lecture 05

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The complement of a meager set is called
\vocab{comeager}.
\end{definition}
\begin{example}
$\Q \subseteq \R$ is meager.
@ -128,7 +127,7 @@
Let $A, B \subseteq X$.
We write $A =^\ast B$
iff the \vocab{symmetric difference},
$A \mathop{\triangle} B \coloneqq (A\setminus B) \cup (B \setminus A)$,
$A \symdif B \coloneqq (A\setminus B) \cup (B \setminus A)$,
is meager.
\end{notation}
\begin{remark}
@ -144,13 +143,14 @@ Note that open sets and meager sets have the Baire property.
% \begin{example}
% $\Q \subseteq \R$ is $F_\sigma$.
%
% $\R \setminus \Q \subseteq \R$ is $G_\delta$.
%
% $\Q \subseteq \R$ is not $G_{\delta}$.
% (It is dense and meager,
% hence it can not be $G_\delta$,
% by the Baire category theorem).
% \end{example}
\begin{example}
\begin{itemize}
\item $\Q \subseteq \R$ is $F_\sigma$.
\item $\R \setminus \Q \subseteq \R$ is $G_\delta$.
\item $\Q \subseteq \R$ is not $G_{\delta}$.
(It is dense and meager,
hence it can not be $G_\delta$
by the Baire category theorem).
\end{itemize}
\end{example}

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\lecture{05}{2023-10-31}{}
\begin{fact}
A set $A$ is nwd iff $\overline{A}$ is nwd.
If $F$ is closed then
$F$ is nwd iff $X \setminus F$ is open and dense.
Any meager set $B$ is contained in
a meager $F_{\sigma}$-set.
\end{fact}
\begin{definition}
A \vocab{$\sigma$-algebra} on a set $X$
is a collection of subsets of $X$
such that:
\begin{itemize}
\item $\emptyset, X \in \cA$,
\item $ A \in \cA \implies X \setminus A \in \cA$,
\item $(A_i)_{i < \omega}, A_i \in \cA \implies \bigcup_{i < \omega} A_i \in \cA$.
\end{itemize}
\end{definition}
\begin{fact}
Since $\bigcap_{i < \omega} A_i = \left( \bigcup_{i < \omega} A_i^c \right)^c$
we have that $\sigma$-algebras are closed under countable intersections.
\end{fact}
\begin{theorem}
\label{thm:bairesigma}
Let $X$ be a topological space.
Then the collection of sets with the Baire property
is a $\sigma$-algebra on $X$.
It is the smallest $\sigma$-algebra
containing all meager and open sets.
\end{theorem}
\begin{refproof}{thm:bairesigma}
Let $\cA$ be the collection of sets with the Baire property.
Since open sets have the Baire property,
we have $\emptyset, X \in \cA$.
Let $A_n \in \cA$ for all $n < \omega$.
Take $U_n$ such that $A_n \symdif U_n$ is meager.
Then
\[
\left( \bigcup_{n < \omega} A_n \right) \symdif \left( \bigcup_{n < \omega} U_n \right)
\]
is meager,\todo{small exercise}
hence $\bigcup_{n < \omega} A_n \in \cA$.
Let $A \in \cA$.
Take some open $U$ such that $U \symdif A$ is meager.
We have $(X \setminus U) \symdif (X \setminus A) = U \symdif A$.
\begin{claim}
\label{thm:bairesigma:c1}
If $F$ is closed,
then $F \setminus \inter(F)$
is nwd.
In particular, $F \symdif \inter(F)$ is nwd.
\end{claim}
\begin{refproof}{thm:bairesigma:c1}
\todo{TODO}
\end{refproof}
From the claim we get that
$X \setminus A =^\ast X \setminus U =^\ast \inter(X \setminus U)$.
Hence $X \setminus A \in \cA$.
It is clear that all meager sets have the Baire property.
Let $A \in \cA$. Then $A = (A \setminus U) \cup (A \cap U)$
for some open $U$
such that $A \setminus U$ is meager.
We have $A \cap U = U \setminus (U \setminus A)$.
Thus we get that $\cA$ is the minimal $\sigma$-algebra
containing all meager and all open sets.
\end{refproof}
%\begin{example}
% Nwd set of positive measure.
% TODO
% remove open intervals such that their length does not add to 0
%
%\end{example}
\begin{theorem}[Baire Category theorem]
Let $X$ be a completely metrizable space.
Then every comeager set of $X$ is dense in $X$.
\end{theorem}
\todo{Proof (copy from some other lecture)}
\begin{proposition}
Let $X$ be a topological space.
The following are equivalent:
\begin{enumerate}[(i)]
\item Every nonempty open set
is non-meager in $X$.
\item Every comeager set is dense.
\item The intersection of countable many
open dense sets is dense.
\end{enumerate}
\end{proposition}
\begin{proof}
\todo{Proof (short)}
(iii) $\implies$ (i)
Let us first show that $X$ is non-meager.
Suppose that $X$ is meager. Then $X = \bigcup_{n} A_n = \bigcup_{n} \overline{A_n}$
is the countable union of nwd sets.
We have that
\[
\emptyset = \bigcap_{n} (X \setminus \overline{A_n})
\]
is dense by (iii).
This proof can be adapted to other open sets $X$.
\end{proof}
% TODO Fubini
\begin{notation}
Let $X ,Y$ be topological spaces,
$A \subseteq X \times Y$,
$x \in X, y \in Y$.
Let
\[
A_x \coloneqq \{y \in Y : (x,y) \in A\}
\]
and
\[
A^y \coloneqq \{x \in X : (x,y) \in A\} .
\]
\end{notation}
The following similar to Fubini,
but for meager sets:
\begin{theorem}[Kuratowski-Ulam]
\label{thm:kuratowskiulam}
Let $X,Y$ be second countable topological spaces.
Let $A \subseteq X \times Y$
be a set with the Baire property.
Then
\begin{enumerate}[(i)]
\item $\{x \in X : A_x \text{ has the BP }\}$
is comeager\footnote{Note that not necessarily all sections
have the BP. For example $\{x\} \times Y$ is meager in $X \times Y$}
and similarly for $y$.
\item $A$ is meager
\begin{IEEEeqnarray*}{rll}
\iff &\{x \in X : A_x \text{ is meager }\}&\text{ is comeager}\\
\iff &\{y \in Y : A^y \text{ is meager }\}& \text{ is comeager}.
\end{IEEEeqnarray*}
\item $A$ is comeager
\begin{IEEEeqnarray*}{rll}
\iff & \{x \in X: A_x \text{ is comeager }\} &\text{ is comeager}\\
\iff & \{y \in Y: A^y \text{ is comeager}\} & \text{ is comeager}.
\end{IEEEeqnarray*}
\end{enumerate}
\end{theorem}
\begin{refproof}{thm:kuratowskiulam}
(ii) and (iii) are equivalent by passing to the complement.
\begin{claim}%[1a]
\label{thm:kuratowskiulam:c1a}
If $F \overset{\text{closed}}{\subseteq} X \times Y$
is nwd,
then
\[
\{x \in X : F_x \text{is nwd}\}
\]
is comeager.
\end{claim}
\begin{refproof}{thm:kuratowskiulam:c1a}
Put $W = F^c$.
This is open and dense in $X \times Y$.
It suffices to show that $\{x \in X : W_x \text{ is dense}\}$
is comeager.
Note that $W_x$ is open for all $x$.
Fix a countable basis $(V_n)$ of $Y$
with $V_n$ non-empty.
We want to show that
\[
\{x \in X: \forall n.~ (W_x \cap V_n) \neq \emptyset\}
\]
is a comeager set.
This is equivalent to
\[
\{x \in X : (W_x \cap V_n) \neq \emptyset\}
\]
being comeager for all $n$,
because the intersection
of countably many comeager sets is comeager.
Fix $n$ and let $U_n \coloneqq \{x \in X: (W_x \cap V_n) = \emptyset\}$.
We will show that $U_n$ is open and dense,
hence it is comeager.
$U_n = \proj_x(W \cap (X \times V_n))$ is open
since projections of open sets are open.
Let $U \subseteq X$ be nonempty and open.
We need to show that $U \cap U_n \neq \emptyset$.
It is
\[
U \cap U_n = \proj_x(W \cap (U \times V_n))
\]
nonempty since $W$ is dense.
\end{refproof}
\begin{claim} % [1a']
\label{thm:kuratowskiulam:c1ap}
If $F \subseteq X \times Y$
is nwd,
then
\[
\{x \in X : F_x \text{is nwd}\}
\]
is comeager.
\end{claim}
\begin{refproof}{thm:kuratowskiulam:c1ap}
We have that $\overline{F}$ is nwd.
Hence by \yaref{thm:kuratowskiulam:c1a}
the set
\[
\{x \in X: \overline{F_x} \text{ is nwd}\} \subseteq
\{x \in X: F_x \text{ is nwd}\}
\]
is comeager.
\end{refproof}
\begin{claim}% [1b]
\label{thm:kuratowskiulam:c1b}
If $M \subseteq X \times Y$ is meager,
then
\[
\{x \in X : M_x \text{ is meager}\}
\]
is comeager.
\end{claim}
\begin{refproof}{thm:kuratowskiulam:c1b}
This follows from \yaref{thm:kuratowskiulam:c1ap}:
Let $M = \bigcup_{n < \omega} F_n$
where the $F_n$ are nwd.
Apply \yaref{thm:kuratowskiulam:c1ap}
to each $F_n$.
We get that
$M_x$ is comeager
as a countable intersection of comeager sets.
\end{refproof}
\todo{Finish proof}
\phantom\qedhere
\end{refproof}
\begin{remark}
Suppose that $A$ has the BP.
Then there is an open $U$ such that
$A \symdif U \mathbin{\text{\reflectbox{$\coloneqq$}}} M$ is meager.
Then $A = U \symdif M$.
\end{remark}

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%\mathbin{\raisebox{1ex}{\scalebox{.7}{$\frown$}}}%
\DeclareMathOperator{\hght}{height}
\DeclareMathOperator{\symdif}{\triangle}
\DeclareSimpleMathOperator{proj}
\newcommand\lecture[3]{\hrule{\color{darkgray}\hfill{\tiny[Lecture #1, #2]}}}

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\input{inputs/lecture_02}
\input{inputs/lecture_03}
\input{inputs/lecture_04}
\input{inputs/lecture_05}