\documentclass{rae}
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\newcommand{\theorem}[2]{\vskip6pt\noindent{\bf{#1}.}{ \sl #2}}
\newcommand{\demrem}[2]{\vskip6pt\noindent{\em{#1}.}{ #2}}
\newcommand{\cov}{\mbox{cov}(L)}
\newcommand{\add}{\mbox{add}(L)}
\newcommand{\ma}{ ($\cal M\cal A$)$_{\aleph_1}$ }
\newcommand{\ch}{ ($\cal C \cal H$) }
\newcommand{\ns}{ ($\cal A\cal D\cal D$)$_{\aleph_1}$ }
\newcommand{\nns}{ $\neg$($\cal A\cal D\cal D$)$_{\aleph_1}$ }
\newcommand{\ce}{ ($\cal C\cal O\cal V$)$_{\aleph_1}$ }
\newcommand{\nce}{ $\neg$($\cal C\cal O\cal V$)$_{\aleph_1}$ }
\newcommand{\de}{ ($\cal D$) }
\newcommand{\nd}{ $\neg$($\cal D$) }
\newcommand{\zf}{ ($\cal Z \cal F \cal C$) }
\newcommand{\ac}{ ($\cal A \cal C$) }
\newcommand{\nac}{ $\neg$($\cal A \cal C$) }
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\title{LIMITS OF TRANSFINITE CONVERGENT SEQUENCES OF
DERIVATIVES}
\author{Martin Dindo\v{s}, Department of Mathematics, University of
North Carolina, Phillips Hall CB \#3250, Chapel Hill, NC 27599,\\
e-mail: {\tt dindos@math.unc.edu}}
\markboth{Santa Barbara Symposium -- M.~Dindo\v{s}}
{Santa Barbara Symposium -- M.~Dindo\v{s}}
\begin{document}\label{s-conf-md}
\maketitle
The covergence of transfinite sequences of functions was introduced in
the paper [Sie]. Let $\Omega$ be the first uncountable ordinal number,
let $I$ be a real interval and \funk{f_\xi}{I}{R}, $1 \le \xi <\Omega$
be a sequence of real functions. We say that \funk f I R is the
pointwise limit of this sequence if $f_{\xi}(x)\to f(x)$ holds for every
$x\in I$, i.e. $$\forall x\in T\quad\forall \varepsilon>0\quad\exists
\eta< \Omega\quad\forall \xi \ge \eta:\quad
|f(x)-f_{\xi}(x)|<\varepsilon$$
An important question is whether the pointwise transfinite convergence
preserves some important properties of functions, e.g., continuity or
first Baire class. These questions were solved positively in the paper
[\v{S}] or [Sie] respectively. In the present paper the question of
preserving the property `being derivative' will be discussed. The
results of this paper can be also used to the solution of the question
of preserving the property `being an approximately continuous function'.
This problem was mentioned in the paper [\v{S}] as open.
Let $\Delta$ denote the set of all derivatives on the interval $I$,
i.e., all functions \funk f I R having primitive functions \funk F I R
such that $f(x)=F'(x)$ for each $x\in I$. Let us denote the next
statements in the following way:
\begin{itemize}
\item[{}]
\begin{itemize}
\item[{\ch}] Continuum hypothesis:\quad $\aleph_1=2^{\aleph_0}$.
\item[{\ma}] Martin's axiom:\quad For a nonempty poset (partialy ordered
set) $P$ having property (CCC)\footnote{Poset $(P,\prec)$, briefly $P$
has property (CCC) if every set $Q\subset P$ which elements are parwise
incompatible is at most denumerable. Two elements $p$,$q$ $(p\ne q)$ of
an poset $P$ are incompatible if there does not exist any element $r\in
P$ such that $p\prec r$ and $q\prec r$.} and a family $\{D_j;j \in J\}$
of dense\footnote{Set $D$ is dense in the poset $P$ if for an arbitrary
$p\in P$ there exists $d\in D$ such that $p\prec d$.} sets in $P$
(card$(J) \le \kappa$) there exists subnet\footnote{$Q$ is subnet of $P$
if $Q\subset P$ and $Q$ is a net i.e. for every elements $p,q\in Q$
there is an element $r\in Q$ such that $p\prec r$ and $q\prec r$.}
$Q\subset P$ such that:$\quad Q \cap D_j \ne \emptyset$. (See [Sch]). We
shall use this axiom for $\kappa=\aleph_1$. \item[{\ns}] The union of
$\aleph_1$ null sets (Lebesgue measure on $R$) has (Lebesgue) measure
zero. This statement can be written as $\add>\aleph_1$ where $\add$ is
the usual notation for the smallest cardinal $\kappa$ with the property
that there are $\kappa$ null sets such that their union is not null.
\item[{\ce}] There are $\aleph_1$ null sets (Lebesgue measure on $R$)
covering $R$. This statement can be written as $\cov=\aleph_1$ where
$\cov$ is the usual notation for the smallest cardinal $\kappa$ such
that the real line is the union of $\kappa$ null sets.
\item[{\de}] If \funk {f_{\xi}} I R, $1 \le \xi <\Omega$ is an arbitrary
transfinite pointwise convergent sequence of derivatives then the limit
function $f = \displaystyle\lim_{\xi<\Omega} f_{\xi}$ is also
derivative, i.e. $f \in \Delta$.
\item[{\ac}] If \funk {f_{\xi}} I R; $1 \le \xi <\Omega$ is an arbitrary
transfinit pointwise convergent sequence of approximately continuous
functions then the limit function $f = \displaystyle\lim_{\xi<\Omega}
f_{\xi}$ is also approximately continuous.
\item[{\zf}] Zermelo-Fraenkel set theory including the axiom of choice.
\end{itemize}
\end{itemize}
Both the continuum hypothesis\ch and Martin's axiom\ma are statements that are
independent with respect to Zermelo-Fraenkel set theory\zf.
The main aim of my presentation is to present the proofs of following
implications
$$\mbox{\zf$+$\nce$\Longrightarrow$\zf$+$\de}$$
$$\mbox{\zf$+$\ce$\Longrightarrow$\zf$+$\nd}$$
which means that \de and\nd are statements that cannot
be derived from\zf because both\zf$+$\de and \zf$+$\nd remain consistent
if\zf is consistent. In addition, the following axiomatic systems are equivalent
$$\mbox{\zf$+$\nce$\Longleftrightarrow$\zf$+$\de}$$
$$\mbox{\zf$+$\ce$\Longleftrightarrow$\zf$+$\nd}$$
These are the main theorems proving the previous implications.
\theorem{Theorem}{Let \funk {f_{\xi}} I R, $1 \le \xi < \Omega$,
be a pointwise convergent transfinite sequence of measurable functions. Let
$f=\displaystyle\lim_{\xi <\Omega} f_{\xi}$. Let $(i)$ or $(ii)$ holds.
$(i)$ Axiom \ns holds.
$(ii)$ Function f is measurable and axiom \nce holds.
\vskip2mm
Then the function $f$ is measurable and there exists an ordinal number
$\eta<\Omega$ such that for every $\eta \le \xi < \Omega$ $\quad f_{\xi}(x)=f(x)$
holds almost everywhere on $I$.}
\theorem{Theorem}{Let \ce. The function \funk f I R is Baire 1 if and only if there
exists a transfinite sequence of derivatives $(f_{\xi})_{\xi<\Omega}$ such that
$\displaystyle\lim_{\xi<\Omega} f_{\xi}=f$.}
\demrem{Remark}{This is a stronger version of a theorem published in [L] where
only the implication `$\Longrightarrow$' was proved with assumption of semi-continuity
of function $f$ instead of the assumption of being Baire 1 function.
This theorem also gives an affirmative answer to the question asked
by author of [L].
This theorem is also an analogue of Preiss's theorem [P]. He proved that each
Baire 2 function is a limit of sequence of derivatives. The assumption \ce provides
us a similar theorem for transfinite sequences.}
\demrem{Remark 6}{Previous proofs can be reformulated to approximately
continuous functions instead of derivatives. Hence also
following statements are true:
$$\mbox{\zf$+$\nce$\Longrightarrow$\zf$+$\ac}$$
$$\mbox{\zf$+$\ce$\Longrightarrow$\zf$+$\nac}$$}
\begin{thebibliography}{999}
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\bibitem[{L}] {L} J. S. Lipinski, {\em On transfinite sequences of approximately
continuous functions} Bul. Pol. Acad. Sci. {\bf 9} (1973), 817--821.
\bibitem[{P}] {P} D. Preiss, {\em Limit of derivatives and Darboux-Baire
functions,} Rev. Roum. Pures et Appl. {\bf 14} (1969), 1201--1206.
\bibitem[{P-L}] {P-L} G. Petruska and M. Laczkovich, {\em Baire 1 functions,
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Academic Press Inc., New York and London, 1968.
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Amer. Math. Monthly {\bf 82} (1975), 610--617.
\bibitem[{Sie}] {Sie} W. Sierpinski, {\em Sur les suites transfinies
convergentes de fonctions de Baire,}
Fund. Math. {\bf 1} (1920), 132--141.
\bibitem[{\v{S}}] {S} T. \v{S}al\'at, {\em On transfinite sequences of $B$-measurable
functions,}
Fund. Math. (1973), 157--162.
\end{thebibliography}
\end{document}