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A problem in the theory of [[Trigonometric series|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 [[#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.

====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 [[#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 $
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. [[CA-set| $  C {\mathcal A} $-
set]]) countable or do they have the cardinality of the continuum? The resolvent  $  E $
of this problem is a [[Luzin set|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|Baire property]]?

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} $
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]].

''B.A. Efimov''

====References====
<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>