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converges almost everywhere on  $  [ 0 , 2 \pi ] $.  
 
converges almost everywhere on  $  [ 0 , 2 \pi ] $.  
The conjecture was made by N.N. Luzin in 1915 in his dissertation (see [[#References|[1]]]). Luzin's problem was solved in 1966 in the affirmative sense by L. Carleson (see [[Carleson theorem|Carleson theorem]]). Until Carleson's paper [[#References|[2]]] it was not even known whether the Fourier series of a continuous function on the interval  $  [ 0 , 2 \pi ] $
+
The conjecture was made by N.N. Luzin in 1915 in his dissertation (see [[#References|[a1]]]). Luzin's problem was solved in 1966 in the affirmative sense by L. Carleson (see [[Carleson theorem|Carleson theorem]]). Until Carleson's paper [[#References|[a2]]] it was not even known whether the Fourier series of a continuous function on the interval  $  [ 0 , 2 \pi ] $
 
converges at least at one point.
 
converges at least at one point.
  
====References====
+
====Comments====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N.N. Luzin,  "The integral and trigonometric series" , Moscow-Leningrad  (1953)  pp. 219  (In Russian)  (Thesis; also: Collected Works, Vol. 1, Moscow, 1953, pp. 48–212)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L. Carleson,  "Convergence and growth of partial sums of Fourier series"  ''Acta Math.'' , '''116'''  (1966)  pp. 135–157</TD></TR></table>
+
See [[Luzin set]] for usual terminology. For other problems of Luzin see [[Luzin theorem]].
  
 
''B.S. Kashin''
 
''B.S. Kashin''
  
One of a number of fundamental problems in set theory posed by N.N. Luzin [[#References|[1]]], for the solution of which he proposed the method of resolvents. Namely, a problem  $  P $
+
One of a number of fundamental problems in set theory posed by N.N. Luzin [[#References|[b1]]], for the solution of which he proposed the method of resolvents. Namely, a problem  $  P $
 
of set theory is posed in a resolvent if one can indicate a set of points  $  E $
 
of set theory is posed in a resolvent if one can indicate a set of points  $  E $
 
such that  $  P $
 
such that  $  P $
Line 53: Line 53:
 
Problem 3. Does there exist a Luzin set without the [[Baire property|Baire property]]?
 
Problem 3. Does there exist a Luzin set without the [[Baire property|Baire property]]?
  
Luzin conjectured that the Problems 1, 2, 3 are undecidable. This conjecture has been confirmed (see [[#References|[3]]], [[#References|[4]]]). Connections between these problems have been established. For example, from the existence of an unmeasurable set of type  $  A _ {2} $
+
Luzin conjectured that the Problems 1, 2, 3 are undecidable. This conjecture has been confirmed (see [[#References|[b3]]], [[#References|[b4]]]). Connections between these problems have been established. For example, from the existence of an unmeasurable set of type  $  A _ {2} $
 
follows the existence of an uncountable set of type  $  C {\mathcal A} $
 
follows the existence of an uncountable set of type  $  C {\mathcal A} $
not containing a perfect subset. I. Novak [[#References|[5]]] obtained an affirmative solution of Luzin's problem about parts of the series of natural numbers, starting from the [[Continuum hypothesis|continuum hypothesis]] or the negation of the [[Luzin hypothesis|Luzin hypothesis]].
+
not containing a perfect subset. I. Novak [[#References|[b5]]] obtained an affirmative solution of Luzin's problem about parts of the series of natural numbers, starting from the [[Continuum hypothesis|continuum hypothesis]] or the negation of the [[Luzin hypothesis|Luzin hypothesis]].
 
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N.N. [N.N. Luzin] Lusin,  "Sur le problème de M. Emile Borel et la méthode des résolvants"  ''C.R. Acad. Sci. Paris'' , '''181'''  (1925)  pp. 279–281</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N.N. Luzin,  "Collected works" , '''2''' , Moscow  (1958)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  P.S. Novikov,  "On the non-contradictibility of certain propositions in descriptive set theory"  ''Trudy Mat. Inst. Steklov.'' , '''38'''  (1951)  pp. 279–316  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  R. Solovay,  "A model of set theory in which every set of reals is Lebesgue measurable"  ''Ann. of Math. (2)'' , '''92''' :  1  (1970)  pp. 1–56</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  J. Novak,  "On some problems of Lusin concerning the subsets of natural numbers"  ''Czechoslovak. Math. J.'' , '''3'''  (1953)  pp. 385–395</TD></TR></table>
 
  
 
''B.A. Efimov''
 
''B.A. Efimov''
 
====Comments====
 
See [[Luzin set|Luzin set]] for usual terminology. For other problems of Luzin see [[Luzin theorem|Luzin theorem]].
 
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  T.J. Jech,   "Set theory" , Acad. Press  (1978)  pp. Chapt. 7  (Translated from German)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  Y.N. Moschovakis,  "Descriptive set theory" , North-Holland  (1980)</TD></TR></table>
+
<table>
 +
<TR><TD valign="top">[a1]</TD> <TD valign="top">  N.N. Luzin, "The integral and trigonometric series" , Moscow-Leningrad  (1953)  pp. 219  (In Russian)  (Thesis; also: Collected Works, Vol. 1, Moscow, 1953, pp. 48–212)</TD></TR>
 +
<TR><TD valign="top">[a2]</TD> <TD valign="top">  L. Carleson, "Convergence and growth of partial sums of Fourier series"  ''Acta Math.'' , '''116'''  (1966)  pp. 135–157</TD></TR>
 +
<TR><TD valign="top">[b1]</TD> <TD valign="top">  N.N. [N.N. Luzin] Lusin, "Sur le problème de M. Emile Borel et la méthode des résolvants"  ''C.R. Acad. Sci. Paris'' , '''181'''  (1925)  pp. 279–281</TD></TR>
 +
<TR><TD valign="top">[b2]</TD> <TD valign="top">  N.N. Luzin, "Collected works" , '''2''' , Moscow  (1958)  (In Russian)</TD></TR>
 +
<TR><TD valign="top">[b3]</TD> <TD valign="top">  P.S. Novikov, "On the non-contradictibility of certain propositions in descriptive set theory"  ''Trudy Mat. Inst. Steklov.'' , '''38'''  (1951)  pp. 279–316  (In Russian)</TD></TR>
 +
<TR><TD valign="top">[b4]</TD> <TD valign="top">  R. Solovay, "A model of set theory in which every set of reals is Lebesgue measurable"  ''Ann. of Math. (2)'' , '''92''' :  1  (1970)  pp. 1–56</TD></TR>
 +
<TR><TD valign="top">[b5]</TD> <TD valign="top">  J. Novak, "On some problems of Lusin concerning the subsets of natural numbers"  ''Czechoslovak. Math. J.'' , '''3'''  (1953)  pp. 385–395</TD></TR>
 +
<TR><TD valign="top">[c1]</TD> <TD valign="top">  T.J. Jech, "Set theory" , Acad. Press  (1978)  pp. Chapt. 7  (Translated from German)</TD></TR>
 +
<TR><TD valign="top">[c2]</TD> <TD valign="top">  Y.N. Moschovakis,  "Descriptive set theory" , North-Holland  (1980)</TD></TR>
 +
</table>

Revision as of 11:07, 9 April 2023

efere

A problem in the theory of trigonometric series. It consists in proving Luzin's conjecture, stating that the Fourier series

$$ \tag{* } a _ {0} ( f ) + \sum _ { n= } 1 ^ \infty \{ a _ {n} ( f ) \cos nx + b _ {n} ( f ) \sin nx \} $$

of a Lebesgue-measurable function $ f $, defined on the interval $ [ 0 , 2 \pi ] $, with finite integral

$$ \int\limits _ { 0 } ^ { {2 } \pi } | f ( x) | ^ {2} dx, $$

converges almost everywhere on $ [ 0 , 2 \pi ] $. The conjecture was made by N.N. Luzin in 1915 in his dissertation (see [a1]). Luzin's problem was solved in 1966 in the affirmative sense by L. Carleson (see Carleson theorem). Until Carleson's paper [a2] it was not even known whether the Fourier series of a continuous function on the interval $ [ 0 , 2 \pi ] $ converges at least at one point.

Comments

See Luzin set for usual terminology. For other problems of Luzin see Luzin theorem.

B.S. Kashin

One of a number of fundamental problems in set theory posed by N.N. Luzin [b1], for the solution of which he proposed the method of resolvents. Namely, a problem $ P $ of set theory is posed in a resolvent if one can indicate a set of points $ E $ such that $ P $ is solved affirmatively every time one can indicate a point of $ E $, and is solved negatively if one can prove that $ E $ is empty. The set $ E $ itself is called the resolvent of the problem $ P $.

Problem 1. Are all co-analytic sets (cf. $ C {\mathcal A} $- set) countable or do they have the cardinality of the continuum? The resolvent $ E $ of this problem is a Luzin set of class at most 3; that is, if one can find a point of $ E $, then there is an uncountable co-analytic set without perfect part, while if $ E $ is empty, then there are no such co-analytic sets.

Problem 2. Do there exists Lebesgue-unmeasurable Luzin sets?

Problem 3. Does there exist a Luzin set without the Baire property?

Luzin conjectured that the Problems 1, 2, 3 are undecidable. This conjecture has been confirmed (see [b3], [b4]). Connections between these problems have been established. For example, from the existence of an unmeasurable set of type $ A _ {2} $ follows the existence of an uncountable set of type $ C {\mathcal A} $ not containing a perfect subset. I. Novak [b5] obtained an affirmative solution of Luzin's problem about parts of the series of natural numbers, starting from the continuum hypothesis or the negation of the Luzin hypothesis.

B.A. Efimov

References

[a1] N.N. Luzin, "The integral and trigonometric series" , Moscow-Leningrad (1953) pp. 219 (In Russian) (Thesis; also: Collected Works, Vol. 1, Moscow, 1953, pp. 48–212)
[a2] L. Carleson, "Convergence and growth of partial sums of Fourier series" Acta Math. , 116 (1966) pp. 135–157
[b1] N.N. [N.N. Luzin] Lusin, "Sur le problème de M. Emile Borel et la méthode des résolvants" C.R. Acad. Sci. Paris , 181 (1925) pp. 279–281
[b2] N.N. Luzin, "Collected works" , 2 , Moscow (1958) (In Russian)
[b3] P.S. Novikov, "On the non-contradictibility of certain propositions in descriptive set theory" Trudy Mat. Inst. Steklov. , 38 (1951) pp. 279–316 (In Russian)
[b4] R. Solovay, "A model of set theory in which every set of reals is Lebesgue measurable" Ann. of Math. (2) , 92 : 1 (1970) pp. 1–56
[b5] J. Novak, "On some problems of Lusin concerning the subsets of natural numbers" Czechoslovak. Math. J. , 3 (1953) pp. 385–395
[c1] T.J. Jech, "Set theory" , Acad. Press (1978) pp. Chapt. 7 (Translated from German)
[c2] Y.N. Moschovakis, "Descriptive set theory" , North-Holland (1980)
How to Cite This Entry:
Luzin problem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Luzin_problem&oldid=47722
This article was adapted from an original article by B.S. Kashin, B.A. Efimov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article