I really owe a large debt. Is it
possible now to buy a copy in Paris? [I do not know if Stanford library has
e- it]. To return to Borel vis-a-vis Lebesgue, now that you have seen the
execution of his ideas by the two portugese [they must have studied in Paris,
perhaps under Borel?], do you not think it is a good time to write a new his-
torical account of the discovery. It is VERY had for anybody to appreciate
a COUP apres temp: Borel's proof that the rational have measure zero. This
alone together with the Borel/Heine lenma merits un article^entier. If only
this would refute the wrong history and idea of Bourbaki on measure, it would
open some innocents' eyes. But who will do it?
One reason for the delay of my writing you is because I went to the
Librairie de France'at the Rockefeller Center in New York which (sousterre)
has a good collection of books, among which Stendhal's Voyage en France, but
-not Voyage en Italie. I had hoped to get the latter there to save trouble,
but was told that if I ordered it from them it would cost double. Moreover
I did get an old edition (printed in Lausanne) of Stendhal par lulmemes at
a low price (under,20 dollars). Do you know that Stendhal loved mathematics
,in the lycee in Grenoble, and got a first prize: a volume by Euler. On a
single page he mentioned the three L's: Legendre* Laplace, Lagrange* with
different evaluations (qua personality) It is a pity that he did not pur-
sue mathematics after he arrived in Paris at age 17. Otherwise there must
result a beautiful theorem of Beyleis. I hope you will not mind sending
the inclosure to any good bookstore that you patronize: with your support*
the rest should go.' If so I may order other books later such as Bachelier's
Calcul. 'But since I do these things only to amuse myself, on'y attends.
Here is a curious historical question (I asked Stigler with no answer):
Who (first) observed the remarkable relation
fp(ti zay) dt = 2nx-yl ? p is normal with var. t mean x,
0 if y varies.
Sand suspected there was some connexion between physics and'probability?
As you know, this ought to be the debut of Brownian motion in Newton-
Coulomb potential. Did any of the ancient greats, even Poincare, realize
I hope to be in Europe the latter part of May* but not likely in
Paris, otherwise I would certainly try to visit with you. With best regards
S/./.f Ctost connu, etvrai (de notre experience t Strasbourg il y a
quai trea ans, qe les petits mbrchands peuvent etre fthants aux e trang-
ers ... e alors-.
June 17-19, 1995
Dear Prof. Bru:
As you see I havebeen to Parma and back. The
Pl iad volume arrived yesterday: merci beaucoup!
The inclosed letter is more than 2 months old.
I am very glad to learn that you have given some
lectures on Bachelier: rien 6crit? But perhaps
you could answer the three questions marked in
red pencil in the copy of my old letter. Now a
fourth, even more important oneA In Bachelier's
Calcul, he has more than one chapter on probabili-
tes connexe = enchain'ea = chainebde Markoff (19-
o7). Since you have obviously read the volume,
could you uncover at least one good result there
which can be classified as a result in M. chains?.
I suspect that he has a law of large numbers for
the dependent case, because that is the title of
hne of his later paper-back monographs. Now B.
wanted to have continuous t and Euclidean space,
which "obscured" the fundamental concepts of M.
chains and led to FUTIIE things like the equations
of Kolmogoroff [sic] whicha/ nothing but exten-
sion of Fokker-Planck. In my opinions Kol.'s
"too famous" paper Ueber die Analytische Wvethoden
had the demerit of fusing the, s as
,well as LmFe ma c ou. what was the true
signif-cance of the Markoff dependence as a /ex-
tension of Pascal-Laplace-loisson. So if one can
uncover in Bachelier a good result for that case,
like iarkoff's limit theorem, that alone will jus
tify a memorial in Besancon.
< ~) c-
STANFORD, CALIFORNIA 94305-2125
DEPARTMENT OF MATHEMATICS
April 7.9, 1995
Dear Professer Bru:
Please regard this as an appendix to my letter of March 26.
I have now leafed through Bachelier's Calcul des probabilities, tone 1,
a very large marvellously printed volume. Please tell me if there is a
tone 2? In his 1938 book Speculation, there is a list of his publications
including the Calcul, but no meationof "toee" er "tomes". It should be
possible to find out i: Paris whether there exists any volume after 1.
Now I have convinced myself that Bachelier did not mention Brownia.
motion by that name, nor did he give any reference to any other authors
such as Einstein. He did discuss Fourier's equation which is called the
heat equation, This is the equation that Einstein obtained (unrigorously!)
and used to derive the,.nrmal density.
I said in my last letter that Levy callei:BM the.Bachelier-Wiener process
He did give in the footnote on p.15: f his PSetMB (1948, i965) the two refer-
ences to Bachelier (1900) and Wiener (192),:rbut I do net know why I said he
called it the B-W process. After a long sesnch I found that in Feller Vol. 2
2nd ed. p. 99. Levy referred t. the priority of Bachelier in ether places,
and discussed this question in his Memorial (1954).where he explained why
1 ''P 4 -
he had overlooked B. before.
It seems that a new biographical essay about L. Bachqeier would be of
very high interest. Judging from his style of writing (which I. enjoyed)
he must be a real numero. If you kno of a worthwhile biography of him pleas
tell me. He thanked Poincare for patronage, which made me think that there
must be some interesting history about this extraordinary savant. Did he spe-
culate a lot ot the stock market? Did he win or lose?
1. "... une erreur strange dans la difiniti initial ..." was men-
tioned there, repeated on p. 97 of Quelques asp ts (1970). What it is I
still do not know. On p. 123, loc. cit., "... L. Bachelier avait, peu avan
la guerre, public un nouveau livre sur le movement brownien." What book?
Could you possibly find out? This is surprising and possibly escaped the
attention of Feller et al.
December 17-19, 1995
Dear Prof. Bru:
I have written what I knew about Bachelier, c'est fait. I had in fact
surmised (1) there is no t. 2i (2) the 1939 book "Nouvelles ..." (whfih I
have, and I think I told you in my pr&or letter) was the book Levy mentioned.
But as far as I can recall, the word "brown(ien)" I did not find in it. Now
you say he used its please specify the page. Of course I knew his usage of
kinetic aind dynamic, but where is '"BRIQN" --- and if there is nono, WHY did
he not cite it? By 1911 he could have read Einstein and the PRENCH Perrin
--- read my account.
The only news wae that error, but I contest Levy's El = oo. This hap-
pens to be true in RK, but it would be false in 1 anrd R2, as i-roved by me
and Puchs some 40 years later. One could also guess the result from the
1922 rolya paper. i3'T RXAD crefuIlly iij discussi-n (tout-a-fait% nouvelle)
that the 3-dierAsional case CAN.CT be deduced from 'oilya's random walk!
See p. 68.
in 19C6 'inctein was also cucc, but fort'inately he did not apply for
a position. See my p. 31 (read Bernstein there). It was always amazing
to me that physicists got away with murder --- some say because they have
the 'obel. Kabane knows nothing of processes, he knows only rfaom variables.
I asked Andr4 Weil about the Euler integral. He la 02a replied that I
had overestimated his knowledge of Euler's works, knowing only his number
thkery -- voici un type modest.
In the not-t.o-likely euent tnat a second edition of my book is feas-
ible, I intend to add quite a few more NOTES (perhaps also subtract a few)
that I had left aside. Thus, any relevant historical information and "cri-
tique of impure reason s" would be welcome. /
Is Gevrey the same as Gevrey-Chambertin? Most Gevrey-Ambertin is not
good enough only hiamrbertin sans prefix is worth the money, but now the
Japs bought them all. Alas. I ,iay stcp in Pari~ next spring, oela de-
"ith best wishes, when are you going to write abou Borel?
Jan. 31, 1997
Dear Prof. Bru,
Thanks for your letter about Borel (ONLY!-'ant-pis) and especial-
ly for the copy of Lebesgue's letter to Borel. Unfortunately I cannot
read it owing to the terrible scribble. If there is atranscribed
printed copy of it in your archives I would like to see it. But still
I enjoy reading L(s "complaint?" about your PAris at the top. I am
r also an4sed to learn that he called Borel simply Mon cher Borel, sans
I' M. I did not know at that epoch academicians in France could be so
informal as nowadays in America(-1-4on ~t-apove- uch-infornma4iy-ea.-r-
p-iedtoo far-I---ekeet-~;i- alge.. This s s remind you to write soon
again on the following topics:
(1) Bachelier's memorial. Did you not see in that Luxembourg volume
by Pier --- qui est ?) in which BAt#ELIER bean thee, 20th Centur :
1900. But the mention of brownian mbtion is WRONG'. -s told you and
wrote in my book, there is not a word (pas un mot) brown.. dans son
livre, meme en 1912. Pier a fa it tort, parceque il ne le sait.
(2) I had sent you, years ago, the page from Schwartz 's text which con-
tains the big mistake: about integrability of BOREL functions! If you
want you can ask Yor to see a copy of his letter.
(3) Inclosed are two (poorly cuti pages from Borel's 1926 Lecons with
Deltheil : Application a l4r thmetque. Even here he stll wrote the
P(ISn-npl>nEl with his oi des ecare as an EQUALITY: ph
so he never admitted that mistake. [For him 10 equ vie 0]. But zou
are absolutely correct to say that this is a petty error or disinvofz-
za whkh any sophomore (anche in Francia, purtroppo) can correct. The
more serious mistake of non-independence was a much curiouser mistake
[a psychological one I think] which he did amen jere where I had pen-
ciled for your benefit. Enfin, Borel ne savai 8n propre 6randchose
limsup En THAT IS WHY HISTORY IS "fun" (American sense).
ms I .ZSleUBiV ,ite91 l ao'l 1o ~38sq B nRt BSqT I :erMn=no3 0 ryo
neqqsd I .iedmjn Lsmmon .eads a nibn'll tdrods "beasid" triad exua
bna (?) enoLieqmsd0 od nevi T HIt asw olqmBxo ns dous tfaii wonI od
Xaev beAtl I doldw) zoedT iedmibl no 26od it iiW-Ybis ed: ni betio
1o0 alls tB JivlitJ TOXI 8t tl (g-tmuix rri OM itrocds essw I naeriw dorm
otf *ud eas9 mesa Xasmn "sebt" edT .4D/beoto evsd ton blrow ybasH eaes-
n~o I batl edf to anoi.tierej i ang \itnlnt scuiipes i to ti X8Iso
t11 swynA ."e9ldaoqmi" ed t im Si tidst 9lma dtiw s99es yliifs
XbisH bib yidW .)ti sea of 91SiL Eliow neridt lqmaxe aid bedaoldcfu .J
nevia bns qu bewollol bad a / e I obarsH wensi I ?oa yse tdiiW
evad blow (ennoszeq asb et)/ nnoa-9q crd ,looq o caifdsdc oc1q-O0
8aaaAHaMOIkar aaTIdIaAIOHM f i irrodtw a nzdt dous Anuob lo bevteonoo
.bebeonoo evsd ion Ji-di rooa emil silot \e9ridw yieal9oq at iBdt -.-
-oaolid q Ste seuptq tuels aid ni Yllu tuicfe eoiw lesio ,~aw edit a
-dmod qeL edt aniuub Aoo etiovtal m asw s4di em t emos o01 --- sedq
estseb Isoifmedism ditw beiod ant sew ed 9 aqsrisb .t1o edt lto gni
tol edt tadt wonx to (?Tavils ed at ysw edt d) IfedtSle bib ydw mdc
Y9iaSl u-JJp9n 'l f rloreE m t!y/llaups na vi ton bib atisoe aeb
fl avswon 8W 90 doidw ni # 2doocfd \l 'IV.ianeqaU nit Ilsteb ni
Pour continue: I read again a part of L's letter, vaguely. I am
sure that L. "bragged" about finding an abs. normal number. I happen
to know that such an example was FIRST given by Champerone (?) and
cited in the Hardy-Wright book on Number Theory (which I liked very
much when I was about 20, in Kunming). It is NOT trivial at all, or
else Hardy would not have cited Ch. The "idea" may seem easy but to
carry it out requires infinitely many repetitions of the kind I can
fairly agree with Emile that it might be "impossible". Anyway if
L. published his example then I would like to see itX. Why did Hardy
Wright say so? I knew Hausdorff et al had followed up and given
NON-pro babilistic proofs, but personnel (je dis personnel) would have
conceived of doing such things without LES PROBABILITES DENOMBRABLES
--- that is precisely where folks like Doob might not have conceded.
By the way, Borel wrote beautifully in his Valeur pratique et p hifoso-
phie --- for some time this was my favorite book during the Jap bomb-
ing of the city. Perhaps he was just bored with mathematical details
but why did Deltheil (by the way is he alive?) not know that the loi
des ecarts did not give an equality! Bernoulli's inequality is Af'i8
in detail in Uspensky's 1937 book, in which Borel was not even
Message for Chung Corinna J L / / &
Date: Wed, Jun 10, 1998 3:10 AM
Subject: Re: chung
To: Chung Corinna
Paris, le 10 juin 1998
University Rent Descartes Paris5
45 rue des Saints PEres
Cher professeur Chung
Peter Doblin est mort il y a environ deux ans et je n'ai pas son adresse *
Philadelphie. A la suite de son deces j'ai de nouveau tcrit t son frEre Claude
qui habite Nice (3 rue Rene Sainson 06000 NICE), pour obtenir son accord pour
l'ouverture du pli cachet dtpost par Wolfgang Doeblin en 1940. Je n'ai rien
obtenu du tout. Recemment 1'Academie des Sciences lui a ecrit de nouveau a ma
demand pour lui rappeler que le pli depose par Wolfgang interessait la
communaute mathematique et que de toutes facons les plis cachetes etaient la
propriete de l'Academie seule, la famille ne disposant que du droit d'ouverture.
Cette demarche n'a pas eu plus de success.
L'adresse de Stephane Doblin est (je crois) la suivante :
Residence Dauphine, 78430-Louveciennes
mais Stephane Doblin ne s'occupe de rien, tout est decide par Claude Doblin et
son avocat (qui est un dur a cuire).
Mon etude sur les chaines d'avant guerre est destine a un volume d'hommage a
Robert Fortet, elle advance si lentement que je prefer vous en envoyer un
morceau preliminaire, provisoire et non termine (il reste encore a parler de
Frechet, Hostinsky et Fortet, pour Doeblin et Kolmogorov il y a heureusement des
etudes assez complettes qu il me suffira de citer, et je n ai pas 1 intention de
parler de l'ecole allemande pour le moment). Vous etes cite a plusieurs endroits
surtout a la fin des notes (note 49 en particulier), vous etes naturellement le
seul a pouvoir decider de l'opportunite et du choix de ces citations.
De toute facon je serais heureux de connaitre votre opinion sur cette version
tres provisoire encore de mon texte (vous verrez que je parle un peu de Borel et
Ltvy). Je vous envoie a titre indicatif la bibliographie mais elle est tres loin
d etre complete, je la complete au fur et a measure quand j ai le temps, ce qui
J'espere que vous allez bien; je vous prie de croire en mes sentiments devoues.
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ir iA.(LA (8A& L v
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May 5, 2000
Dear Professor Bru:
It is with the greatest gratitude and pleasure that I received a
few minutes ago your kind letter including the photo copy of Borel's 1894
C. R. Note. It was most frustrating on my part that I was unable to find
the copy of my last letter to you asking for this favor (and also asking
the field of mathematics Madame is engaged in). Could you possibly trou-
ble to send me a copy (of my last letter) or perhaps simply return it to
me for my dossier? Je Vous en prie. Voici une grande decouverte spell---
inT5mag e;rifid) inspired by what you sent me. I well remembered that Borel
stated that lemma in his study of XX/XfX'/ functions of a complex variable
not in his theory of measure, but I failed until I saw the C. R. Note to
locate that same Lemma in the very first paper reproduced in the Selecta
don'tt nous parlions): it is on ,j more or less with the same words
as in the C. R. H4las --- etMerci a vous.
Now I have re-read (after c. 60 years) his discussion, it becomes ob-
bious that he was also indicating a proof by ordinals (in fact, Cantor was
cited there). In his typical, inimitable style (faqon) as you and Mme so
well described in that historic article, he has really given the intuitive
proof that I had the audacity to indicate in my textbook (new edition of
A Course ...). Whathe announced is: if an intervalu(open, closed, or
semi-open) has length L and if a family of intervals (arbitrarily inter-
secting and arbitrarily placed with respect toth1-c fir stone!) have the
sum of their total lengths strictly smaller than L, then there must be at
least one point (and thereforein~inn' eiTy many point) in the fir i-nar-
a- that is not in any of the aeeend family of intervals. In this formula-
tion it seems to me (pour ce moment ci) the Uaual Covering Theorem is more
clearly indicated. In the formulation of the countable additivity of mea-
sure over the pre-tribe generated by ]a,b] to be given in my text, Borel's
argument must be more or less what I had written on the inclosed 3 sheets.
As you see, I have also given the habitual argut~j available in all books.
En fin de compete, Borel l'a su. "Sans trop dire?" --- as you said.
Naturally I would appreciate your eminent judicious opinion. En
attendant, mes sentiments plus distingues,
P. S. Please don't forget to send copy of my last letter?
Bernard Bru Paris, le 30 mars 2001
11 rue Monticelli
Cher Professeur Chung
J'ai bien requ votre lettre du 19 mars et je vous en remercie vivement. J'ai attend pour
vous r6pondre d'avoir recu des exemplaires du "pli cachet" de W. Doeblin, pour vous en offrir
Comme vous le voyez, le pli a 6t6 publiC par 1'Acad6mie en un temps record. C'est tres
6tonnant quand on salt, et vous le savez tres bien, les r6ticences de nos collogues franqais a
participer au colloque Doeblin de 1991. C'est peut 8tre l'indication d'un changement de mentality
dans le milieu math6matique official, mais je n'en suis pas tres str.
J'ai reproduit les commentaires que vous aviez bien voulu me donner, j'espere n'avoir pas
trop d6form6 votre pens6e. Quoiqu'il en soit, je vous suis tres reconnaissant de votre aide. Je
dois d'ailleurs dire que vous etes la seule personnel a avoir propose des am6liorations au texte.
J'attends maintenant les critiques qui, j'en suis convaincu, seront, elles, abondantes et
argument6es. II aurait fallu faire ce que vous avez fait pour la theorie g6n6rale des chaines mais
je ne m'en suis pas senti capable et Yor n'avait pas le temps. le principal est que le texte soit
public dans de bonnes conditions.
Je m'inquietais de ne plus avoir de vos nouvelles mais j'imaginais bien que vous 6tiez tres
occupy. Je suis tres heureux d'apprendre que votre livres "Green, Brown ..." va 8tre r66dit6e, c'est
un tres beau livre qui m'a beaucoup appris et qui restera longtemps sans equivalent.
Les lettres de Lebesgue a Borel n'ont toujours pas paru. Il y a deux ans elles devaient
paraitre incessamment a Geneve comme tome VI des oeuvres de Lebesgue et puis le comit6 de
r6daction de 'Enseignement math6matique a inform Pierre Dugac qu'un referee anonyme
consid6rait que le travail 6tait "bacle" et qu'il ne m6ritait pas d'8tre public. On nous reprochait
d'avoir corrig6 les fautes d'orthographe et de ponctuation sans le signaler en note. Ces
corrections avaient 6t6 faites automatiquement, lors de la transcription du texte tap6 a la machine
(IBM a boules) en TEX par l'interm6diaire d'un scanner et nous n'en savions rien, mais apres
tout ce n'6tait pas plus mal et puis nous aurions pu remettre les fautes si c'6tait la le voeu des
6diteurs. Rien n'y a fait et nous en sommes la. C'est Choquet qui s'est charge du dossier, il m'a
dit r6cemment qu'il ne d6sesp6rait pas d'obtenir que le comit6 genevois revienne sur sa decision.
Je pense qu'il est tres optimiste et qu'il vaudrait mieux chercher un autre 6diteur mais on ne sait
Je suis convaincu que Chatterji n'est absolument pour rien dans le refus des 6diteurs, il a
toujours 6t6 tres honn8te avec nous. Certes, Chatterji a protest (avec vigueur) contre notre
analyse de la pol6mique de Lebesgue avec Borel. II parait convaincu pour sa part que Borel a
voulu s'approprier l'int6grale de Lebesgue et s'est obstin6 A penser, contre toute evidence, que sa
measure est non contradictoire et bien construite ind6pendamment de la demonstration de
Lebesgue. Visiblement Chatterji, comme Lebesgue, aime pol6miquer. On sent qu'il aimerait
comee Lebesgue) punir Borel d'avoir cherch6 a diminuer les m6rites de Lebesgue a son profit.
Sa position est tres int6ressante parce qu'il semble dans cette affaire se confondre avec Lebesgue
et qu'en tout cas il r6agit avec la meme passion, cela donne a posteriori une explication
suppl6mentaire de la pol6mique. En fait nous avions 6crit que pour nous il s'agissait plut6t d'une
rupture d'amiti6 et que, dans cette affaire, la pol6mique scientifique est tres secondaire, d'autant
que Lebesgue, emport6 dans son 61an et oubliant ce qu'il devait a Borel sur le plan scientifique et
professionnel, d6toume syst6matiquement le sens des phrases de Borel et transforme les 6loges
les plus 6vidents en mensonges perfides et les mises au point en attaques sournoises. Lebesgue
est coutumier de ce type de reaction ; c'est dans son caractere et je suis convaincu qu'il se l'est
reproch6 quelques ann6es plus tard. Cela n'6te rien a son g6nie math6matique qui est
Sur le plan scientifique, si je l'ai bien compris, Chatterji considere que Borel n'a jamais
d6montr6 la non contradiction de sa measure et que seul Lebesgue l'a fait. Il semble meme penser
que la demonstration de Lebesgue est la seule vraiment possible, en tout cas la plus important.
La il y va sans doute un peu fort ; come vous me l'avez appris, la demonstration de Borel par
recurrence transfinie se tient. De plus Borel, dans son m6moire de 1912 (reproduit en note dans
la second edition de ses Legons sur la th6orie des functions), propose une second
demonstration (assez 6nigmatique il est vrai) par completion m6trique a partir des reunions
finies d'intervalles. C'est une demonstration classique qu'on trouve en exercices dans le premier
livre de Neveu et sans doute dans beaucoup d'autres livres. En revanche Chatterji reconnait a sa
just valeur l'une des constructions bor6liennes de l'int6grale de Lebesgue d'une function f,
come limited des int6grales des polyn6mes convergeant en measure vers f. I1 s'6carte en cela de
Lebesgue qui considere lui que cette m6thode (reprise par F. Riesz) n'est qu'une "remarque"
J'ai la plus grande estime pour l'int6grit6 de Chatterji qui a toujours d6fendu l'id6e d'une
publication des lettres de Lebesgue comme tome VI des oeuvres et il n'est en aucune fagon
responsible du refus de l'Enseignement math6matique, refus qui, pour moi, reste inexpliqu6.
Sij'ai d'autres nouvelles a ce sujet je vous en informerai. L'int6grale des lettres a d6jA 6t6
publi6e, par Dugac et moi-m8me, sous forme polycopi6es, avec plus de mille notes, dans les
"Cahiers du s6minaire d'histoire des math6matiques" de 1991. Malheureusement il s'agit d'un
tirage limit etje n'en ai plus d'exemplaires. I1 ne doit en exister aucun exemplaire en Califomie.
C'est a partir de cette edition que Choquet et Dugac ont constitu6 l'ouvrage actuellement en
Je vous signal A ce propos que Choquet a fait republier dans les Comptes rendus de
1'Acad6mie (num6ro de f6vrier 2001) la note de Lebesgue du 29 avril 1901 pr6sentant son
int6grale. I1 y aura 6galement un colloque organism a Lyon les 27 et 28 avril prochains, en
l'honneur de l'int6grale de Lebesgue. Vous pouvez done rassurer Chatterji: Lebesgue est c616br6
et reconnu a sa just valeur (ce qui n'est pas vraiment le cas pour Borel).
J'ai toujours grand plaisir a correspondre avec vous.
Je vous prie de croire, Cher Professeur Chung, en mes sentiments tres cordiaux.
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7 July 26, Sat., 200"
Dear Prof. Bru:
Your letter gave me something to think about in my ennui plus malaise
N'en parlon pas, mais Grazier
Your letter made me pick up my copy of Levy's Calcul 1925. Later
I found several letters from Levy and I read them again. Tant pis his
,uilles are too fragile for our fax copier but I'll have copies made
commercially and send one or two to you: "pour vos yeux seulement".
Inw the mean time I will draft a short "history" about my encontre
with Levy II saw in Barbut's article that he did not write LAvy as
I usua lly do]. Beginning with my war years when I must have read
parts of kis Calcul (1925) and Addition (1937). When I met Hunt in
Princeton (who got his degree in the same year as I did: 1947 summ,
I learned that he read the Addition and he said to me that his e4a.
teacher von Neumann said of Levy: "It is clear that he (Levy) thin
71" in'a different way fromare". At the Levy memorial (1987?) Lau-
rent Schwartz also fold some stories +o that effect. In 1950 at
the Second Berkeley Symposium (did you attend any?) Levy was giv
a course which I attended. I also visited him at his office and
must be then that I asked him the problem of pij(t) being either
0 for all t or >0 for all t. La gave a roof which contained
7 an error (about a random Lime not optional). That was the origi
S of his 1951 paper in which surpassed Doo et al. He also public
ed a C. R. Note (I am not sure wbar) acknowledging the error
whtch made his proof valuable when all states are 'stable only.
Austin first proved the result brilliantly that L. fikedV.
Doob and I invited the LevySto a dinner in Chinatown. Doob wa
so sloppy(kb)t L. later complained to me that they had to eat
something prior to pu icking them up. I hope your friend
student won't mind this episode (Doob would'not but I'll send
[i1 him a copy). In 1967 when we (my wife and son plus mo$) spent
a quarter at Strasbourg we visited Paris, We were invited to
dinner chez eux with Chevally (my teacher in Princeton) and
the ScfWartzes. He gave me a little port safut and warned m
of its parfum. We correspoded also when they were in Menton
w W In 1976 I woi*e a paper on Brpwnian Excurs ions that was inspired by
I d.nh, an important result proved by L. by using his idea of max B(s)
a I found hard to understand (as Khintchine also experienced c. 1935
L's great theorems on stable and inf-div.laws. I think this was
August 29, 2003
Dear Prof. Bra:
It is such a pleasure to get your letter which inspired me to try
to make a copy of my article (inclosed)in the Math Monthly some years
ago. That journal is so POOR that no reprints are given and my skill at
our small fax machine is not adequate. Please excuse the disorder and
poor printing/ I have also written an article on my former colleague Polya
published in a collection, so it g6ves me pleasure to write about Levy, and
a little about Frechet. The latter will also be mentioned in my very short
"Postscript" on Hausdorff's posthumous article being edited by mon ancient
eleve (mais pas doctorate, because I went to Chicago then). Unfortunate-
ly ne is not a good correspondent like you and I cannot depend on him. You
may recall that I (we) disagree# with him on Borel-Lebesgue?
I had indeed seen the bad news in France, 15(000 dead in heat. Uere
the summer has been very fine and rather not hot enough for me, because I
spent my youth in insai which is famous for summer heat (and winter cGld)
but it WAS a truly beautiful place. I spent 10 days there in 1999 alone
and six weeks at Mao's former villa in 1988, immediately after my reti'ment.l
Allow me to wish you a happy retirement with Madame next year (?).
Re Meyer, years after the incident, I wrote a "nice" letter to him
via his student hene Inow in Irnine) and received a proper "excusep-moi".
He even told me tha his son (whom I saw as a boy) would visit USA wit s
Taiwanese wife. I invited thaEbut never heard form them. As for Marc, may
I tell you a discreet story? Before I met him in Paris, Meyer warned me
thaPYor was the kind of person that the police would always stop and harass
on account partly 4 eaete of his behavior. C,&"e vous sembldTvrai? I asked
him to prove the Pareto type of tail behavior (proved by Levy by path-analy-
sis as well as the Fourier tranforn) directly and "insisted" that he and
maybe Biane should work on that ancient theorem. Zero answer up to now.
Feller in his Vol. 2 tried hard to do it and succeeded somewhat but still
using Laplace transform and Tauberian. I am almost sure it can be done more
directly. Dommage I did not ask Erdbs in the early years, and perhaps also
Donald Austin. Now I am proposing this 4n another channel.
As Steal well knew, c'est LENNUI that is the big "problem". Your
correspond ce is a tr s agreeable aide, Merci encore.
4c^ ? /
STANFORD, CALIFORNIA 94305-2125
September 11, 2003
4 /Y6O) -OZ02 0 ATTENTION! Le JOUR
Dear Prof. Bru:
I am glad to receive the fax from M. Maaliak. Since you seem
favorably inclined to my contribution, and M. Barbut is mentioned, let
me propose an addition to be insertef&here you see fiti It was in my
5th Er 6th draft but omitted for fear of gplet You may edit it with M.
Barbut (who knows everything about Pareto) anyway you
e- a-- ., decide.
r 05ib L
I take this opportunity to prose a recherch f one of
L1vy's not-so-known results. Using his definition of stable
laws thae he invented, see p. 9# 94, at the beginning of
Chapiter V, prove directlytthe result below that he proved
by analysis of thier Fourier transfrom: for any stable law
L. we have
I ~J I'AJJ:..L....
U I I 'uu,'r~n.
tai 1 L(x) = Const. x-a
------- -n where O except the normal law which also satisfies his definition
but ~t well known to have Exxxam a much faster asymtotic
"tail". If the general case is too hard, just do it in
? the symmtric case. Then if we take the part of L for
x>x >0, there results the famigerata Pareto's law (of
income distribution (without debitsa) The latter inte-
rested ecomonists like M. Debreu, who asked me about
it when we met in Stanford when he was at Berkely.
CA y'est. Indeed many other economists asked me the same question but
I have forgotten if they won the noble prize (yet).
I hope you arecmused? Merci and best wishes for you atd Madame,
It is a pity you do not have a fax, and I refuse to touch oir
computer. /C u"
j -fc ? ) ko- "-foa,
p~. Af. Al C't ^
h 'et, f^go h
Drear Prof. Bru:
You wrote me after returning home, with much to do for your house.
Trust all went well and you and Madame are now enjoying the autumn
SSome time ago I received a fax from Barbut who has seen my MS.
Please thank him on my behalf. Before him I received also a fax
from Monsieur M. )please excuse me for not recalling his complete
name with "z" and "liak" in it?). Your pupil of his dissertation
under your direction the Levy-Frechet correspondence will be publish-
ed, must have returned to Paris but I have not heard from him. This
of course is of no importance and is mentioned here only for the re-
cord, since it is quite possible some fax or mail be lost. When
that book is ready for lim5r? I should be glad to correct the
galley of my piece. -ei/l
In my last letter to you I asked if you would be interested
to re-do the Appendix 4 for the stable-Pareto connexion? The pos-
sibility of such a revision will depend on the sale of the book, and
would not be possible before say, a year from now. In any case
your idea of revision will certainly be followed if for any reason
you prefer not to write it yourself. In the last few months I have
made propaganda among a few young competent probabilists to put
the mathematics (all due to Levy) on a more direct and elementary
level in the hope that at least a good idea of the theory may be pre-
sented in the book, for those "happy few" ( a la Beyle) who care to
learn. So far three good mathematicians have taken this up and some
progress has been made. It is clear that the venerable Feller had
made the same effort with some success (see Chpater 8, Vol. 2 of
his Tome, scarcely completed before his untimely passing).. Again
and again, Levy's original construction is resorted to by these
baav& jeune hommes. On the other hand, have hear notriing Irom
uu ire ami .aaru ari ULIi ._ asKedn u- LO bur ueo LW- uinger _iaii,.
11 : ;.lage [9J'._ -1 pixctl:
201 487 9856
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Green, Brown, and Probability
Brownian Motion on the Line
Kui aii Chung
(I '2 21ii l1 t \.\!
1 ___ _
Dear Prof. Bru:
Trust all is well with you and Madame since your rentr chez vous.
May I inquire about the state of affairs of your student's book of corres-
pondence of Levy-Frechet? I have not heard from him but I did from Barbut.
Please thank him for his kindnesses. In particular the problem of a rnw
approach to Pareto-Stable, proposed by me to several young probabilists,
has made some progress. The most interesting result is due to Giorgio Letta
(at Pisa) and Luca Pratelli. They are able to prove "half" of the result,
i. e. for any stable law, we have
symmetric PX>x Cx as x->aD
This shows that the tail decreases to zero infintely more slowly than the
normal. exponential speed. Perhaps Irof. Barbut can comment on this? Their
proof begins with the main property for a stable law, namely the Nual sum
Sn normedd" by division by nl/a has the same law as each individual Xk, l
is excluded. I do not know if there is any easy proof that a cannot be
greater than 2. These results were Ontained in heavy details by Khintchine
(and others?), and given in the book by Gnetenko-Kolmogorov.
For Levy's complete general result, the two Italians have also suc-
ceeded in putting the result in explicit expressions involving Poissonian
jumps as in L6vy's original construction (beginning with has long paper
Iropri'etes asymptotiques referred to by Barbut, and reproduced in l'Addition
Fitzsimmons (at Univ. of Cali;. La Jolla) was able to generalize their way
to the assymetric case. The details are beyond me but I hope will eventually
be available in print, "for the happy few" (Beyle). Although I first pro-
posed this problem to notre ami l Marc when he was at Berkeley, and suggest-
ed also Biane
for possible suctor, I regret to report that I have not heard from them at
all. In the meantime I have sent an announcement of this "New Solution of
Famous Theorem" in a Ckinese journal published in Dalian (site of great
naval war between the Russians and the Japs). After all, the Chinese have
put a man in Space and retrived him sans pine, non?
With best wishes as ever,
6i TLSGZL09 TE0B t0t 3WVN/'ON XVJ
6t0:60 62/t 3Wr1'3iV
6t0:60 EBBZ/6Z/0TB 3WI1
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