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\author{
Marek Balcerzak,
Institute of Mathematics,
{\L}\'od\'z Technical University, al. \\
Politechniki 11, 90-924 {\L}\'od\'z, Poland 
(mbalce@@krysia.uni.lodz.pl)\\
Krzysztof Ciesielski%
\thanks{This work was partially supported by NSF Cooperative 
Reasearch Grant INT-9600548 with its Polish part financed by KBN.\endgraf 
Papers authored or co-authored by a Contributing Editor are managed 
by a Managing Editor or one of the other Contributing Editors.},
Department of Mathematics, West Virginia University, \\
Morgantown, WV 26506-6310, USA
(KCies@@wvnvms.wvnet.edu) }
\title{On sup-measurable functions problem}
\date{}
\MathReviews{Primary: 26B30; Secondary: 04A15, 28A05, 54H05.}
\keywords{Lebesgue measurability, Baire property, sup-measurable function.}

\markboth{Marek Balcerzak and Krzysztof Ciesielski}{On sup-measurable 
functions problem}

%End of proof marker
\def\qed{\hfill$\Box$}
%Proof
\def\proof{\noindent{\sc Proof.} }

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\newtheorem{cor}{Corollary}
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\newtheorem{rem}{Remark}

\begin{document}


\maketitle
\begin{abstract}
We show some results connected with the problem whether it is consistent 
that every sup-measurable function $F\colon\R^2\to\R$ is measurable. 
We will also relate this problem to
a von Weizs\"{a}cker problem concerning
a generalization of Blumberg theorem. 
\end{abstract}

\bigskip
We use standard set-theoretical notation as in~\cite{Ci:book}.
Symbol $|X|$ will stand for the cardinality of a set $X$. 
The cardinality of the set $\R$ of real numbers is denoted by $\co$.
For a set $E\sb\R^2$, we denote by $\dom(E)$ and $\ran(E)$ 
its projections on the first and on the second axis, respectively. 
A function $f$
from a subset of $\R$ into $\R$ will be identified with its graph. 
If $\I$ is a proper
ideal of subsets of $\R$, by $\non (\I )$ we denote the minimal 
cardinality of a set that is not in $\I$.
Symbols $\M$ and $\N$ will stand for the ideals of meager sets 
and of null sets in
$\R$, respectively.

We say that a function $F\colon\R^2\to\R$ is {\em sup-measurable\/} 
if the function $F_f\colon\R\to\R$ given by $F_f(x)=F(x,f(x))$, $x\in\R$, 
is measurable for each measurable function $f\colon\R\to\R$. 
We will also consider a dual
category analog notion that is obtain from the above by replacing 
the requirement of measurability of functions with the requirement 
that the appropriate functions have the Baire property. 
If we want to distinguish these two notions, 
we will use the phrases ``Lebesgue sup-measurability'' and 
``Baire sup-measurability.''
Observe that if in the above definitions we require that $F_f$ 
is measurable (or has the Baire property) only for every Borel function $f$
then we will still obtain the same notions. (See~\cite[Lemma~1.1]{B}.) 

It is not hard to find measurable functions which are not sup-measurable. 
(See \cite{Sr} or \cite[Cor.~1.4]{B}.)
Under the continuum hypothesis
CH or some weaker set-theoretical assumptions, 
nonmeasurable sup-measurable functions were constructed 
in \cite{GL}, \cite{Kh1}, \cite{Kh2}, and \cite{B}. 
(Similarly, for the category case. See \cite{GG}.) 
These rather exotic functions are called ``monsters'' 
in the literature. (See \cite{KP}.) It is interesting 
to know whether the existence of such functions can be 
proved in ZFC or 
whether there is a model of ZFC with no such examples. 
(See \cite{Kh1}, \cite{Kh2}, \cite{B}, and \cite{C} 
where this question was mentioned.)
In our article we discuss these problems. 

Note that the existence of nonmeasurable sup-measurable 
functions has some consequences
in the theory of 
ordinary differential equations, which was described in
detail in the recent article of Kharazishvili \cite{Kh2}. 
Namely, let $G\colon\R^2\to\R$
and $\lew x_0,y_0\r\in\R^2$. We say that the Cauchy problem
\begin{equation} \label{Cy}
y'=G(x,y),\;\;\;y(x_0)=y_0
\end{equation}
has a (unique) solution in the class $\AC_l$ 
of locally absolutely continuous functions on $\R$ 
if there exists a (unique) function $f\in\AC_l$
such that $f(x_0)=y_0$ and $f'(x)=G(x,f(x))$ for 
almost all $x\in\R$. (Recall that
$f\in\AC_l$ if each $x\in\R$
has a neighbourhood $V$ such that 
the restriction $f|V$ of $F$ to $V$ is absolutely continuous.)
We will describe several nonmeasurable (sup-measurable) 
functions $G$ for which (\ref{Cy}) 
has a unique locally absolutely continuous
solution.

In \cite[Prop. 1.7]{B}
the author constructs (in ZFC)  
a nonmeasurable set $H\sb\R^2$ such that the characteristic 
function $F$ of $H$ satisfies the condition
\begin{equation} \label{cont}
|\{x\in\R\colon F(x,f(x))\neq 0\}|<\ce
\end{equation}
for every Borel function $f\colon\R\to\R$. 
Then $F$ is nonmeasurable and its sup-me\-a\-su\-ra\-bi\-li\-ty 
is implied by $\non (\N )=\ce$. Consider the Cauchy problem (\ref{Cy})
where $G(x,y)=F(x,y)+g(x)$
and $g\colon\R\to\R$ is a fixed locally Lebesgue integrable function. 
Note that $G$ is nonmeasurable. Using $\non (\N )=\ce$ and (\ref{cont}) 
we can easily check that the function 
$\vp (x)=y_0+\int_{x_0}^xg(t)dt$, $x\in\R$, is
a locally absolutely continuous 
solution of (\ref{Cy}). 
This solution is unique since if $\ps\in\AC_l$ is an arbitrary 
solution of (\ref{Cy}) then $\ps (x_0)=y_0$ and
\begin{equation} \label{rozw}
\ps '(x)=F(x,\ps (x))+g(x)
\end{equation}
for almost all $x$. But then $F_{\ps}$ is measurable. 
So the set 
$\{x\in\R\colon F_{\psi}(x)\neq 0\}$ is 
measurable and, by (\ref{cont}), it 
is of measure zero. 
Hence $F_{\ps}$ equals zero almost everywhere and 
from (\ref{rozw}) we infer that 
$$
\ps(x)=y_0+\int_{x_0}^xg(t)dt=\vp (x)
$$ 
for every $x\in\R$. Note that in that last
argument we do not use any extra set-theoretical assumptions. 
Also the whole power of sup-measurability of $F$ is not needed
to obtain the solution $\vp$ of (\ref{Cy}). In fact, it
suffices to know that $F_f$ equals zero almost everywhere for every 
Borel function $f$, which is implied by $\non (\N )=\ce$. 

The article \cite{Kh2}
contains a construction similar to that from \cite{B} but its 
advantage appears in special problems of type (\ref{Cy}). 
Namely, in \cite[Th.3]{Kh2} the respective nonmeasurable set 
$H\sb\R^2$ additionally meets every straight line in at most two points. 
In that case
\begin{equation} \label{2}
|\{ x\in\R\colon F(x,f(x))\neq 0\}|\le 2 \end{equation}
provided that $F$ is the characteristic
function of $H$ and $f$ is of the form $f(x)=ax+b$, $x\in\R$. 
Thus the Cauchy problem (\ref{Cy}),
where $G(x,y)=F(x,y)+a$,
has the unique locally absolutely continuous solution 
$\vp (x)= ax+(y_0-ax_0)$, $x\in\R$, and this (by (\ref{2}) 
and the previous argument) can be proved in ZFC. (See \cite[Th.4]{Kh2}.)
However, we do not know whether this $F$ is sup-measurable.
(Though it follows from (\ref{cont}) and $\non(\N)=\ce$.)

Now, let us recall the following simple lemma. 

\begin{lem} \label{lem1} {\rm \cite[Prop. 1.5]{B}}
The following conditions are equivalent: 
\begin{description}
\item[(I)] there is a nonmeasurable sup-measurable function $F\colon\R^2\to\R$;

\item[(II)] there is a nonmeasurable set 
$H\sb\R^2$ such that $\dom(H\cap f)$ is measurable for each Borel 
function $f\colon\R\to\R$; 

\item[(III)] there is a nonmeasurable sup-measurable 
function $F\colon\R^2 \to\{ 0,1\}$.
\end{description}
\end{lem}

\begin{rem}\label{rem1a}{\rm
We can add
three more conditions in Lemma~\ref{lem1}: 
\begin{description}
\item[(I$^\prime$)] there is a nonmeasurable function 
$F\colon\R^2\to\R$ such that $F_f$ is measurable for 
each continuous function $f\colon\R\to\R$; 

\item[(II$^\prime$)] there is a nonmeasurable set 
$H\sb\R^2$ such that $\dom(H\cap f)$ is measurable 
for each continuous function $f\colon\R\to\R$; 

\item[(III$^\prime$)] there is a nonmeasurable 
function $F\colon\R^2\to\{ 0,1\}$
such that $F_f$ is measurable for each continuous 
function $f\colon\R\to\R$. 
\end{description}
}\end{rem}

Among the implications
$$
\mbox{(I$^\prime$)}\Rw\mbox{(I)}\Rw\mbox{(II)} 
\Rw\mbox{(II$^\prime$)}\Rw\mbox{(III$^\prime$)}\Rw\mbox{(I$^\prime$)}
$$ 
only the first is
nontrivial and it follows from \cite[Lemma 1]{Kh2}. 

\begin{rem}\label{rem1b}{\rm
If we consider the category version of Lemma~\ref{lem1} 
(which holds true, see~\cite[Prop.~1.5]{B}) and category 
versions of (I$^\prime$), (II$^\prime$), and (III$^\prime$), 
Baire class~1 functions should be used 
in (I$^\prime$), (II$^\prime$), and (III$^\prime$)
instead of continuous ones,
and then all the conditions are equivalent. 
This follows from the fact that if $F_f$ has the 
Baire property for every Baire class~1 function 
$f\colon\R\to\R$ then $F$ is Baire sup-measurable. 
To show this, consider a function $g\colon\R\to\R$ with 
the Baire property. Then $g|A$ is continuous for a $G_{\delta}$ 
comeager set $A\sb\R$. We can extend $g$ to a function 
$f\colon\R\to\R$ of Baire class~1 \cite[\S 35,~VI]{Ku}.
Thus $F_g(x)=F_f(x)$ for each $x\in A$. 
Hence $F_g$ has the Baire property. 
}\end{rem}

\begin{rem}\label{rem1c}{\rm
Observe that there is an $F\colon\R^2\to\R$ such that 
$F_f$ has the Baire property for each continuous 
$f\colon\R\to\R$ but $F$ is not Baire sup-measurable.
}\end{rem}
To see it, first notice that the existence of an $F$ 
as above follows from the existence of a continuous 
function $g\colon[0,1]\setminus D\to\R$, where $D$ is 
the set of all dyadic numbers, such that 
\begin{equation}\label{eqNEW}
\text{the set $\{x\in[0,1]\setminus D\colon f(x)=g(x)\}$ is nowhere dense}
\end{equation}
for every continuous $f\colon\R\to\R$.

Indeed, if $g$ is such a function, take a subset $B$ of $[0,1]\setminus D$ 
without the Baire property and let $F\colon\R^2\to\R$ be the characteristic 
function of $g|B$. Then $F$ is not Baire sup-measurable since 
$F_{\bar g}^{-1}[\{ 1\}]=B$, where $\bar g\colon\R\to\R$ is a 
Borel extension of $g$.
On the other hand $F_f$ has the Baire property for every 
continuous function $f\colon\R\to\R$ since, by (\ref{eqNEW}), 
there exists a nowhere dense set $N$ such that $F_f(x)=0$ for 
all $x\in\R\setminus N$.

Now, function $g$ satisfying (\ref{eqNEW}) is constructed as follows. For
$x\in[0,1]\setminus D$
let $\la i_n(x)\in\{0,1\}\colon n<\omega\ra$ 
be a sequence from the unique binary representation of $x$, 
that is, such that $x=\sum_{n<\omega}i_n(x) 2^{-(n+1)}$.
The binary representation of $g(x)$ is obtain from 
the binary representation of $x$
by leaving the digits on the odd places unchanged and by 
interchanging all $0$'s and $1$'s at even places, that is, we put
\[
g(x)=\sum_{k<\omega}i_{2k+1}(x) 
2^{-(2k+2)}+\sum_{k<\omega}(1-i_{2k}(x)) 2^{-(2k+1)}.
\]
Now, $g$ is continuous, since
$|g(x)-g(x')|=|x-x'|$ for any $x,x'\in [0,1]\sm D$. 
To finish the argument for (\ref{eqNEW}) take continuous function
$f\colon\R\to\R$
and put $N=\{x\in[0,1]\setminus D\colon f(x)=g(x)\}$. 
The set $N$ is nowhere dense
since for every $d\in D$ there exists
an $h>0$ such that
\[
\text{either $N\cap(d,d+h)=\emptyset$ or $N\cap(d-h,d)=\emptyset$.}
\]
For suppose not, and let $y\in D$ witness for it. 
Then, there are sequences $h_j\to y$ and $k_j\to y$ 
in $N$ such that the first is increasing and the second is decreasing. 
Since $y\in D$, there exist sequences:
$\la i^+_n(y)\in\{0,1\}\colon n<\omega\ra$ eventually equal to $0$, 
and $\la i^-_n(y)\in\{0,1\}\colon n<\omega\ra$ eventually 
equal to $1$ such that
$y=\sum_{n<\omega}i^+_n(y) 2^{-(n+1)}=\sum_{n<\omega}i^-_n(y) 2^{-(n+1)}$. 
Let
\[
g^+(y)=\sum_{k<\omega}i^+_{2k+1}(y) 
2^{-(2k+2)}+\sum_{k<\omega}(1-i^+_{2k}(y)) 2^{-(2k+1)}
\]
and
\[
g^-(y)=\sum_{k<\omega}i^-_{2k+1}(y) 
2^{-(2k+2)}+\sum_{k<\omega}(1-i^-_{2k}(y)) 2^{-(2k+1)}.
\]
Then
\[
\lim_{n\to\infty}f(h_n)=\lim_{n\to\infty}g(h_n)
=g^-(y)\neq g^+(y)=\lim_{n\to\infty}g(k_n)=\lim_{n\to\infty}f(k_n) 
\]
contradicting continuity of $f$.

\bigskip

We are going to study the question
whether condition (I) given in Lemma~\ref{lem1} is independent of ZFC. 
However, the equivalent condition (II) will be more convenient.
In the sequel it will be denoted by 
{\bf (Nsup)}. As we have mentioned before, 
(Nsup) holds true in some models of ZFC where CH or some weaker 
conditions (e.g. MA or $\non (\N )=\ce$) are assumed. 
A.~Ros{\l}anowski and I.~Rec{\l}aw (oral communication) 
have observed independently that a Luzin set $H\sb\R^2$ fulfills 
(Nsup). In fact, we will try to show that some versions of (Nsup), 
which seem stronger, are independent of ZFC. They are the following
\begin{description}
\item[(Nsup1)] there is a nonmeasurable set $H\sb\R^2$ which is a function from
$\R$ to $\R$ such that
$\dom(H\cap f)$ is measurable for each Borel function $f\colon\R\to\R$; 

\item[(Nsup2)] there is a nonmeasurable set $H\sb\R^2$ which 
is a function with a nonmeasurable domain $\dom(H)$ and such 
that $\dom(H\cap f)$ is measurable for each Borel function 
$f\colon\R\to\R$. \end{description}
Conditions (Nsup), (Nsup1), and (Nsup2) have their category 
analogues and if we want to distinguish them, 
we write L-(Nsup) (like Lebesgue) or B-(Nsup) 
(like Baire), and similarly for (Nsup1) and (Nsup2). 
To show that (Nsup1) and (Nsup2) hold true in some 
models of ZFC we recall the construction
from~\cite[Prop.~1.7]{B}
in a modified, sharper version. 
For this, we need the following definitions. 

A Borel set $B\sb\R^2$ is called {\em big\/} 
if $|\{x\in\R\colon |B_x|>\om\} |>\om$ 
where $B_x=\{ y\in\R\colon \lew x,y\r\in B\}$ 
for $x\in\R$. Observe that, by the Fubini theorem and 
its category analog, all Borel sets of positive measure and all 
Borel nonmeager sets in $\R^2$ are big.
Also, we will say that a set $T\sb\R$ is {\em thick\/} 
provided it intersects every perfect subset of $\R$. 
(Such an intersection must then have cardinality $\co$.) 
Recall that a set $S\sb\R$ is a {\em Bernstein set\/} 
if both $S$ and $\R\setminus S$ are thick.

\begin{theor} \label{th1}
Assume that $T\sb\R$ is thick. 
Then there exists a set $H\sb T\times \R$ such that 
\begin{itemize}
\item $|H_x|=1$ for each $x\in T$, that is, 
$H$ is a function from $T$ into $\R$,

\item $|\dom(H\cap f)|<\ce$ for each Borel function $f\colon\R\to\R$, 

\item $H\cap B\neq\emptyset$ for each big set $B\sb\R^2$. 
\end{itemize}
In particular $H$
is neither measurable nor has the Baire property 
(as a subset of $\R^2$). 
\end{theor}

\proof 
Note that if $B\sb\R^2$ is a big set 
then $|B_x|>\om$ is equivalent to $|B_x|=\ce$ since $B_x$ is Borel.
Also $\{x\in\R\colon |B_x|>\om\}$ is 
analytic \cite[\S 39,~VII,~Th.~3]{Ku} and since it is uncountable,
it must contain a perfect subset. In particular, 
\begin{equation}\label{eqAAA}
|\{x\in T \colon |B_x|>\om\} |=\ce.
\end{equation}
That will be used in the construction.

First, we arrange, respectively, all numbers from $T$, all big subsets of $\R^2$, and all Borel functions from $\R$ to $\R$, into one-to-one transfinite sequences \begin{equation}
\la x_{\alpha}\colon \alpha <\ce\ra, \label{w1} \end{equation}
\begin{equation}
\la B_{\alpha}\colon \alpha <\ce\ra, \label{w2} \end{equation}
\begin{equation}
\la f_{\alpha}\colon\alpha <\ce\ra.\label{w3} \end{equation}
For $\alpha<\ce$ we
will define recursively the real numbers $s_{\al}$, 
$y_{\al}$, $t_{\al}$, and $z_{\al}$ as follows. 
Let $s_0=x_0$ and pick an arbitrary $y_0\in\R$. 
Let $t_0$ be the first number in (\ref{w1}) 
different from $s_0$ and such that $(B_0)_{t_0}\neq\emptyset$. 
Then pick an arbitrary $z_0\in (B_0)_{t_0}$. 

Next, assume that $0<\be<\ce$ and that
the points $s_{\al}$, $y_{\al}$, $t_{\al}$, and $z_{\al}$ 
are already constructed for all $\al<\be$. 
We will define $s_{\be}$, $y_{\be}$, $t_{\be}$, and $z_{\be}$. 
So, let $s_{\be}$ be the first number in (\ref{w1}) different 
from all $s_{\al}$ and $t_{\al}$ for $\al<\be$ and pick 
$$
y_{\be}\in\R\sm\{f_{\al}(s_{\be})\colon \al<\be\}. 
$$
Choose $t_{\be}$ as the first number in (\ref{w1}) 
different $s_{\be}$ and from all $s_{\al}$ and 
$t_{\al}$ for all $\al<\be$ and such that $|(B_{\be})_{t_{\be}}|=\ce$. 
Such a point exists by (\ref{eqAAA}).
Then pick
$$
z_{\be}\in (B_{\be})_{t_{\be}}\sm\{f_{\al}(t_{\be})\colon\al<\be\}. $$
This finishes the recursive construction of 
numbers $s_{\al}$, $y_{\al}$, $t_{\al}$, and $z_{\al}$ 
for all $\al <\ce$. Now, we put
\begin{equation}\label{r4}
H=\{\lew s_{\al},y_{\al}\r\colon\al<\ce\} 
\cup\{\lew t_{\al},z_{\al}\r\colon\al<\ce\}. 
\end{equation}
It follows from the construction that
$\bigcup_{\al<\ce}\{s_{\al},t_{\al}\}=T$ and that 
$H$ meets every big set $B_{\al}$. It is also clear 
that $|H_x|=1$ for each $x\in T$.
Moreover, for each $\al<\ce$ we have
$$
\dom(H\cap f_{\al})\sb \{ s_{\ga} \colon \ga <\al\}
\cup \{ t_{\ga}\colon \ga <\al\}.
$$
Hence $|\dom(H\cap f_{\al})|<\ce$.

Finally, if we suppose that $H$ is measurable then, 
by the Fubini theorem, $H$ should be a null set. 
But then there is a big set (of full measure) outside $H$ 
which contradicts our construction. Similarly $H$ does not 
possess the Baire property. 
\qed 

\begin{cor}\label{cor1}
Let $H$ be a set from Theorem~\ref{th1} used with $T=\R$. 
\begin{description}
\item[(a)] If $\non(\N)=\ce$ then $H$ witnesses {\rm L-(Nsup1)}. 
\item[(b)] If $\non(\M)=\ce$ then $H$ witnesses {\rm B-(Nsup1)}. 
\end{description}
\end{cor}

\begin{cor}\label{cor2}
Let $H$ be a set from Theorem~\ref{th1} used with a Bernstein set $T$. 
\begin{description}
\item[(a)] If $\non(\N)=\ce$ then $H$ witnesses {\rm L-(Nsup2)}. 
\item[(b)] If $\non(\M)=\ce$ then $H$ witnesses {\rm B-(Nsup2)}. 
\end{description}
\end{cor}

\begin{rem}\label{rem2a}{\rm
In the construction given in 
Theorem~\ref{th1} we can ensure that $y_{\be}$ 
and $z_{\be}$ are chosen so that the points 
$\lew s_{\be},y_{\be}\r$ and $\lew t_{\be},z_{\be}\r$ are 
not colinear with any two
of points from $H$ that are constructed earlier. 
Thus our set meets every straight line in at most 
two points and has all nice applications described 
in \cite{Kh2} and mentioned above. 
}\end{rem}

\begin{rem}\label{rem2b}{\rm
By Remark~\ref{rem1a} we may use only continuous 
functions $f$ in the measure formulation of (Nsup1) and (Nsup2).
For category, by Remark~\ref{rem1b},
we may use functions $f$ of Baire class~1. 
}\end{rem}
Now we shall prove a characterization of (Nsup2) 
connected with the discontinuity of restricted functions. 

\begin{theor} \label{t3}
Condition {\rm L-(Nsup2)} is true if and only if 
\begin{description}
\item[($*$)]
there exists a nonmeasurable set $H\sb\R^2$ which 
is a function with a nonmeasurable domain $\dom(f)$ and 
such that for each $Z\sb\dom(H)$, with $Z\notin\N$, 
the function $H|Z$ is not continuous. 
\end{description}
The analogous characterization in the category case is also true. 
\end{theor}

\proof  First we will show that L-(Nsup2) implies the condition ($*$).
So, let $H\sb\R^2$ satisfy
L-(Nsup2) and put $S=\dom(H)$. Consider two cases. 

Case 1. The only measurable subsets of $S$ are null sets. 
(In other words, the measurable kernel of $S$ is empty.) 
Suppose the assertion is false.
Thus there is a set $Z\sb S$, $Z\notin\N$, such that the 
function $H|S$ is continuous. Consider a Borel extension 
$f\colon\R\to\R$ of $H|Z$. (See \cite[Th.~1,~\S 35,~I]{Ku}.) Then
$Z\sb\dom(f\cap H)\sb S$, so $\dom(f\cap H)$ cannot be measurable, 
a contradiction. 

Case 2. The measurable kernel $K$ of $S$ is nonempty. 
Observe that then $H^{*}=H\sm (K\ti\R)$ can play the role of $H$ in 
L-(Nsup2) and thus Case~1 works. So, there is a set $Z\sb S\sm K$, 
$Z\notin\N$, such that the function $H^{*}|Z=H|Z$ is continuous. 

The proof of the category version of this implication is essentially the same. 

Now, to prove that ($*$) implies L-(Nsup2) let $H\sb\R^2$ satisfy ($*$) and
suppose that L-(Nsup2) is false.
Thus there is a continuous function $f\colon\R\to\R$ such that 
$\dom(H\cap f)$ is nonmeasurable. (Compare Remark~\ref{rem2b}.) 
Obviously for $Z=\dom(H\cap f)$ the function $f|Z$ is continuous. 
Thus $H|Z$ is continuous, a contradiction. 

In the category case we proceed similarly. Namely, we suppose that 
there is a function $f\colon\R\to\R$ with the Baire property such 
that $\dom(H\cap f)$ does not have the Baire property. 
Hence there exists a comeager $A\sb\R$ such that $f|A$ is continuous. 
Thus $f|Z$ is continuous for
$Z=A\cap\dom(H\cap f)\notin\M$, a contradiction. 
\qed 

\medskip


A function $g$ from $S\sb\R$ to $\R$ will be called {\em ugly\/} 
if $S$ is nonmeasurable and $g$ is nonmeasurable as a subsets of $\R^2$. 
Thus, by Theorem~\ref{t3}, condition $\neg$(L-(Nsup2)) is equivalent to 
the following statement.
\begin{quotation}
\noindent For every ugly function $g$ there exists a set $Z\sb\dom(g)$, 
$Z\notin\N$, such that $g|Z$ is continuous. 
\end{quotation}
The category case is analogous.

A similar statement for functions $g\colon \R\to\R$ is 
connected with the following open problem
of von Weizs\"{a}cker. (See \cite[Problem~AR(a)]{F} 
or \cite[Problem~1]{C}.) 
\begin{quotation}
\noindent Is it consistent that every function $g\colon\R\to\R$ 
restricted to some set of positive outer measure is continuous? 
\end{quotation}
The property considered in the above question will be denoted by (vWH), 
that is,
\begin{description}
\item[(vWH)] For every function $g\colon\R\to\R$ 
there exists a set $Z\notin\N$ such that $g|Z$ is continuous. 
\end{description}

\begin{rem}\label{rem3a}{\rm
From (vWH) it follows that the same statement is true provided 
$g\colon S\to\R$ and $S\sb\R$ is an $F_{\sigma}$ set such that 
$S\notin\N$. 
}\end{rem}

Indeed, each of the spaces $\R$ and $S$ can be partitioned 
into a countable number of uncountable Polish spaces with 
nonzero finite Lebesgue measure. In the case of $S$ we consider 
$\inte S$ (the interior of $S$) and divide $S\sm\inte S$ into 
a countable number of closed parts. Next, if necessary, 
we divide $\inte S$ into uncountable parts and add to 
them countable parts from the division of $S\sm\inte S$. 
Then we consider a Borel isomorphism $\vp$ from $\R$ onto $S$ 
which transforms Polish parts of $\R$ onto the respective 
Polish parts of $S$ and such that for each $E\sb\R$
the conditions $E\in\N$ and $\vp [E]\in\N$ are equivalent. 
This can be deduced from the fact that any two Polish probability 
spaces (where the probabilities vanish on the singletons) are 
Borel isomorphic. (See e.g. \cite[Th.~4.20]{CKW}.)
Now, if $g\colon S\to\R$, we apply (vWH) to $g\circ\vp$ and find 
a set $Z\notin\N$ such that $(g\circ\vp )|Z$ is continuous. 
Thus $\vp [Z]\notin\N$. By the Luzin theorem we find a sequence 
$\{F_n\}_{n=1}^\infty$ of pairwise disjoint closed sets of 
positive measure such that $\R\sm\bigcup_{n=1}^\infty F_n\in\N$ 
and $\vp^{-1}|F_n$ is continuous for every $n$. Since for the outer 
Lebesgue measure $\lambda^*$ we have 
$$
\lambda^* (\vp [Z])=\sum_{n=1}^\infty\lambda^*(\vp [Z]\cap F_n), 
$$
therefore $W=\vp [Z]\cap F_k\notin\N$ for some $k$ and 
$\vp^{-1}|W$ is continuous. 
Thus $(g\circ\vp\circ\vp^{-1})|W=g|W$ is continuous. 

\begin{rem}\label{rem3b}{\rm
The category analogue of (vWH) is consistent with ZFC which 
was shown by Shelah~\cite{Sh}.
}\end{rem}

\begin{rem}\label{rem3c}{\rm
From the category analogue of (vWH) it follows its 
version where $g\colon S\to\R$ and $S\sb\R$ is a $G_{\delta}$ 
set such that $S\notin\M$. }\end{rem}

The proof is similar to that presented for Remark~\ref{rem3a}. 
We use the fact that there is a Borel isomorphism $\vp$ 
from $\R$ onto $S$ such that for each $E\sb\R$ 
the conditions $E\in\M$ and $\vp [E]\in\M$ are equivalent. 
(See e.g. \cite[Th.~3.15]{CKW}.)
If $g\colon S\to\R$, we find a set $Z\notin\M$ such that 
$(g\circ\vp )|Z$ is continuous. Hence $\vp [Z]\notin\M$.
Note that $\vp^{-1}|A$ is continuous for a comeager set $A\sb\R$. 
Thus $W=\vp [Z]\cap A \notin\M$ and $(g\circ\vp\circ\vp^{-1})|W=g|W$ 
is continuous. 

\begin{theor}	\label{t4}
Condition {\rm (vWH)} implies condition
$\neg${\rm (L-(Nsup1))}.
Moreover, the analogous implication for the category case is also true. 
\end{theor}

\proof 
We have to show that for each function $H\colon\R\to\R$ which is 
a nonmeasurable subset of $\R^2$ there exists a Borel 
function $f\colon\R\to\R$ such that $\dom (H\cap f)$ is nonmeasurable.
Thus let $H\colon\R\to\R$ be a function such that $H\sb\R^2$ 
is nonmeasurable. Applying (vWH) to $g=H$ we obtain a set $Z\notin\N$ 
such that $H|Z$ is continuous. Consider a Borel extension 
$f\colon\R\to\R$ of $H|Z$. Then $\dom(H\cap f)\notin\N$. 
If $\dom(H\cap f)$ is nonmeasurable, the proof is finished. 
So assume that
$\dom(H\cap f)$ is measurable and include it in a $G_{\delta}$ 
set $A$ of the same measure. Let $B=\R\sm A$. Then $B\notin\N$ 
since otherwise $H=f$ almost everywhere, and thus $H$ would be 
a measurable subset of $\R^2$. Now, we apply the version of (vWH) 
from Remark~\ref{rem3a} to the function $H|B$. Then there is a set 
$Z_1\sb B$, $Z_1\notin\N$, such that $H|Z_1$ is continuous. 
Consider a Borel extension $f_1\colon B\to\R$ of $H|Z_1$. Then
$\dom (H\cap f_1)\notin\N$. If $\dom (H\cap f_1)$ is nonmeasurable, 
the proof is finished since, for the Borel function $h\colon\R\to\R$ given by 
\[
h(x)=\left\{
\begin{array}{ll}
f(x)&\mbox{for $x\in A$}\\
f_1(x)&\mbox{for $x\in B$,}\\
\end{array}
\right.
\]
the set $\dom (H\cap h)$ is nonmeasurable. 
So assume that $\dom (H\cap f_1)$ is measurable and include 
it in a $G_{\delta}$ set $A_1$ of the same measure. Let $B_1=B\sm A_1$.
Then $B_1\notin\N$ since otherwise,
$H=h$ almost everywhere (for $h$ given above), 
and thus $H$ would be a measurable subset of $\R^2$. 
We proceed inductively, constructing the sets $A_\xi$ 
and $B_\xi$ for $\xi<\omega_1$.
In some step before $\omega_1$
we obtain the required Borel function since otherwise, 
the sets $A,A_1,A_2,\ldots,A_\xi,\ldots$ would exhaust 
$\R$ in a countable number of steps (that is, their union 
would be of full measure in $\R$) and consequently $H$ 
would be a measurable subset of $\R^2$. 

In the analogous proof for the category analog we use 
Remark~\ref{rem3c}. 
\qed 

\medskip

From Remark~\ref{rem3b} and the category part of 
Theorem~\ref{t4} we infer the following.
\begin{cor}\label{cor3}
$\neg${\rm (B-(Nsup1))} is consistent with {\rm ZFC}. 
\end{cor}

From Corollaries~\ref{cor1}(b) and~\ref{cor3} we obtain also 
\begin{cor}
{\rm B-(Nsup1)} is independent of {\rm ZFC}. 
\end{cor}

\begin{thebibliography}{abc}

\bibitem%[B]
{B} M. Balcerzak, {\it Some remarks on sup-measurablilty}, 
Real Anal. Exchange
{\bf 17} (1991--92), 597--607.

\bibitem%[CKW]
{CKW} J. Cicho\'n, A.~B. Kharazishvili, B. W\c{e}glorz, 
{\it Subsets of the Real
Line}, \L\'od\'z Academic Press, \L\'od\'z 1995. 

\bibitem%[C1]
{Ci:book} K.~Ciesielski,
{\it Set Theory for the Working Mathematician}, 
London Math. Soc. Student Texts {\bf 39}, Cambridge Univ. Press 1997.

\bibitem%[C2]
{C} K. Ciesielski, {\it Set theoretic real analysis}, 
J. Appl. Anal. {\bf 3} (1997), 1--48, in print. 
(Preprint available in the electronic form. It
can be accessed from the 
{\it Set Theoretic Analysis Web Page:}
http://www.math.wvu.edu/homepages/kcies/STA/STA.html)

\bibitem%[F]
{F} D.~H. Fremlin, {\it Problems}, circulated notes (1996). 

\bibitem%[GG]
{GG} E. Grande, Z. Grande,
{\it Quelques remarques sur la superposition F(x,f(x))}, 
Fund. Math. {\bf 121} (1984), 199--211. 

\bibitem%[GL]
{GL} Z. Grande, J. Lipi\'nski, 
{\it Un example d'une fonction sup-mesurable qui n'est pas mesurable}, 
Colloq. Math. {\bf 39} (1978), 77--79. 

\bibitem%[Kh1]
{Kh1} A.~B. Kharazishvili,
{\it Some questions from the theory of invariant measures}, 
Bull. Acad. Sci. Georgian SSR {\bf 100} (1980) (in Russian). 

\bibitem%[Kh2]
{Kh2} A.~B. Kharazishvili, {\it Sup-measurable and 
weakly sup-measurable
mappings in the theory of ordinary differential equations}, 
J. Appl. Anal. {\bf 3} (1997), to appear. 

\bibitem%[KP]
{KP} M.~A. Krasnosel'ski\u{\i}, A. V. Pokrovski\u{\i}, 
{\it Systems with Hysteresis}, Springer, Berlin, 1988. 

\bibitem%[Ku]
{Ku} K. Kuratowski, {\it Topology}, vol.1, Academic Press, 
New York 1966.

\bibitem%[Sh]
{Sh} S. Shelah, {\it Possibly every real function is 
continuous on a non-meagre set}, Publ. Inst. Mat. (Beograd) (N.S.) 
{\bf 57} (1995), 47--60.

\bibitem%[Sr]
{Sr} J.~W. \u{S}ragin, {\it Conditions for measurability 
of superpositions},
Dokl. Akad. Nauk SSSR {\bf 197} (1971), 295--298 (in Russian). 
\end{thebibliography}

\end{document}