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\begin{document}




\title{Universally meager sets}

\author{Piotr Zakrzewski}
\address{Institute of Mathematics, University of Warsaw, ul. Banacha 2,
02-097 Warsaw, Poland}
\email{piotrzak@mimuw.edu.pl}

\thanks{Partially supported by
KBN grant  2 P03A 047 09 and by the Alexander von Humboldt Foundation.}


\subjclass{Primary 03E20,54E52; Secondary  54G99, 28A05}



\date{}

\commby{Carl G. Jockusch, Jr.}




\keywords{measure and category, Borel sets, Baire
property, \s-algebra, \s-ideal}

%\maketitle

\begin{abstract}
We study  category counterparts of the notion of a universal
measure zero set of reals.

We say that a set $A\sub {\mathbb R}$ is universally meager, if
every Borel isomorphic image of $A$ is meager in ${\mathbb R}$. We
give various equivalent definitions emphasizing analogies with the
universally null sets of reals.

In particular, two problems emerging from an earlier
work of Grzegorek are solved.


\end{abstract}

\maketitle


\section{Introduction}


\medskip
Since special subsets of the real line are the subject of this note,
a good general reference is Miller's article \cite{mi2}.

A subset $A$ of the reals or, more generally, of a  perfect (i.e.
with no isolated points) Polish (i.e. separable, metrizable)
topological space $X$, is a {\it universal measure zero} set if it
has (outer) measure zero with respect to every Borel measure on
$X$. By a Borel measure we mean a countably additive, continuous
(i.e., points have measure zero)  finite measure defined on the
\s-algebra $\bor(X)$ of Borel subsets of $X$. It is well known and
easy to prove that
%  , given a Borel measure $\mu$ on $X$,
$A\sub X$ is a
universal measure zero set if and only if either of the following
conditions  holds:

\begin{enumerate}

\item every Borel isomorphic image of $A$ in ${\mathbb R}$ has (outer) Lebesgue
measure zero,

\item $A$ does not contain any Borel one-to-one image of a set of
positive outer Lebesgue measure.

\end{enumerate}


A subset $A$ of the reals or, more generally, of a perfect  Polish
space $X$, is {\it perfectly meager} if for all perfect subsets $P$ of
$X$, the set $A\cap P$ is meager relative to the topology of $P$.
Quoting Miller \cite{mi2} we may note that this  notion is ``somewhat
analogous" to the one of a universal measure zero set. However, the
%characterizations above are not true for
category counterparts of the conditions above do not characterize
 perfectly meager sets since,
assuming the continuum hypothesis (CH), there exists a set of reals which is
perfectly meager but has a non-meager image under a Borel isomorphism
(see \cite{g2}).
%%% sprawdzic!
 This led to the study of the following two apparently different classes
of small sets which was undertaken by Grzegorek in \cite{g1}, \cite{g2},
\cite{g3}.

Using Grzegorek's definitions,  we will call a set $A\sub X$:

\begin{enumerate}

\item  {\it absolutely of the first category} if
every Borel isomorphic image of $A$ in $X$ is meager (see \cite{g1}),

\item an $\overline{\hbox{AFC}}$ set, if it does not contain any Borel
one-to-one image of a non-meager set (see \cite{g2}).

\end{enumerate}


Clearly, every $\overline{\hbox{AFC}}$ set is absolutely of the first
category. The natural question, whether the converse is true, seems to
have been left open.

\medskip

In this note we give a simple proof that the two definitions above
describe just the same class.  We  characterize it in topological and
Boolean-algebraic terms collecting further evidence that its members
are
``more analogous" to universal measure zero sets and should therefore be
called {\it universally meager} sets. In
particular, the product of two universally meager sets is
universally meager. This was earlier proved by Grzegorek \cite{g2} for
$\overline{\hbox{AFC}}$ sets using CH. On the
other hand Rec\l aw \cite{re} showed that assuming CH there exist two
perfectly meager sets whose product is not perfectly meager.





\section{Properties of universally meager sets}

The equivalent statements  constituting the theorem below, are
motivated by their respective counterparts concerning universal
measure zero sets. In particular, a subset $A$ of a perfect Polish
space $X$ is a universal measure zero set if $\mu (A) =0$ for
every (countably additive, continuous,  finite) measure defined on
the \s-algebra $\bor(A)$ of (relative) Borel subsets of $A$.
Equivalently, there is no \s-ideal $\mathcal{I}$ in  $\bor(A)$
such that $\bor(A)/\mathcal{I}\cong \hbox{\bf R}$, the factor
algebra of $\bor({\mathbb R})$ modulo the \s-ideal of Lebesgue
null sets. By a \s-ideal in  $\bor(A)$ we mean here a proper
subfamily of $\bor(A)$ containing all singletons which is closed
under taking subsets and countable unions.

Let $\M(A)$  denote the collection of all meager subsets of $A$.
If $\tau$ is a topology on $A$ then $\bor(A,\tau)$ and
$\M(A,\tau)$ denote the  respective families of Borel and meager
subsets of $A$ in the topology $\tau$. The factor algebra of
$\bor({\mathbb R})$ modulo the \s-ideal $\M({\mathbb R})$ will be
denoted by $\hbox{\bf C}$; it is the unique, up to an isomorphism,
complete, atomless Boolean algebra with a countable dense subset.

\begin{theorem} For a subset $A$ of a perfect Polish space $X$, the
following are equivalent:

(i) $A$ is an $\overline{\hbox{AFC}}$ set.

(ii) $A$ is absolutely of the first category.

(iii) For every \s-ideal $I$ in $\bor(X)$ such that $\bor(X)/ I
\cong \mbox{\bf C}$ there is a Borel set $B\in I$ such that $A\sub
B$.

(iv) $A$ is meager in every  Polish topology $\tau$
on $X$ such that $X$ has no isolated points and $\bor(X,\tau)=\bor(X)$.

(v) $A$ is meager in every  second countable Hausdorff topology
$\tau$ on $X$ such that $X$ has no isolated points and all Borel sets
(in the original Polish
topology) have Baire Property in the topology $\tau$.


(vi) There is no \s-ideal $\mathcal{I}$ in $\bor(A)$ such that
$\bor(A)/\mathcal{I}\cong \hbox{\bf C}$.

(vii) $A$ is meager in every second countable Hausdorff topology
$\tau$ on $A$ such that $A$ has no isolated points and all Borel
subsets of $A$ (in the topology
inherited from the original Polish
topology on $X$) have Baire Property in the topology $\tau$.

(viii) $A$ is meager in every separable metrizable topology
$\tau$ on $A$ such that $A$ has no isolated points and
$\bor(A)=\bor(A,\tau)$.



 \end{theorem}

\begin{proof} $\hbox{(i)}\Rightarrow\hbox{(ii)}$: Obvious.

\smallskip

$\hbox{(ii)}\Leftrightarrow\hbox{(iii)}\Leftrightarrow\hbox{(iv)}
\Leftrightarrow\hbox{(v)}$: It suffices to notice that:

\begin{itemize}

\item Due to Sikorski's theorem (see \cite{ke}, 15.10), if $I$ is a
\s-ideal in $\bor(X)$ such that $\bor(X)/ I \cong \hbox{\bf C}$,
then there is Borel automorphism $\Phi: X \w X$ such that $$
\forall A\in\bor(X) \ A\in\M(X) \iff \Phi [A]\in I $$
\item If $\tau$ is a  second countable Hausdorff topology
on $X$ such that $X$ has no isolated points, all Borel sets (in
the original Polish topology) have Baire Property in the topology
$\tau$ and $A\not\in\M(A,\tau)$, then $$ \bor(X)/
{(\M(X,\tau)\cap\bor(X))} \cong \hbox{\bf C}. $$
\end{itemize}

\smallskip

$\hbox{(iii)}\Rightarrow\hbox{(vi)}$: Clear, since if $\mathcal{I}$
is a \s-ideal in $\bor(A)$, then
$$
I=\{B\in\bor(X):\ B\cap A\in \mathcal{I}\}
$$
is a \s-ideal in $\bor(X)$ with $\bor(A)/\mathcal{I}\cong\bor(X)/ I$.

\smallskip

$\hbox{(vi)}\Rightarrow\hbox{(iv)}$: Suppose that $A\not\in\M(X,\tau)$
for a certain  Polish topology $\tau$
on $X$ such that $X$ has no isolated points and $\bor(X,\tau)=\bor(X)$.

Let $\mathcal{I}=\M(X,\tau)\cap \bor(A).$ But then $\mathcal{I}$
is a \s-ideal in $\bor(A)$ with $\bor(A)/\mathcal{I}\cong
\hbox{\bf C}$ (see \cite{z2}, Lemma 2.2).

\smallskip

$\hbox{(vi)}\Rightarrow\hbox{(vii)}$: It is easy to see that if
$\tau$ is a second countable Hausdorff topology on $A$ such that
$A$ has no isolated points, all sets from $\bor(A)$ have Baire
Property in the topology $\tau$ and  $A\not\in\M(A,\tau)$, then $$
\bor(A)/{(\M(A,\tau)\cap\bor(A))}\cong \hbox{\bf C}. $$

\smallskip

 $\hbox{(vii)}\Rightarrow\hbox{(viii)}$: Obvious.

\smallskip

$\hbox{(viii)}\Rightarrow\hbox{(vi)}$: Suppose that $\mathcal{I}$
is a \s-ideal in $\bor(A)$ such that $\bor(A)/\mathcal{I}\cong
\hbox{\bf C}$. We will find a   separable metrizable topology
$\tau$ on $A$ such that $A$ has no isolated points,
$\bor(A)=\bor(A,\tau)$ and $A\not\in\M(A,\tau)$.


Define a \s-ideal $J$ in $\bor(X)$ by:
$$
 B\in J \iff B\cap A \in \mathcal{I}, \hbox{ for } B\in\bor(X).
$$ Since $$ \bor(X)/J \cong \bor(A)/{\mathcal{I}}\cong\hbox{\bf
C}, $$
 there is a Polish topology
$\tau_{1}$ on $X$
for which we have $\bor(X)=\bor(X, \tau_{1})$ and
$J=\M(X, \tau_{1})\cap \bor(X)$ (see \cite{ke}, 15.10).

 Let $\tau=\tau_{1}|A$ be the topology on $A$
inherited from the topology $\tau_1$. Clearly, $\tau$ is second
countable and $\bor(A)=\bor (A,\tau)$.

 Also, the topological space $(A,\tau)$ is perfect.
Otherwise, there is a $U\in\tau_1$, $U\neq\0$ with $|A\cap U|=1$. But
then $U\setminus A\not\in\M(X,\tau_1)$ as a non-empty open subset of a Polish
space contradicting the fact that $U\setminus A\in J$ and $J=\M(X,
\tau_{1})\cap \bor(X)$.

Finally, $A\not\in\M(A,\tau)$. Otherwise, $A\in\M(X,\tau_1)$, so there
is $B\in\bor(X)$ with $A\sub B$ and $B\in\M(X,\tau_1)$. Then $B\in J$
which means $A\in\mathcal{I}$ -- a contradiction. (It could even be
proved that $\M(A,\tau)= \mathcal{I}$.)

\smallskip


$\hbox{(iii)}\Rightarrow\hbox{(i)}$: Suppose $A$ is not an
$\overline{\hbox{AFC}}$ set. Let  $f:Z \w A$ be a Borel one-to-one
function with $Z\not\in \M(X)$. We will find a \s-ideal $I$ in
$\bor(X)$ with $\bor(X)/ I \cong \hbox{\bf C}$ and such that no
Borel set $B\in I$ covers $A$.

Define a \s-ideal $I$ in \bor(X) by letting $$ B\in I \iff
f^{-1}[B]\in  \M(X), \hbox{ for } B\in\bor(X). $$ Note that $f$
induces a complete embedding of the  Boolean algebra $\bor(X)/I$
into the algebra  $\bor(Z)/(\M(X)\cap\bor(Z))$. Moreover,
$\bor(Z)/{(\M(X)\cap\bor(Z))}\cong \hbox{\bf C}$ (see \cite{z2},
Lemma 2.2). It follows that the algebra $\bor(X)/ I$ is also
isomorphic to $\hbox{\bf C}$ as a  complete subalgebra of the
latter which has no atoms. To verify the last statement, suppose
that there is $C\in \bor(X) \setminus I$ such that for every $B\in
\bor(X)$ either $B\cap C\in I$ or $C \setminus B \in I$. Then,
since $I$ contains all singletons, every point $x$ of $C$ has an
open neighbourhood  $V_x$ with  $V_x\cap C\in I$. This immediately
implies that $C \in I$ -- a contradiction.

 Finally,
since $f^{-1}[A]=Z\not\in\M(X)$, $A$ cannot be covered by a Borel
set from $I$.


\end{proof}


From now on a subset $A$ of a perfect Polish space $X$ will be called
{\it universally meager}, if it satisfies any of the equivalent conditions
above.

\medskip

 Clearly, every universally meager set is perfectly meager but
(at least consistently) not vice versa (see \cite{g2}). In
particular, the two classes differ as far as the closure under products
is concerned. Assuming CH there exist two
perfectly meager sets whose product is not perfectly meager (cf.
\cite{re}). On the other hand the following result (and its proof)
closely resembles
Marczewski's theorem (and its proof -- see \cite{mi2}, Theorem 8.1)
that the product of two universal measure zero
sets has universal measure zero. It
improves Grzegorek's theorem that, assuming CH, the class of
$\overline{\hbox{AFC}}$ sets is closed under taking products.

\begin{theorem}
 The product of two universally meager
sets is universally meager.
\end{theorem}

\begin{proof} Let $D$ and $E$ be universally meager subsets of
perfect Polish spaces $X$ and $Y$, respectively. Suppose, towards
a contradiction, that $A=D\times E$ is not universally meager. By
Theorem 2.1(vi) this means that there is a \s-ideal $\mathcal{J}$
in $\bor(A)$ such that $$ \bor(A)/\mathcal{J}\cong \hbox{\bf C}.
$$ Define a \s-ideal  $\mathcal{I}$ in $\bor(E)$ by: $$ B\in
\mathcal{I}, \ \hbox{ if }\  D\times B \in \mathcal{J}\ \hbox{ for
}\ B\in\bor(E). $$ Since $D$ is universally meager, so is $D\times
\{y\}$ for $y\in Y$; it follows that $D\times \{y\}\in\mathcal{J}$
and so $\{y\}\in \mathcal{I}$. This implies that the quotient
algebra $\bor(E)/\mathcal{I}$ is atomless. Moreover, using the
formula $$ [B]_{\mathcal{I}} \mapsto [D\times B]_{\mathcal{J}} $$
we can  completely embed it into the algebra
$\bor(A)/\mathcal{J}$. But, since the latter is isomorphic to
$\hbox{\bf C}$, so is the algebra $\bor(E)/\mathcal{I}$, contrary
to the fact that $E$ is universally meager.

\end{proof}

Despite the differences between universally meager and perfectly
meager sets indicated above, in all standard examples  uncountable
perfectly meager sets turn out to be universally meager as well.
These examples include, in particular, a selector of the
constituents of a nontrivial coanalytic set (see \cite{mi2},
Theorem 5.3), a Hausdorff $(\omega_1,\omega_1^{*})$-gap  and a
tower, i.e., any set well-ordered by a Borel relation (see
\cite{ple}).

Recall that a set $A\sub 2^\omega$ has the {\it Hurewicz property}, if
every continuous image of $A$ in $\omega^\omega$ is bounded, i.e.,
contained in a $K_{\sigma}$ subset of $\omega^\omega$.
 In Theorem 5.5 of Just-Miller-Scheepers-Szeptycki \cite{jmss} it was
shown that if $A$ has the Hurewicz property and contains no perfect
subset, then $A$ is perfectly meager. This can be strengthened as follows.

\begin{proposition} If a set $A\sub 2^\omega$ has the Hurewicz property
and contains no perfect
subset, then $A$ is universally meager.
\end{proposition}

\begin{proof} By a result of Nowik (see \cite{nw}, the proof of
Theorem 2), a sufficient condition for $A$ to be universally
meager is that for every countable family $\{ Q_n:n\in{\mathbb
N}\}$ of perfect subsets of $2^\omega$ there exists an $F_\sigma$
set $F$ such that $A\sub F$ and $Q_n\not\subseteq F$ for any
$n\in\omega$.

So let $Q_n,\ n\in{\mathbb N},$ be perfect subsets of $2^\omega$.
Since $A$ contains no perfect subsets, for each $n\in{\mathbb N}$
there is an open set $U_n$ such that $A\sub U_n$ but
$Q_n\not\subseteq U_n$.

Let $G=\bigcap_{n\in{\mathbb N}}U_n$. Now, since $A$ has the
Hurewicz property, Theorem 5.7 of Just-Miller-Scheepers-Szeptycki
\cite{jmss} tells us that there is a $F_\sigma$-set $F$ such that
$A\sub F\sub G$. Clearly, $Q_n\not\subseteq F$ for any
$n\in{\mathbb N}$.

\end{proof}

Let us finally indicate that the problem, whether one can prove in
ZFC alone that the classes of perfectly meager and universally
meager sets are different, seems to be open.

The same question is apparently open for the
classes of universal measure zero sets and universally meager sets,
though both inclusions between the two are known to be relatively
consistent. By the results of Miller \cite{mi}, the models are the
Cohen real model, in which every
universally meager set has size less than continuum and has therefore
universal measure zero, and the random real model, in which the situation
is symmetric.

Note, that as a simple consequence of the equivalence ``$\hbox{(ii)}
\Leftrightarrow  \hbox{(vi)}$" from
Theorem 2.1 and the well
known dual fact concerning universal measure zero we have the following

\begin{proposition}

The following  are equivalent:

(i) The classes of universal measure zero sets and universally meager
sets are equal.

(ii) If a measurable space $(X,\mathcal{A})$ is countably
generated and separates points, then there exists a \s-ideal
$\mathcal{I}$ in $\mathcal{A}$ such that
$\mathcal{A}/\mathcal{I}\cong \hbox{\bf C}$ if and only if there
exists a \s-ideal $\mathcal{J}$ in $\mathcal{A}$ such that
$\mathcal{A}/\mathcal{J}\cong \hbox{\bf R}$.

\end{proposition}



\bibliographystyle{amsplain}

\begin{thebibliography}{16}


\bibitem{g1} E. Grzegorek \textit{Solution to a problem of Banach on
\s-fields without continuous measures}, Bull. Acad. Pol. Sci.
\textbf{28} (1980), 7--10.

\bibitem{g2} E. Grzegorek \textit{Always of the first category sets}, Rend.
Circ. Mat. Palermo, II. Ser. Suppl. \textbf{6} (1984), 139--147.

\bibitem{g3} E. Grzegorek \textit{Always of the first category sets. II},
Rend. Circ. Mat. Palermo, II. Ser. Suppl. \textbf{10} (1985), 43--48.

\bibitem{jmss} W. Just, A.W. Miller, M. Scheepers, P.J. Szeptycki,
\textit{The combinatorics of open covers (II)}, Topology Appl.
\textbf{73} (1996), 241--266.


\bibitem{ke} A. S. Kechris, \textit{Classical descriptive set theory},
Graduate Texts in Math. 156, Springer-Verlag 1995.

\bibitem{mi} A.W. Miller, \textit{Mapping a set of reals onto the reals},
J. Symb. Logic \textbf{48} (1983), 575--584.

\bibitem{mi2} A.W. Miller, \textit{Special subsets of the real line} in
\textit{Handbook of set--theoretic topology}, North-Holland 1984, 201--233.

\bibitem{nw} A. Nowik, T. Weiss, \textit{Not every $Q$-set is perfectly
meager in the transitive sense}, to appear in  Proc. Amer. Math.
Soc.

\bibitem{ple} S. Plewik, \textit{Towers are universally measure zero
and always of first category},
Proc. Amer. Math. Soc. \textbf{119(3)} (1993), 865--868.


\bibitem{re} I. Rec\l aw, \textit{Products of perfectly meager sets},
Proc. Amer. Math. Soc. \textbf{112(4)} (1991), 1029--1031.


\bibitem{re-za} I. Rec\l aw and P. Zakrzewski, \textit{Strong Fubini
properties of ideals},  Fund. Math. \textbf{159} (1999), 135--152.

\bibitem{z2} P. Zakrzewski, \textit{Extending Baire property by
countably many sets}, to appear in Proc. Amer. Math. Soc.



\end{thebibliography}






\end{document}
