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\begin{document}
\bigskip
\begin{center}
{\Large Souslin's Hypothesis and Convergence in Category}
\end{center}
\bigskip
\begin{center}
by Arnold W. Miller\footnote{
I want to thank Krzysztof Ciesielski for many
helpful conversations.
The results presented in this paper were obtained
during the Joint US--Polish Workshop in Real Analysis,
{\L}{\'o}d{\'z}, Poland, July 1994.
The Workshop was partially
supported by the NSF grant INT--9401673.
AMS Subject Classification. Primary: 28A20; Secondary: 03E65, 54E52.
This appears in
Proceedings of American Mathematical Society, 124(1996), 1529-1532.
}\\
\end{center}
\begin{quote}
Abstract: A sequence of functions $f_n: X \to {\Bbb R}$ from a Baire
space $X$ to the reals ${\Bbb R}$ is said to converge in category
iff every subsequence has a subsequence which converges on all
but a meager set. We show that if there exists a Souslin Tree, then
there exists a nonatomic Baire space X such that every sequence which
converges in category converges everywhere on a comeager set.
This answers a question of Wagner and Wilczynski who proved the
converse.
\end{quote}
Suppose that $S\subseteq P(X)$ is a $\sigma$-field of
subsets of $X$ and $I\subseteq S$ is a $\sigma$-ideal.
If $I$ has
the countable chain condition (ccc), i.e., every family of
disjoint sets in $S\setminus I$ is countable, then
$S/I$ is a complete boolean algebra. A boolean algebra
is atomic iff there is an atom beneath every nonzero element.
A function $f:X\to \reals$ is $S$-measurable iff
$f^{-1}(U)\in S$ for every open set $U$.
A sequence of $S$-measurable functions $f_n:X\to \reals$
converges $I$-a.e. to a function $f$ iff there exists $A\in I$ such
that $f_n(x)\to f(x)$ for all $x\in (X\setminus A)$.
If $(X,S,\mu)$ is a finite measure space, then a sequence of
measurable functions $f_n:X\to \reals$ converges in measure
to a function $f$ iff for any $\epsilon>0$ there exists $N$ such
that for any $n>N$:
$$\mu(\{x\in X: |f_n(x)-f(x)|>\epsilon\})<\epsilon.$$
In this case if $I$ is the ideal of measure zero sets, then
$f_n$ converges to $f$ in measure iff
every subsequence $\{f_n:n \in A\}$ (where $A\subseteq \naturals$)
has a subsequence $B\subseteq A$ such that $\{f_n: n\in B\}$
converges $I$-a.e. This allows us to define convergence in
measure without mentioning the measure, only the
ideal $I$. So in the abstract setting define the following:
$f_n$ converges
to $f$ with respect to $I$ iff every subsequence $\{f_n:n \in A\}$
has a subsequence $B\subseteq A$ such that $\{f_n: n\in B\}$
converges $I$-a.e. (where $A$ and $B$ range over infinite sets of
natural numbers.) For more background on this subject in
case $I$ is the ideal of meager sets, see
Poreda, Wagner-Bojakoska, and Wilczy\'{n}ski [PWW] and
Ciesielski, Larson, and Ostaszewski [CLO].
Marczewski [M] showed that
if $(X,S,\mu)$ is an atomic measure and $I$ the
$\mu$-null sets, then
`$I$-a.e. convergence' is the same as `convergence with respect to $I$'.
Gribanov [G] proved the converse,
if $(X,S,\mu)$ is a finite measure space and $I$ the
$\mu$-null sets, then
if `$I$-a.e. convergence' is the same as `convergence with respect to $I$'
then $\mu$ is an atomic measure.
Souslin's Hypothesis (SH) is the statement that there are no
Souslin lines. It is known to be independent (see Solovay and
Tennenbaum [ST]). It was the inspiration for Martin's Axiom.
\begin{theorem} (Wagner and Wilczy\'{n}ski [WW]) \label{WW}
Assume SH.
Then for any $\sigma$-field $S$ and ccc $\sigma$-ideal $I\subseteq S$
the following are equivalent:
\begin{itemize}
\item `$I$-a.e. convergence' is the same as `convergence with respect to $I$'
for $S$-measurable sequences of real-valued functions, and
\item the complete boolean algebra $S/I$ is atomic.
\end{itemize}
\end{theorem}
At the real analysis meeting in \L\'{o}d\'{z} Poland in July 94,
Wilczy\'{n}ski asked whether or not SH is needed for the Theorem
above. We show here that the conclusion of Theorem \ref{WW} implies
Souslin's Hypothesis.
\begin{theorem}
Suppose SH is false (so there exists a Souslin tree). Then
there exists a regular topological space $X$ such that
\begin{enumerate}
\item $X$ has no isolated points,
\item $X$ is ccc (every family of disjoint open sets is countable),
\item every nonempty open subset of $X$ is nonmeager, and
\item if $I$ is
the $\sigma$-ideal of meager subsets of $X$, then
`$I$-a.e. convergence' is the same as `convergence with respect to $I$'
for any sequence of Baire measurable real-valued functions.
\end{enumerate}
Hence if $S$ is the $\sigma$-ideal of sets with the property of Baire
and $I$ the $\sigma$-ideal of meager sets, then $S/I$ is ccc and
nonatomic, but the two types of convergence are the same.
\end{theorem}
\proof
Define $(T,<)$ to be an $\omega_1$-tree iff it is a partial order and
for each $s\in T$ the set $\{t\in T: t< s\}$ is well-ordered
by $<$ with some countable order type, $\alpha<\omega_1$.
We let
$$T_\alpha=\{s\in T: \{t\in T: t