300 lines
8.8 KiB
TeX
300 lines
8.8 KiB
TeX
\lecture{05}{20231031}{}




\begin{fact}


\begin{itemize}


\item A set $A$ is nwd iff $\overline{A}$ is nwd.


\item If $F$ is closed then $F$ is nwd iff $X \setminus F$ is open and dense.


\item Any meager set $B$ is contained in a meager $F_{\sigma}$set.


\end{itemize}


\end{fact}


\gist{%


\begin{proof}


\begin{itemize}


\item This follows from the definition as $\overline{\overline{A}} = \overline{A}$.


\item Trivial.


\item Let $B = \bigcup_{n < \omega} B_n$ be a union of nwd sets.


Then $B \subseteq \bigcup_{n < \omega} \overline{B_n}$.


\end{itemize}


\end{proof}


}{}




\gist{%


\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 \gist{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,


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}


$F \setminus \inter(F)$ is closed,


hence $\overline{F \setminus \inter(F)} = F \setminus \inter(F)$.


Clearly $\inter(F\setminus\inter(F)) = \emptyset$.


\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]


\yalabel{Baire Category Theorem}{Baire Category Theorem}{thm:bct}


Let $X$ be a completely metrizable space.


Then every comeager set of $X$ is dense in $X$.


\end{theorem}


\gist{%


\todo{Proof (copy from some other lecture)}


}{Not proved in the lecture.}


\begin{theoremdef}


Let $X$ be a topological space.


The following are equivalent:


\begin{enumerate}[(i)]


\item Every nonempty open set


is nonmeager in $X$.


\item Every comeager set is dense.


\item The intersection of countable many


open dense sets is dense.


\end{enumerate}


In this case $X$ is called a \vocab{Baire space}.%


\footnote{cf.~\yaref{s5e1}}


\end{theoremdef}


\begin{proof}


(i) $\implies$ (ii)


\gist{%


Consider a comeager set $A$.


Let $U\neq \emptyset$ be any open set. Since $U$ is


nonmeager, we have $A \cap U \neq \emptyset$.


}{The intersection of a comeager and a nonmeager set is nonempty.}




(ii) $\implies$ (iii)


The complement of an open dense set is nwd.


\gist{%


Hence the intersection of countable


many open dense sets is comeager.


}{}








(iii) $\implies$ (i)


Let us first show that $X$ is nonmeager.


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}




\begin{notation}


Let $X ,Y$ be topological spaces,


$A \subseteq X \times Y$


and


$x \in X, y \in Y$.




Let


\[


A_x \coloneqq \{y \in Y : (x,y) \in A\}


\]


and


\[


A^y \coloneqq \{x \in X : (x,y) \in A\} .


\]


\end{notation}




\gist{%


The following similar to Fubini,


but for meager sets:


}{}




\begin{theorem}[KuratowskiUlam]


\yalabel{KuratowskiUlam}{KuratowskiUlam}{thm:kuratowskiulam}


Let $X,Y$ be second countable topological spaces.


Let $A \subseteq X \times Y$


be a set with the Baire property.%


\footnote{It is important that $A$ has the Baire property (cf. \yaref{s5e4}).}




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}


\gist{


(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 $V_n$ nonempty.


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}


}{}






% \phantom\qedhere


% \end{refproof}


% TODO fix claim numbers




