From 0b659968db287d8f5cf6f5b75b7d8775726ef2cc Mon Sep 17 00:00:00 2001 From: Josia Pietsch Date: Tue, 31 Oct 2023 11:53:03 +0100 Subject: [PATCH] lecture 05 --- inputs/lecture_04.tex | 24 ++-- inputs/lecture_05.tex | 271 ++++++++++++++++++++++++++++++++++++++++++ logic.sty | 2 + logic3.tex | 1 + 4 files changed, 286 insertions(+), 12 deletions(-) create mode 100644 inputs/lecture_05.tex diff --git a/inputs/lecture_04.tex b/inputs/lecture_04.tex index d849794..ceca77d 100644 --- a/inputs/lecture_04.tex +++ b/inputs/lecture_04.tex @@ -119,7 +119,6 @@ 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} diff --git a/inputs/lecture_05.tex b/inputs/lecture_05.tex new file mode 100644 index 0000000..161ec91 --- /dev/null +++ b/inputs/lecture_05.tex @@ -0,0 +1,271 @@ +\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} + + diff --git a/logic.sty b/logic.sty index 97f3eb7..2c60891 100644 --- a/logic.sty +++ b/logic.sty @@ -131,5 +131,7 @@ %\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]}}} diff --git a/logic3.tex b/logic3.tex index 2f5d966..56808ff 100644 --- a/logic3.tex +++ b/logic3.tex @@ -28,6 +28,7 @@ \input{inputs/lecture_02} \input{inputs/lecture_03} \input{inputs/lecture_04} +\input{inputs/lecture_05}