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\begin{document}




\title{Extending Baire Property by countably many 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 03E35, 54E52; Secondary 28A05}



\date{}

\commby{Carl G. Jockusch, Jr.}

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


\begin{abstract}
We  prove  that  if ZFC is consistent so is ZFC + ``for any sequence
$( A_{n})$ of subsets of a Polish space $\langle X,\tau\rangle $
there exists a separable metrizable topology $\tau'$ on $X$ with
$\bor (X,\tau)\sub\bor (X,\tau')$, $\M (X,\tau')\cap \bor(X,\tau)=
\M (X,\tau)\cap \bor(X,\tau)$ and $A_{n}$ Borel in $\tau'$ for all
$n$.''  This is a category analogue of  a theorem of Carlson on
the possibility
 of extending
Lebesgue measure to any countable collection of sets. A uniform
argument is presented, which gives a new proof of the latter as well.

Some consequences of these extension properties are also studied.


\end{abstract}

\maketitle


\section{Introduction}


\medskip
Most of our terminology and notation is standard and essentially agrees with
\cite{ke}. Throughout the paper $(X,\tau)$ is a fixed perfect Polish
space.


The results of this article are  the category analogues of some
known theorems on  extending a given finite,  countably additive,
atomless  measure $\mu$
 defined on the \s-algebra
$\bor(X,\tau)$ (denoted also by $\bor(X)$) of Borel
subsets of   $X$ (shortly: a Borel measure) to a
countably additive measure $m'$ defined on a \s-algebra
$\mathcal{A}$ containing $\bor(X)$ and such that ${\mu}'|{\mathcal{A}}=\mu$ (it is
well-known that without loss of generality $\mu$ can be assumed to be
the Lebesgue measure on the reals -- see \cite{ke}, 17.41).

\L o\'s and Marczewski \cite{l-m} proved that if $\A$ is generated (as
a \s-algebra) by
$\bor(X)$ plus any
finite collection of new subsets of $X$, then such an  extension
exists. Banach and Kuratowski
\cite{ba-ku} showed that assuming CH there is a countable collection
of subsets of $X$ which cannot be included in the domain of any
extension of $\mu$. On the other hand, Solovay \cite{S} proved that the
possibility of extending $\mu$ to $\Pot(X)$, the power set of $X$, is
equiconsistent with the existence of a measurable cardinal. Finally, Carlson
\cite{ca} obtained the consistency of ``if $\A$ is generated  by
$\bor(X)$ plus any
countable collection of new subsets of $X$, then there is an extension
of $\mu$ defined on $\A$ ''.

Observe that we can formulate results about extensions of measures in
terms of the
related \s-ideals and quotient \s-algebras. Let $\N_\mu$ denote the
\s-ideal of
all subsets of $X$, having outer
measure zero with respect to a given Borel measure $\mu$ on $X$. For an
infinite cardinal $\kappa$ let $\hbox{\boldmath R}_{\kappa}$ denote the measure
algebra of $2^{\kappa}$, i.e., the \s-algebra generated by the
basic open sets modulo null  sets with respect to the usual product measure.



\begin{proposition}[folklore?]  Let $\mu$ be a finite, atomless Borel measure on
$X$.  For a $\s$-algebra
$\mathcal{A}$ containing $\bor(X)$  the following are equivalent:


(i) There exists an extension of $\mu$ to a measure defined on $\A$.

(ii) There exists a \s-ideal $\mathcal{I}$ in $\A$ such that:

\begin{enumerate}

\item $\mathcal{I} \cap \bor(X)=\N_{\mu}\cap\bor(X)$,

\item The Boolean algebra $\A/\mathcal{I}$ can be completely
embedded in the algebra $\hbox{\boldmath R}_{\kappa}$ for some infinite
cardinal $\kappa$.

\end{enumerate}


Moreover, if $\A$ is countably generated, then condition 2 can be
replaced by:



$2'.$ $\A/\mathcal{I}\cong \hbox{\boldmath R}_{\omega}.$


\end{proposition}



\begin{proof} $\hbox{(i)}\Rightarrow\hbox{(ii)}.$ Just let ${\mathcal{I}
}=\{A\in\A:{\mu}'(A)=0\}$, where $\mu'$ is a given extension of
$\mu$ defined on \A. Then condition 1 is obvious and Maharam's theorem (see
\cite{frem}) implies the rest.

$\hbox{(ii)}\Rightarrow\hbox{(i)}.$ Let $\nu$ be a probability measure on \A\
with $\mathcal{I}=\{A\in\A:\nu(A)=0\}$. Then the Radon--Nikodym theorem
tells us that there is a Borel function $f$ on $X$ such that
$$
\mu(B)=\int_B f\,d\nu,  \hbox{ for every } B\in\bor(X).
$$
Hence the formula
$$
\mu'(B)=\int_B f\,d\nu,  \hbox{ for } B\in\A
$$
defines an extension of $\mu$ to a measure $\mu'$ on \A.

Finally, the ``moreover'' part follows from the fact that any atomless
countably completely generated complete subalgebra of $\hbox{\boldmath
R}_{\kappa}$ is isomorphic to $\hbox{\boldmath R}_{\omega}$ (see \cite{frem}).~\end{proof}



This motivates the following definition. Let $\M(X,\tau)$ (or simply
$\M(X)$) be the \s-ideal of
all meager subsets of $X$. For an
infinite cardinal $\kappa$ let $\hbox{\boldmath C}_{\kappa}$ denote the
category algebra of $2^{\kappa}$, i.e., the \s-algebra generated by the
basic open sets modulo meager sets.



\begin{definition}
Suppose     $\mathcal{A}$ is a   $\s$-algebra
containing $\bor(X)$. We say that the  Baire Property can be extended
to $\A$, if   there exists a \s-ideal $\mathcal{I}$ in $\A$ such that:

1. $\mathcal{I} \cap \bor(X)=\M(X)\cap\bor(X)$,

2. The Boolean algebra $\A/\mathcal{I}$ can be completely
embedded in the algebra $\hbox{\boldmath C}_{\kappa}$ for some infinite
cardinal $\kappa$.
\end{definition}


Kamburelis \cite{kam2} proved that the
possibility of extending the Baire Property to $\Pot(X)$ is
equiconsistent with the existence of a measurable cardinal.
When $\A$ is countably generated, which is the case dealt with in
this note, the definition above takes a nicer form. Recall that
$\hbox{\boldmath C}_{\omega}$ is the unique, up to an isomorphism, complete,
atomless Boolean algebra with a countable dense subset.



\begin{proposition}

  If $\A$ is a countably generated $\s$-algebra
containing $\bor(X,\tau)$, then the following are equivalent:

(i)  The Baire Property can be extended
to $\A$;

(ii) There exists a \s-ideal $\mathcal{I}$ in $\A$ such that:

\begin{enumerate}

\item $\mathcal{I} \cap \bor(X,\tau)=\M(X,\tau)\cap\bor(X,\tau)$,

\item  $\A/\mathcal{I}\cong\hbox{\boldmath C}_{\omega}$;

\end{enumerate}

(iii) There exists a separable metrizable topology $\tau'$ on $X$ with
$\A=\bor (X,\tau')$ and $\M (X,\tau')\cap \bor(X,\tau)=
\M (X,\tau)\cap \bor(X,\tau)$.
\end{proposition}



\begin{proof} $\hbox{(i)}\Rightarrow \hbox{(ii)}$  follows from the fact that any
atomless countably completely generated complete subalgebra of
$\hbox{\boldmath C}_{\kappa}$ is isomorphic to $\hbox{\boldmath
C}_{\omega}$.

To get $\hbox{(ii)}\Rightarrow \hbox{(iii)}$
it suffices to prove that if $\mathcal{I}$ is a \s-ideal in a
countably generated \s-algebra $\A$ that  separates points in $X$ and
$\A/\mathcal{I}\cong \hbox{\boldmath C}_{\omega}$, then there exists a
separable metrizable topology
$\tau'$ on $X$ with
$\A=\bor (X,\tau')$ and $\mathcal{I}=\M (X,\tau')\cap \A$.


First note that since $\A$ is
countably generated and  separates points in $X$, there is a
separable metric $d$ on $X$ such that
$\A=\bor(X,d)$ (see \cite{ke}, 12.1). Let $(\hat X, \hat d)$ be the
completion of $(X,d)$.
Define a \s-ideal $J$ in $\bor(\hat X, \hat d)$ by:
$$
 A\in J \iff A\cap X \in \mathcal{I}, \hbox{ for } A\in\bor(\hat X, \hat d).
$$
Now, since
$$
\bor(\hat X, \hat d)/J \cong \A/{\mathcal{I}}\cong\hbox{\boldmath C}_{\omega},
$$
 there is a Polish topology
$\tau_{1}$ on $\hat X$
for which we have $\bor(\hat X, \hat d)=\bor(\hat X, \tau_{1})$ and
$J=\M(\hat X, \tau_{1})\cap \bor(\hat X, \tau_{1})$ (see \cite{ke}, 15.10).
 Let $\tau'=\tau_{1}|X$ be the relative topology on $X$  treated as a
subspace of $(\hat X, \tau_{1})$. Since, clearly, $\A=\bor (X,\tau')$,
to complete the proof it is enough to check that
$\mathcal{I}=\M (X,\tau')\cap \A$. In view of the definitions of the \s-ideals
involved, it suffices to prove that for any $A\in \bor(\hat X,
\tau_{1})$
$$
A\in\M(\hat X,\tau_1) \iff A\cap X \in \M (X,\tau').
$$
This, in turn,  reduces to the fact that if $A$ is closed
in $\tau_1$, then it is nowhere dense in $\tau_1$ if and only if
$A\cap X$ is nowhere dense in $\tau'$. But for the latter note that if
$U\in \tau_1$ and $U\cap X=\emptyset$, then $U\in J=\M(\hat X,
\tau_{1})$, so $U=\emptyset$.

Finally, $\hbox{(ii)}\Rightarrow \hbox{(i)}$ is trivial and
$\hbox{(iii)}\Rightarrow \hbox{(ii)}$ follows from the fact, that
 if $\tau'$ is a second countable topology on $X$ with no isolated
points and
$X\not\in\M (X,\tau')$, then
$$
\bor(X, \tau')/{(\M(X, \tau')\cap \bor(X, \tau'))}
\cong\hbox{\boldmath C}_{\omega}.
$$
\end{proof}

The main result of this paper shows that it is consistent,
relative to the consistency of ZFC, that the Baire Property can be
extended to any countably generated \s-algebra $\A$ containing
$\bor(X)$. In Section 2 we show that the latter follows from a
statement concerning special sets of reals, dual version of which
implies that any Borel measure  on $X$ can be extended to a
measure on such an $\A$. In Section 3 we indicate some
consequences of the extension properties in question.

\smallskip

In the sequel $I$ always stands  either for $\M(X)$ or
$\N_{\mu}$ for a given Borel measure $\mu$ on $X$.




\section{Main results}



 Recall that an $I$-Lusin set is an
uncountable subset of $X$
which has countable intersection with  every element
of $I$. If $I=\M(X)$ ($I=\N_\mu$, resp.), then
an $I$-Lusin set is called a Lusin set
(a Sierpi\' nski set, resp.).

We shall need two auxiliary lemmas.  The first one was proved  by
Rec\l aw and the author in \cite{re-za} (see \cite{re-za}, Lemma
3.2).



\begin{lemma}

 Let \A\ be  any countably generated \s-algebra
  containing $\bor(X)$.

If there exists a $I$-Lusin set of cardinality the continuum and
every subset of $X$ of cardinality the continuum contains a
one-to-one Borel image  of a set not in $I$, then
there exist
 a set $Z\sub X,\ Z\not\in I$ and
a $\bor(Z)$-\A\ measurable function $\varphi:Z \w X$ such that:

\begin{enumerate}

\item $\forall A\in I\ \varphi^{-1}[A]\in I$,

\item $\forall A\in(\bor(X)\setminus I)\ \varphi ^{-1}[A]\not\in I.$

\end{enumerate}
\end{lemma}






\begin{lemma}

Suppose that $Z\sub X$ and $Z\notin I$. Then
$$
\bor(Z)/{(I\cap\bor(Z))}\cong
\bor(X)/(I\cap\bor(X)).
$$
\end{lemma}


\begin{proof} Let $Z^{*}\in\bor(X)$ be such that $Z\sub Z^{*}$ and
$\forall B\in\bor(X)\ B\sub Z^{*}\setminus Z \Rightarrow B\in I$.


Note that the function $F:\bor(Z)/(I\cap\bor(Z))\w
\bor(Z^{*})/(I\cap\bor(Z^{*}))$ given by
$$
F([B\cap Z]_{I\cap \hbox{\scriptsize B}(Z)}) =
[B\cap Z^{*}]_{I\cap\hbox{\scriptsize B}(Z^{*})}
\hbox{ for } B\in\bor(X),
$$
is an isomorphism of the corresponding Boolean algebras.

But   the
algebras $\bor(Z^{*})/(I\cap\bor(Z^{*}))$ and
$\bor(X)/(I\cap\bor(X))$ are isomorphic, since the latter is homogeneous.

\end{proof}

Now we are ready to state the main result.


\begin{theorem}

 Let \A\ be  any countably generated \s-algebra
  containing $\bor(X)$.

If there exists a $I$-Lusin set of cardinality the continuum and
every subset of $X$ of cardinality the continuum contains a
one-to-one Borel image  of a set not in $I$, then
there exists a \s-ideal $\mathcal{I}$ in $\A$ such that:

\begin{enumerate}

\item $\mathcal{I} \cap \bor(X)=I\cap\bor(X)$,

\item  $\A/\mathcal{I}\cong \bor(X)/(I\cap\bor(X))$.

\end{enumerate}

\smallskip

In particular, if there exists a Lusin set (a Sierpi\'nski set, resp.) of
cardinality the continuum and
every set of reals of cardinality the continuum contains a
one-to-one Borel image of a non-meager set  (a  set of positive
 outer Lebesgue measure, resp.), then   the Baire Property (the
Lebesgue measure, resp.) can be extended
to any countably generated  \s-algebra containing Borel sets.


\end{theorem}

\begin{proof}  Let $Z\not\in I$ and $\varphi:Z \w X$ be the objects whose
 existence is guaranteed by Lemma 2.1. Define $\mathcal{I}$ by letting
$$
A\in \mathcal{I} \iff \phi^{-1}[A]\in  I, \hbox{ for } A\in\mathcal{A}.
$$
Condition (1) is clearly satisfied. Moreover,
$\phi$ induces a complete embedding of the  Boolean algebra
$\mathcal{A}/\mathcal{I}$ into the algebra  $\bor(Z)/(I\cap\bor(Z))$ which in turn,
by  Lemma 2.2, is isomorphic to $\bor(X)/(I\cap\bor(X))$. It follows that the
algebra $\A/\mathcal{I}$ is also isomorphic to
$\bor(X)/(I\cap\bor(X))$
as an atomless,  complete subalgebra of
the latter.

\end{proof}

If $I=\M(X)$ ($I=\N_\mu$, resp.), then the hypotheses of Theorem
2.3 are true in the Cohen real model  (random real model, resp.),
i.e., the model obtained by adding $\kappa>\omega_1$ Cohen reals
(random reals, resp.) to a model of CH (see Miller \cite{mi}). The
natural question, to what extent these conditions are also
necessary,  will be briefly discussed in the next section.


\section{Corollaries and  remarks}



It seems to be of some interest to study consequences of
statements like ``Lebesgue measure can be extended by any countable collection
 of sets" and its category counterpart, dealt primarily with in this
paper. The first one implies for instance that if $Z\subseteq X,\
Z\notin \N_\mu$ and a family $\{D^y:y\in Z\}$ consists of non-null
measurable sets,  then there exists $E\subseteq Z,\ E\notin
\N_\mu$ such that $\bigcap_{y\in E} D^y\neq\emptyset$ (see
\cite{z}). Dually, we have (compare \cite{kam1}, Proposition 2,
where a different proof of a similar result is given).


\begin{theorem}

 Assume that the Baire Property  can be extended
to any countably generated  \s-algebra  containing $\bor(X)$.

If $Z\subseteq X,\ Z\notin \M(X)$ and a family $\{D^y:y\in Z\}$
consists of non-meager sets with the Baire property, then there
exists $E\subseteq Z,\ E\notin \M(X)$ such that $\bigcap_{y\in E}
D^y\neq\emptyset$.

\end{theorem}

\begin{proof} Equivalently, we shall prove that given a set $D\sub X\times X$
with all sections $D_x= \{y:\ \langle x,y\rangle\in D\}$
 meager and all
sections $D^y=\{x:\ \langle x,y\rangle\in D\}$ having the Baire property,
then $D^y\in\M(X)$ for every $y$ outside a meager set.

To begin with, cover $D$ by a set $C$ with $C_x\in
\hbox{F}_{\sigma}\cap \M(X)$ for every $x\in X$. It is easy to see
that $C\in \A\otimes \bor(X)$ for a certain countably generated
\s-algebra \A. Since the Baire Property  can be extended to \A, we
have $\A=\bor(X,\tau')$ for a certain second countable topology
$\tau'$ on $X$ with  $\M (X,\tau')\cap \bor(X,\tau)= \M
(X,\tau)\cap \bor(X,\tau)$.

Now the Kuratowski--Ulam theorem (see \cite{ke}, 8.41), applied to
$C$ considered as a Borel subset of the product of the spaces
$(X,\tau')$ and $(X,\tau)$, gives $$ \{y\in X: C^y\not\in
\M(X,\tau')\} \in\M(X,\tau). $$ But $$ \{y:D^y\not\in
\M(X,\tau)\}\sub \{y\in X: C^y\not\in \M(X,\tau')\}, $$ due to the
fact that $\M (X,\tau')\cap \bor(X,\tau)= \M (X,\tau)\cap
\bor(X,\tau)$ and $D^y$ has the Baire property in $(X,\tau)$, for
every $y\in X$.

\end{proof}

 Theorem  3.1 and its dual  may be considerably generalized, if we take
into account   the results obtained by Rec\l aw and the author in
\cite{re-za}. The point is that the technical condition expressed in
Lemma 2.1
 turns out to be an equivalent formulation of the extension property in
question. More precisely, we have


\begin{proposition}

  Let \A\ be  a countably generated \s-algebra
containing $\bor(X)$. The following are equivalent:

\smallskip

 (i) There exists a \s-ideal $\mathcal{I}$ in $\A$ such that:

\begin{enumerate}

\item  $\mathcal{I} \cap \bor(X)=I \cap \bor(X)$,

\item $\A/\mathcal{I}\cong \bor(X)/(I \cap \bor(X))$.

\end{enumerate}

\smallskip

(ii) There exist
 a set $Z\sub X,\ Z\not\in I$ and
a $\bor(Z)$-\A\ measurable function $\varphi:Z \w X$ such that:

\begin{enumerate}

\item $\forall A\in I\ \varphi^{-1}[A]\in I$,

\item $\forall A\in(\bor(X)\setminus I)\ \varphi ^{-1}[A]\not\in I.$

\end{enumerate}

\end{proposition}

\begin{proof} Only the implication $\hbox{(i)}\Rightarrow\hbox{(ii)}$ requires
proof,
since the other one was established  in the course  of proving Theorem 2.3.

As in the proof of Proposition 1.3 ($\hbox{(ii)}\Rightarrow\hbox{(iii)}$)  find a
separable metric $d$ on $X$ such that
$\A=\bor(X,d)$,  let $(\hat X, \hat d)$ be the
completion of $(X,d)$ and
define a \s-ideal $J$ in $\bor(\hat X, \hat d)$ by:
$$
 A\in J \iff A\cap X \in \mathcal{I}, \hbox{ for } A\in\bor(\hat X, \hat d).
$$
Since
$$
\bor(\hat X, \hat
d)/J \cong \A/\mathcal{I}\cong \bor(X,\tau)/(I \cap
\bor(X,\tau)),
$$
there is a Borel isomorphism $\psi:\hat X \w X$ between $(\hat
X,\hat d)$ and $(X,\tau)$ such that
$$
 A\in J \iff \psi[A]\in \M(X,\tau), \hbox{ for } A\in\bor(\hat X, \hat d).
$$
(see \cite{ke}, 15.10).
Finally define $Z=\psi[X]$ and $\varphi=\psi^{-1}|Z$. It is easy
to check that this works.

\end{proof}

As corollaries we obtain the following reformulations of theorems 3.3
and 3.4 from \cite{re-za}.



\begin{theorem}

 Assume that the Baire Property  can be extended
to any countably generated  \s-algebra  containing $\bor(X)$.

Let $Y$ be a Polish space and suppose that $J$ is a \s-ideal on
$Y$ generated by a hereditary $\hbox{\boldmath $\Pi^1_1$}$ (in the
Effros Borel structure, see \cite{ke}, \S \hbox{\rm 35}) family of
closed subsets of $Y$.

If $Z\subseteq Y,\ Z\notin J$ and a family $\{D^y:y\in Z\}$
consists of non-meager sets with the Baire property, then there
exists $E\subseteq Z,\ E\notin J$ such that $\bigcap_{y\in E}
D^y\neq\emptyset$.


\end{theorem}

\begin{theorem}

Assume that the Lebesgue measure  can be extended
to any countably generated  \s-algebra  containing $\bor(X)$.

Let $Y$ be a Polish space and suppose that $J$ is a \s-ideal on $Y$
generated by any of the following families of closed subsets of  $Y$:

(i) all compact sets (in this case $Y$ is assumed to be non-\s-compact),

(ii) all closed sets in $\N_\nu$ for a \s-finite Borel continuous
measure $\nu$ on $Y$,

(iii) all closed subsets of a $\hbox{\boldmath $\Pi^1_1$}$ set $A\sub
Y$.

If $Z\subseteq Y,\ Z\notin J$ and a family $\{D^y:y\in Z\}$
consists  of non-null measurable sets,
 then there exists
$E\subseteq Z,\ E\notin J$ such that $\bigcap_{y\in E}
D^y\neq\emptyset$.

\end{theorem}


Let us now discuss the hypotheses of Theorem 2.3.  It is well--known that
 a set $A\sub X$ contains a Borel one-to-one image of a set of
positive outer $\mu$-measure if and only if it is not a universal
measure zero set, i.e.,  $A\not\in \N_{\nu}$ for a certain finite
atomless Borel
 measure $\nu$.
 The non-existence of  universal measure zero
sets of size continuum means, in turn, that if \A\ is a
$\s$-algebra  generated by $\bor(X)$ plus any
countable collection of new subsets of $X$, then \A\ carries a
finite, countably additive measure, vanishing on singletons -- see
\cite{b-c}
 (every such
measure clearly extends a certain Borel measure but this does not
necessarily
mean that there is one which extends the measure $\mu$, fixed in
advance). On the category side,  sets  that do not contain  Borel
one-to-one images of
 non-meager sets were studied by Grzegorek in \cite{grze1},
\cite{grze2}  and called
there $\overline{\hbox{AFC}}$ sets.
One can prove that the non-existence of $\overline{\hbox{AFC}}$
sets of size continuum means  that if \A\ is a countably generated
$\s$-algebra  containing $\bor(X,\tau)$,
then  there exists a separable metrizable topology $\tau'$ on $X$ such
that   $X$ is dense in itself,
$\A=\bor (X,\tau')$ and $X\not\in\M (X,\tau')$ -- see \cite{z2} (it is
easy to see that for each such topology $\tau'$ there is
a certain Polish topology $\tau''$ on $X$ such that
$\bor(X,\tau'')=\bor(X,\tau)$  and
$\M (X,\tau')\cap \bor(X,\tau)=\M (X,\tau'')\cap \bor(X,\tau)$.
 but this does not
necessarily
mean that there is one  for which $\M (X,\tau')\cap \bor(X,\tau)=
\M (X,\tau)\cap \bor(X,\tau)$,  where $\tau$ is the Polish topology
fixed in advance).

In view of the above we see that the non-existence of
  $\overline{\hbox{AFC}}$ sets
 (universal
measure zero sets, resp.) of size continuum is a necessary
condition for the respective extension principles and it  seems
conceivable that it is also sufficient. The latter, however, is
not the case.


\begin{proposition}


The following two statements are consistent with ZFC:

(i) There are no   $\overline{\hbox{AFC}}$ sets of size continuum
but for any Polish topology $\tau$ on $X$ there is a countably
generated \s-algebra \A\ containing $\bor(X,\tau)$ such that the Baire
Property cannot be extended to \A.

\smallskip


(ii) There are no universal measure zero sets of size continuum
but for any finite, atomless Borel measure $\mu$ on $X$ there is a
countably generated \s-algebra \A\ containing $\bor(X)$ such that
there is no extension of $\mu$ to a measure defined on \A.


\end{proposition}

\begin{proof} A model for both, (i) and (ii), is the iterated
perfect set model, i.e.,  the model obtained by the countable
support iteration of length $\omega_2$ of the Sacks forcing over a
model of CH.
We give an argument for the category case only -- the measure case is
symmetric.

So  let us work in this model. Then, by a result of Miller
\cite{mi}, there are no
 $\overline{\hbox{AFC}}$ sets of size continuum. On the
other hand,  $X$ is the union of $\omega_1$ many meager sets (see \cite{b-j}, 7.6.2). Using
this, it is easy to  construct a family $\{D^y:y\in X\}$
 of
co-meager sets such that for every
$E\notin \M(X)$, $\bigcap_{y\in E} D^y\in\M(X)$.
Now the second clause of (i) follows from Theorem 3.1.

\end{proof}

 In order to complete the discussion of the duality between
measure and category, as far as the extension properties dealt with in
this paper are concerned, let us remark that category versions of
both the \L o\' s--Marczewski and the Banach--Kuratowski  theorems are
also true.

 For the latter, just take the entries of an
 Ulam $(\omega_{1},\omega)$-matrix, which under CH can be identified with
sets of reals. It is well-known that together with Borel sets
they generate a countably
generated \s-algebra, say $\mathcal{A}$, with the property that for no
\s-ideal $\mathcal{I}$ in \A\ containing singletons, the Boolean algebra
$\mathcal{A}/\mathcal{I}$ satisfies the countable chain condition. Hence
neither Lebesgue measure nor  Baire Property can be extended to \A.

Finally, the fact that if $\A$ is generated  by
$\bor(X)$ plus any
finite collection of subsets of $X$, then the Baire Property can be
extended to \A, may be established by a simple argument that covers the
measure case as well.


\begin{proposition}

If a \s-algebra $\A$ is generated  by
$\bor(X)$ plus a
finite collection $\{Z_1,\ldots,Z_n\}$ of subsets of $X$, then
there exists a \s-ideal $\mathcal{I}$ in $\A$ such that:

\begin{enumerate}

\item $\mathcal{I}
 \cap \bor(X)=I\cap\bor(X)$,

\item  $\A/\mathcal{I}\cong \bor(X)/(I\cap\bor(X))$.

\end{enumerate}

\end{proposition}

\begin{proof} Without loss of generality we can assume that
$\{Z_1,\ldots,Z_n\}$ is a partition of $X$ into sets not in $I$.
Now it suffices to let $\mathcal{I}=I\cap\A$. Indeed, we have
$$
\A/\mathcal{I} \cong
\bor(Z_1)/{(I\cap\bor(Z_1))}\times\cdots\times\bor(Z_n)/{(I\cap\bor(Z_n))}.
$$
But
$$\displaylines{
\bor(Z_1)/{(I\cap\bor(Z_1))}\times\cdots\times\bor(Z_n)/{(I\cap\bor(Z_n))}
\hfill\cr
\hfill \cong\bor(X)/{(I\cap\bor(X))}\times\cdots\times\bor(X)/{(I\cap\bor(X))}
\cong\bor(X)/{(I\cap\bor(X))},
}$$
due to Lemma 2.2 and the homogeneity of the algebra $\bor(X)/{(I\cap\bor(X))}$.

\end{proof}



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