Mikhail Y. Suslin and Lebesgue's error

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Dave L. Renfro

Jul 29, 2006, 2:21:40 PM7/29/06
William Elliot wrote (in part):

thread = More on Souslin (= analytic) sets (was: V = L)
date = July 29, 2006

> M. Souslin is from Moscow university. That's all I could
> find out as links to his biographies were faulty and he's
> not in the two math biography sites I know. Another Abel
> lost to the world?

I'm starting a new thread because what I'm posting is likely
to be of wider interest.

Here's a biography of Suslin:

V. I. Igoshin, "A short biography of Mikhail Yakovlevich
Suslin", Russian Mathematical Surveys 51 #3 (1996), 371-383.

I thought I'd type some of this up for the ever-growing sci.math
archive, seeing as how it's Saturday morning (at least, when I
started this) and I don't have to go to work today.

Before getting started, here are some of the relevant papers
mentioned below.

[1] Mikhail Y. Suslin, "Sur une définition des ensembles
mesurables B sans nombres transfinis", Comptes Rendus
Académie des Sciences (Paris) 164 (1917), 88-91.

Available on the internet -- click on "Facsimiles" at the
JFM 46.0296.01 web page above.

[2] Mikhail Y. Suslin, "Probleme 3", Fundamenta Mathematicae 1
(1920), 223.

Page 223 of Volume 1 of Fund. Math. is NOT at digitized at

[15] Henri L. Lebesgue, "Sur les fonctions représentables
analytiquement", Journal de Mathématiques Pures et Appliquées
(6) 1 (1905), 139-216.

Lebesgue's error is on pp. 191-192. Lebesgue incorrectly
asserted that the projection of a Borel set in R^n onto
R^i is a Borel set in R^i. Suslin showed, in fact,
that even the projection of a G_delta set in R^2 onto R^1
does not have to be a Borel set in R^1. The following English
translation of the relevant passage is given in Comment (3)
on p. 12 of "Preface to Lusin's book ...", Russian Mathematical
Surveys 40 #3 (1985), 9-14: "Given a set E of points (x_1,
x_2, ..., x_n), as we know, the projection of this set onto
the manifold x_{i+1} = x_{i+2} = ... = x_{n} = 0 is the name
for the set e of all collections of corresponding values
(x_1, x_2, ..., x_i). I intend to prove that if E is Borel
measurable, then its projection is Borel measurable. This is
obvious if E is an interval: then e is also an interval.
But every set that is Borel measurable is obtained from
intervals by the repeated application of operations I and II,
which are preserved by projection; {footnote: Which was not
true for operation II.} the conjecture is proved."

Lebesgue's paper is available on the internet.
Select "1905 (Série 6 / T. 1)" at

[40] Mikhail Y. Suslin, "Sur un corps dénombrable de nombres
réels" [based on a posthomous memoir of M. Souslin by C.
Kuratowski], Fundamenta Mathematicae 4 (1923), 311-315.

Available on the internet at

(p. 371): "The period of independent creative activity
of Mikhail Yakovlevich Suslin only lasted for two or three
years. There are just three short articles published in his
name, and only one of these was published during his lifetime.
However, such fundamental discoveries were made in one of
these articles, and such deep ideas lay in another, that one
can say without exaggeration that they had a revolutionary
impact on the 20th century development of set theory, one
of the fundamental branches of mathematics."

(pp. 371-372): "In the only article published during his
lifetime [1] Suslin discovered a new class of sets (later
called _Suslin_sets_) which became for many years to come
a very important object of research in descriptive set theory.
This discovery brought the Moscow School of the theory of
sets and functions to a leading position in the world, and
gave it a new and rich range of research topics. These areas
lay beyond the traditional topics of the French Mathematical
School and extended the capacity of mathematical analysis
as a science. The significance of Suslin's discovery was fully
recognized by his contemporaries. In his other article [2]
a problem was posed which touched upon such deep connections
between mathematical structures that it turned out to be
beyond the power of Suslin and his contemporaries to solve
it. The problem remained unresolved for more than forty years,
until the beginning of the 1960's. It had a powerful stimulating
impact on the development of mathematics by generating new
concepts, methods and theories. Mathematicians of the world
were giving Suslin's name to the new concepts connected with
the problem he posed. Even now the name of Suslin is often
mentioned in a wide range of publications concerning set
theory and model theory: Suslin sets, the Suslin criterion,
the Suslin property, Suslin number [3]. This scientist was
highly honoured: the concepts connected with his name are
often now used without reference to the publications of
Suslin. This concepts treated in this way in mathematics
are those which have entered its gold reserves for ever [sic]
and are connected with classical names: Fourier series, the
Laplace operator, the Lebesgue integral, and so on. History
has therefore added Suslin to the list of classical names of
mathematical science."

(p. 372): "It was the hand of fate that the journey of his
life traversed a circular path through Balashov boys' grammar
school, Moscow University and the Ivanovo-Voznesensk Polytechnic
Institute, and finally back to his birthplace -- the village
of Krasavka in the Balashov region of the Saratov province.
{footnote: _Volost_, _uyezd_ and _guberniya_ were administrative
divisions of Tsarist Russia. _Raion_ and _oblast_ were
administrative divisions of the Soviet Union. We have kept
_volost_, for which there seems to be no translation; we have
translated _uyezd_ and _raion_ as region, _guberniya_ as province,
and _oblast_ as district.} He was born in this village on 15
November (3 November in the old style) in 1894 and was the only
child of poor peasants Yakov Gavrilovich and Matrena Vasil'evna


(p. 373): "In the year he was born his father was 36 and his
mother 35 years old. They had a little shed where the father
kept a small shop. From there he traded various small items --
salt, paraffin, matches, spoons, buttons, thread, and horse
harnesses. This little shop did not bring wealth to Yakov
Gavrilovich, nor even a sufficient income to enable him to be
comfortably off. It just allowed him to make ends meet."


(p. 373): "There were two primary schools in Krasavka. Krasavka
1st primary school of the zemstvo {footnote: _Zemstvo_ was an
elective district council in Russia, 1864-1917.} was founded
in 1872 and the second on 11 October 1899. they taught God's
Law, Slavonic and Russian reading, writing and arithmetic [7].
Krasavka 2nd primary college of the zemstvo was situated in a
small house on the right bank of the Elan', next to the church,
exactly opposite the house of Yakov Gavrilovich. In the autumn
of 1903, at the age of 9, Mikhail Suslin stepped over its
threshold. In the spring of 1904 Mikhail successfully passed
his leaving tests. In the council register of Balashov region
there is a record of certificates of graduation from local
primary schools during the period of 1901-1911. It is written
there that Suslin, Mikhail Yakovlevich, of the orthodox faith,
born in 1894, on the 3rd day of November, was given in 1904 a
graduation certificate from Krasavka 2nd school, number
13227 [8]. His teacher Vera Andreevna Teplogorskaya-Smirnova
not only planted in him the first grain of knowledge, but she
also noticed and encouraged his original talent. She also put
a lot of effort into trying to persuade the wealthy men of
the village that Mikhail should continue his studies further.
To this end these men gave financial help to Yakov Gavrilovich
and Matrena Vasil'evna to send their son to the city of Balashov."

(pp. 373-374): [snip] "Mikhail Suslin was admitted to the
preparatory class of this educational establishment. On 1 July
1910 this establishment was transformed into a government grammar
school. On that very day Alexandr Ivanovich Rozanov was appointed
the teacher of mathematics, and also had to perform the duties
of headmaster of the grammar school. This man was a Councillor
of State, and was 36 years old. He had graduated in 1902 with
a Diploma of 1st degree from Kazan Imperial University, Department
of Mathematics of the Physical-Mathematical Faculty [10]."

(p. 374): "While a grammar school student, Mikhail Suslin took
lodgings with a merchant called Bezborodov. The trader quickly
recognized the intellectual strength of his young lodger, and
more than once offered him the hand of his daughter, promising
a substantial dowry of a million roubles. Mikhail smiled and
answered: "I will wait a while, I have to conquer science first".
He found his studies at the grammar school quite easy. During
his free time, in order to earn money, he gave lessons and
prepared the children of the wealthy for examinations. In this
way he himself gradually became able to pay for his studies in
the grammar school -- the fees were about 50 roubles per year."

(p. 374): "On 30 May 1913 Mikhail Suslin brilliantly finished
the full eight-year course at Balashovskaya grammar school,
having demonstrated excellent knowledge of all subjects studied.
He was also granted awards when he transferred from the 6th form
to the 7th and from the 7th to the 8th. The following was written
on his school leaving certificate: "In view of his constant
excellent behaviour and diligence and his excellent successes
in sciences, especially in mathematics, the Pedagogical Council
have decided to award him a gold medal ..." [11]."

(p. 374) "On 22 July 1913 Suslin sent a request from Krasavka to
the rector of Moscow Imperial University concerning his admission
as a first-year student to the Department of Mathematics of the
Faculty of Physics and Mathematics. On 25 July Suslin's request
was received by Moscow University and on 7 August 1913 it was
approved by an aide of the rector who wrote: "Admitted as a
student of the Department of Mathematics" [12]. (A corresponding
member of the Academy of Sciences of the USSR D. E. Men'shov,
who began his second year studies that autumn, recalled [13]:
"There were no entry exams ... at that time. It sufficed to have
a school leaving certificate from a grammar school. There were
not that many people wishing to apply.")"

(p. 374): "From his first year on, Suslin attended the scientific
student seminars led by Professor D. F. Egorov. From the autumn
of 1914 Suslin together with other students -- D. E. Men'shov,
A. Ya. Khinchin, P. S. Aleksandrov -- began to work under the
direction of a young senior lecturer called Nikolai Nikolaevich
Luzin. They formed the first generation of pupils of the young
mathematician (who would soon become a professor). They jokingly
called their friendly and purposeful group the 'Luzitania'.
Luzin had just returned from a long business trip abroad during
which he did his training in Göttingen and participated in the
work of the Mathematical Congress in London. His deep scientific
knowledge, his skill in delivering it to an audience, and his
being easily approachable immediately attracted active students to
him. The young scientist passionately promoted a new mathematical
discipline -- the theory of sets and functions. It was problems
from this theory which he posed to his students."

(pp. 374-375): "The student P. S. Aleksandrov proved in 1915 that
any Borel set (B-set) can be obtained by a single application of
the operation which he introduced to closed sets. He gave a talk
about it at the student mathematical circle on 13 October 1915
([14], p. 235). Suslin suggested naming the new operation the
A-operation after Aleksandrov and the sets obtained by its
application to closed sets A-sets. Thus in the terminology
suggested by Suslin the result obtained by Aleksandrov stated
that every Borel set (B-set) is an A-set. After that Luzin
persistently suggested to Aleksandrov and Suslin that they
concentrate on solving the converse problem -- to find out if
every A-set is a Borel set, since he correctly perceived this
problem as being the cornerstone of descriptive set theory. As
Men'shov recalled ([13], p. 323) Aleksandrov told him that
trying to solve the problem of the cardinality of the Borel
sets took tremendous concentration. [Renfro -- He is referring
to the problem of proving that every uncountable Borel set has
cardinality c, not the (much easier and due to, I think, Lebesgue,
in Lebesgue's famous 1905 paper) problem of proving that there are
c many Borel subsets of the real line.] Nevertheless Aleksandrov
devoted the winter of 1915-1916 and all the following summer to
seeking the solution of this problem ([14], p. 235). Yet the
problem did not yield to him."

(p. 375): "At the same time Suslin, also at Luzin's suggestion,
was studying the work of the French mathematician H. Lebesgue
published in a French mathematical journal [15]. The work was
dedicated to analytically representable functions. Suslin found
that the statement of one of the lemmas given by Lebesgue without
a proof was erroneous. Suslin confirmed it by a relevant example.
After thinking further about this problem, in the summer of 1916
Suslin constructed a non-Borel set which was a projection of a
Borel set. Soon a direct connection became clear between this
example and the original problem which Luzin posed to Aleksandrov
and Suslin, and also with the A-operation: Suslin constructed
a set which was not a Borel set but which was obtained from closed
sets (intervals) by applying the A-operation. In other words there
are A-sets which are not Borel sets (B-sets), that is to say, the
class of A-sets is significantly broader than the class of B-sets.
As a result Suslin discovered a new class of sets which became
known as A-sets, or analytical or Suslin sets. Aleksandrov
recalled his attempts to solve the problem of the relations
between the classes of A-sets and B-sets: "My extremely persistent
thoughts stopped only in the early autumn of 1916 when it became
known that during the summer Suslin had constructed an example
of an A-set which was not a B-set, and by doing that he opened
up a new phase in the development of the whole of descriptive
set theory" ([14], p. 235). (Concerning the terminology of open
sets see [14], p. 235-236.)" [Renfro -- I believe "open set" here
means its early usage as "arbitrary set", and not its present usage
as set that is equal to its interior.]

(p. 375): "Suslin also established a number of properties of these
sets and thereby laid the foundation of their theory which was
later developed more deeply by Luzin and his students, by the
Polish mathematician W. Sierpinski, and by the German mathematician
F. Hausdorff, and is most fully represented in the manuscripts
[16], [17]."

(p. 375): "In the preface to the French edition of the first of
these books H. Lebesgue acknowledges: "The source of all the
problems described here has turned out to be a crude mistake
in my memoire on analytically representable functions. It was
a fruitful mistake, and I was simply inspired to make it!"
([18], p. 9)."

(p. 375): "During those years the Polish mathematician W. Sierpinski
was in Moscow and also participated in the work of the scientific
seminars of Luzin. He recalled: "I happened to be a witness of
how Suslin told Luzin of his observation and handed to him the
manuscript of his first work. Luzin took the report of the young
student very seriously and confirmed that the student had indeed
found a mistake in the work of a famous scientist" ([19],
pp. 189-190, [20], pp. 33-34)."

(pp. 375-376): "Suslin gave a talk about his discovery entitled
"B-measurable sets" on 28 November 1916 in the scientific student
seminar ([21], p. 198). The results which laid the foundations
of the theory of analytic (Suslin) sets were published by Suslin
in his work in _Comptes_Rendus_ of the Paris Academy of Sciences
on 8 January 1917 [1]. It was presented for publication by the
famous French mathematician J. Hadamard. This was the only work
of Suslin published during his lifetime."

(p. 376): "The impression which Suslin made on his fellow student
and scientific colleague Aleksandrov, who later became an
academician, was as follows. "Already in his early student years
Suslin appeared an interesting and unusual person. When he was
18 or 19 he had already composed for himself the programme of
his further intellectual development. Mathematics was only the
beginning of this programme. The second stage was going to be
physics and chemistry and then biology was supposed to follow.
The conclusion of the programme was medical science, to which
Suslin was intending to dedicate the rest of his life. As we
will see later, Suslin did not go further than his first step
in the implementation of his programme. He died a mathematician,
a bright and original mathematician, and one of the creators of
modern descriptive set theory" ([14], p. 232)."

(p. 376): "Mikhail Suslin went through his Moscow University
course and passed the state examinations brilliantly in February-
March 1917. In his student's record book and his examination report
there are no other marks except 'very satisfactory' (the top mark
in the three grade system adopted in the pre-revolutionary
universities of Russia) [22], [23]. He graduated from Moscow
University with a Diploma of the first degree."

(p. 376): "On 22 March 1917 Extraordinary Professor N. N. Luzin
sent the following request to the Faculty of Physics and

I have the honour to humbly ask the Faculty of Physics and
Mathematics that Mikhail Yakovlevich Suslin, who graduated
with the Diploma of the first degree, be given leave to
remain at the university for two years without pay in
preparation for a professorship.
During his studies at the university Suslin was mainly
interested in the theory of functions of a real variable.
Concerning this subject he studied in detail the works of
Hausdorff, Baire and Lebesgue as well as following an
appropriate special course.
As a result of his systematic study he made the discovery
of an important class of non-measurable sets [Renfro -- He
means non-Borel measurable.] which are definable in a finite
way. Their existence until then had been denied by the French
mathematical school following the errors in the classical
memoir of Lebesgue which were also revealed by Suslin.
This work of his, which has attracted general attention,
and which in my view has many interesting mathematical and
philosophical consequences, was published in _Comptes_Rendus_
of the Paris Academy on 8 January of that year.
Suslin knows the French and German languages. A reprint of
his work together with an instruction for future studies is
attached herewith [24]."

(p. 376): "On 27 April 1917 an administrator of the Moscow
educational district, S. Chaplygin, granted permission to stay
at the university in preparation for a professorship for the
period of two years without financial support to Mikhail Suslin
in the department of pure mathematics from 28 March 1917 [25]."

(pp. 376-377): "However, Mikhail Suslin's state of material
well-being, since he only had leave to remain at the university,
was very unstable. On 20 September 1917 Luzin again wrote to the
Faculty of Physics and Mathematics asking for assistance for
Suslin [26]:

Herewith I have the honour to humbly ask the Faculty of
Physics and Mathematics to plead for a studentship from
ministerial funds for Mikhail Yakovlevich Suslin for the
period of two years at 1200 roubles per year. This student
has leave to remain at the university from spring of this
year in the department of pure mathematics.
During his study at the university Suslin took part in
the mathematical seminars on the theory of functions organized
by Professor D. F. Egorov and myself. In addition Suslin
attended all recommended theoretical courses taught at the
university during his time here.
As a result of his studies he has produced extremely
valuable independent work on Baire's classification; a short
list of the results of this work is published in "Comptes
Rendus de l'Academie des Sciences de Paris" at the meeting
of the Academy of 8 January 1917 under the title "Sur une
définition des ensembles mesurables B sans nombres
In this important work Suslin eliminates the use of
transfinite numbers when obtaining B-measurable sets.
The use of transfinite numbers was previously regarded
as unavoidable in this instance. Having given a new method
of obtaining B-measurable sets Suslin discovered a new
class of sets which are no longer B-measurable and yet
which can be obtained as a result of simply analytic
procedures. Thus, for example, the set of values of the
sum of polynomials is sometimes not a B-measurable set.
This result of his is a real discovery which has major
significance for set theory and which has already attracted
general attention. The existence of such sets was denied
by French mathematicians until recently (until the work
of Suslin).
As a result of the extraordinary talent displayed by
Suslin and his ability to work hard I put forward a
petition to the faculty for a studentship for Suslin."

(p. 377): "On 27 September 1917 the dean of the faculty L. K.
Lakhtin supported this request in his petition to the rector of
the university and on 29 September the latter issued a resolution
'to recommend'."

(pp. 377-378): "The revolution, civil war and intervention made
life in Moscow very difficult and many scientists left the capital
hoping to find better working conditions in provincial towns. In
particular V. V. Golubev and I. I. Privalov started to work at
Saratov University. Luzin was offered the directorship of the
department of pure mathematics in the re-opened Ivanovo-Voznesensk
Polytechnic Institute. He accepted the offer and invited his
students D. E. Men'shov, A. Ya. Khinchin, V. S. Fedorov, V. N.
Veniaminov and others to join the department. One of the organizers
of the institute, Professor of Building Mechanics V. M. Keldysh
(the father of M. V. Keldysh who later became the president of
the Academy of Sciences of the USSR) told Lyusternik about his
negotiations with Suslin concerning the latter joining the
institute. These negotiations finished with a letter from Suslin
in which he explained about his illness from which he could not
hope to "climb out" [27]. Yet Suslin agreed to move to Ivanovo
and work at the institute. In the academic year 1918-1919 he
worked in the Chemistry Faculty having the post of Extraordinary
Professor in the pure mathematics department [28]. His colleague
from the department Vladimir Semenovich Fedorov recalled: "As a
professor Suslin left in his audiences the clear and definite
memory of his distinct and rigorously paced lecturing, being
infallibly methodical, being able to make students work and
having compassion to the needs and demands of the audience" [29]."

(p. 378): "The winter of 1918-1919, which Suslin spent in
Ivanovo-Voznesensk, turned out to be very difficult and possibly
fatal for him. When moving from Moscow his basket containing
linen and underwear was stolen [30]. In the middle of December
he sought permission to go home to Krasavka to get food [31],
since the shortage of nourishment in the city was becoming
critical. It is not known whether this trip of his took place
but even if it did it brought him no relief. In January he
(together with Luzin, Fedorov and other professors) was moved
from the centre of the city, where the institute was located,
to the outskirts [32]. The coming summer of 1919 did not bring
him any relief either. The cold and hungry winter had broken
his already weak health. On 14 June 1919 Suslin sent a request
addressed to the Presidium of the Council responsible for
establishing the Ivanovo-Voznesensk Polytechnic Institute asking
to be allowed a holiday for the whole period of the summer vacation
so that he could at least restore his health a little by going to
stay with his parents. On 20 June the Presidium refused Suslin's
request and agreed to give him a holiday of just one month from
28 June to 28 July [33]. In August he was supposed to coach the
students who were falling behind. On the same day, 20 June 1919,
Suslin decided to leave the Inanovo-Voznesensk Polytechnic
Institute and wrote the following letter [34]:"


(p. 379): "During his year of work at the institute the new
professor gained the students' affection and when the students
learned of the intention of their beloved lecturer to leave the
institute, they issued the following statement [35]:

To the Council of the Ivanovo-Voznesensk Polytechnic Institute
from the group of students who attended the mathematics
tutorials of M. Ya. Suslin.

A statement

We students, having learned that M. Ya. Suslin has asked to
resign, herewith state to the Council of Professors that in
the person of Mikhail Yakovlevich we have a wonderful teacher.
Our studies with him have been extremely interesting and rich
in content. We students have attended his classes with great
joy. Those of us who have had temporarily to stop studying
under him regret it very much, and are hoping to continue to
study under his guidance. Therefore we all wish to have M. Ya.
Suslin among our lecturers in the future, and we ask for the
cooperation of the Council of Professors.
Ivanovo-Voznesensk, 14 July 1919."

(pp. 379-380): "During the summer months Suslin had to give the
mathematics tutorials against which he had protested so vigorously
and unsuccessfully [37]. From 1 September 1919 Suslin was released
from his duties at the Ivanovo-Voznesensk Polytechnic Institute
at his own request. P. S. Aleksandrov recalls: "... Suslin did
not get on in Ivanovo and soon lost his job there. In view of
this, thanks to the initiative of V. V. Golubev and I. I. Privalov,
a plan emerged to get a professorship at Saratov University for
Suslin. A recommendation by N. N. Luzin was expected. Luzin did
not give it and did not support the idea of giving Suslin a
lectureship at Saratov University. Having failed to obtain this
position, Suslin went back to his village (in the Saratov province).
He fell ill soon after with typhus and died. Thus was written one
of the most tragic chapters in the history of Soviet mathematics.
A portrait of Suslin stood on Luzin's writing table until the end
of his life. It was the only portrait of Suslin I have every seen"
([14], p. 241)."

(p. 380): "After the death of Suslin, in 1920 in the first volume
of the newly created Polish mathematical journal _Fundamenta_
_Mathematicae_ (one of its founders was W. Sierpinski) ten problems
were published. The third one was due to Suslin: "Let a linearly
ordered set without gaps and jumps possess the property that every
set of disjoint non-empty intervals is at most countable. Will
this set necessarily be an (ordinary) linear continuum?" [2].
A set satisfying these conditions was given the name of a Suslin
continuum. At the beginning of the 40's it was proved that the
existence of a Suslin continuum is equivalent to the existence
of a special ordered set called a Suslin tree. The solution to
Suslin's problem was clarified only at the beginning of the 60's
when the American mathematician P. Cohen invented an essentially
new method of proof -- the method of forcing. At the International
congress of Mathematicians in Moscow in 1966 for this achievement
he was awarded the highest honour available to mathematicians --
the Fields medal. Suslin's conjecture, like Cantor's continuum
hypothesis, turned out to be independent of the other axioms of
set theory. Also the mutual independence of the two conjectures
themselves -- of Suslin and of Cantor -- was established [39],
[3]. Questions connected with Suslin's conjecture continue to
be studied in numerous works concerning set theory. Generalizations
of this conjecture are considered, new constructions and concepts
connected with it are introduced and named after Suslin. These
constructions are widely used not only in set theory, they penetrate
the neighbouring areas -- model theory and set-theoretic topology."

(p. 380): "The third article published under the name of Suslin [40]
remains slightly in the shade. As written in its title it is
composed by the Polish mathematician K. Kuratowski (1896-1980)
"on the basis of a posthumous memoir of Mikhail Suslin". It
contains an elegant construction of a non-countable subfield
of the real numbers which does not coincide with the field of
real numbers itself. One would like to hope that these ideas of
Suslin also await a happy destiny."

[snip details of various commemorative awards and conferences]

Dave L. Renfro

Dave L. Renfro

Jul 29, 2006, 7:33:25 PM7/29/06
Dave L. Renfro wrote (in part):

I thought it would be useful to post some additional information
about Lebesgue's error.

There are three things involved. A theorem Lebesgue was trying
to prove, a lemma that Lebesgue was trying to prove as part of
the proof of the theorem, and a proof (attempt) of the lemma.

THEOREM (TRUE): (roughly) If f is an invertible Baire function,
then the inverse of f is a Baire function.

LEMMA (FALSE): The projection of a Borel set is a Borel set.

We first look at the part of the proof of the lemma that
is invalid.

Because the hierarchy of Borel sets is defined by (transfinite)
induction using the countable union operation (Lebesgue's
"operation I") and the countable intersection operation
(Lebesgue's "operation II"), Lebesgue sought to prove the
lemma by (transfinite) induction. Lebesgue thought this was
immediate by noting that projections preserve both unions
and intersections. However, while it is true that

proj(union) equals union(proj),

it is possible for

proj(inter) not equal inter(proj).

Indeed, if A = {(x,0): x in R}, B = {(x,1): x in R}, and proj
means projection to the x-axis, then

proj(A inter B) = empty set


proj(A) inter proj(B) = R.

We can also show the non-commutativity of 'proj' and 'inter' by
using a nonempty collection of disjoint intervals on the y-axis
and projecting to the x-axis.

NOTE 1: This non-commutativity arises from a non-commutativity
of the logical quantifiers "for all" and "there exists".
The projection of E in R^2 to the x-axis is the set
{x: there exists y with (x,y) in E}, and the intersection
of a collection of sets is associated with "for all".
Because (there exists)(for all) ==> (for all)(there exists),
proj(inter) is a subset of inter(proj).

NOTE 2: It can be shown that nothing new arises if we replace
the operation of projection with the operation of taking
continuous images. Thus, the Suslin sets in R are the
sets f[B] = {y: there exists x with y = f(x)} where B
varies over the Borel sets in R and f varies over the
continuous functions from R to R. This substitution of
continuous images for projections can be used throughout
the projective set hierarchy, I believe.

TECHNICAL NOTE: Actually, Lebesgue's "operation II" was the
operation of intersection applied to countable
decreasing sequences of sets. That is, the
intersection of E_1, E_2, E_3, ... when
E_1 superset E_2 superset E_3 superset ...
However, by letting E_n = {(0,y): 0 < y < 1/n},
we can still get a counterexample.

[Incidentally, Lebesgue used II for the countable intersection
operation and II' for the operation II applied to a decreasing
sequence of sets. I should have written II' in the English
translation of the relevant passage of Lebesgue's paper. The
correct symbol II' is used in the Russian Mathematical Surveys
English translation but, when I was typing it, I assumed the
prime symbol was a printer's error and left it out.]

Of course, all this shows is that the *method* of proof that
Lebesgue used fails. There still remains the possibility that
the lemma itself is true. However, Suslin showed that the
lemma is false by giving a counterexample. Indeed, Suslin's
counterexample occurs at the earliest place in the Borel
hierarchy where such an example is possible, namely at the
first place the operation of countable intersection is employed,
by showing there exists a G_delta set in the plane whose projection
is not a Borel set. [Since translations and rotations do not
affect the Borel class a set belongs to, we can assume that
all projections are to the x-axis.]

It turns out that the theorem Lebesgue was attempting to prove
about Baire functions is true. A special case of Lebesgue's
theorem is: "If f: R^2 --> R is a Baire function such that
for each y there exists a unique x such that f(x,y) = 0,
then x as a function of y is Baire." Lusin later gave a
correct proof of Lebesgue's theorem.

Dave L. Renfro

William Elliot

Jul 30, 2006, 5:22:46 AM7/30/06
On Sat, 29 Jul 2006, Dave L. Renfro wrote:
> William Elliot wrote (in part):
> > M. Souslin is from Moscow university. That's all I could
> > find out as links to his biographies were faulty and he's
> > not in the two math biography sites I know. Another Abel
> > lost to the world?
> I'm starting a new thread because what I'm posting is likely
> to be of wider interest.
> Here's a biography of Suslin:
Thanks Dave, it was excellent read.

Riddle of the day: why is history so much fun?

.... because truth is stranger than fiction

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