\relax 
\providecommand\zref@newlabel[2]{}
\providecommand\hyper@newdestlabel[2]{}
\providecommand\HyField@AuxAddToFields[1]{}
\providecommand\HyField@AuxAddToCoFields[2]{}
\@writefile{toc}{\contentsline {section}{\numberline {1}Lecture 13 -- The Ergodic Theorem}{1}{section.1}\protected@file@percent }
\newlabel{sec:lec13}{{1}{1}{Lecture 13 -- The Ergodic Theorem}{section.1}{}}
\newlabel{sec:lec13@cref}{{[section][1][]1}{[1][1][]1}{}{}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {1.1}Measure-preserving maps, invariance, ergodicity}{1}{subsection.1.1}\protected@file@percent }
\newlabel{def:measure-preserving}{{1.1}{1}{Measure-preserving map}{tcb@cnt@definition.1.1}{}}
\newlabel{def:measure-preserving@cref}{{[tcb@cnt@definition][1][1]1.1}{[1][1][]1}{}{}{}}
\newlabel{def:invariant}{{1.2}{1}{Invariant set, invariant function}{tcb@cnt@definition.1.2}{}}
\newlabel{def:invariant@cref}{{[tcb@cnt@definition][2][1]1.2}{[1][1][]1}{}{}{}}
\newlabel{def:ergodic}{{1.3}{1}{Ergodic map}{tcb@cnt@definition.1.3}{}}
\newlabel{def:ergodic@cref}{{[tcb@cnt@definition][3][1]1.3}{[1][1][]1}{}{}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces Orbit of a point under the rotation \(T(x)=x+a\bmod  1\): for irrational \(a\) the orbit is dense, the dynamics is ergodic, and Birkhoff's theorem says time averages equal space averages.}}{2}{figure.caption.1}\protected@file@percent }
\providecommand*\caption@xref[2]{\@setref\relax\@undefined{#1}}
\newlabel{fig:orbit-rotation}{{1}{2}{Orbit of a point under the rotation \(T(x)=x+a\bmod 1\): for irrational \(a\) the orbit is dense, the dynamics is ergodic, and Birkhoff's theorem says time averages equal space averages}{figure.caption.1}{}}
\newlabel{fig:orbit-rotation@cref}{{[figure][1][]1}{[1][2][]2}{}{}{}}
\newlabel{prop:ergodic-facts}{{1.4}{2}{Two basic facts}{tcb@cnt@definition.1.4}{}}
\newlabel{prop:ergodic-facts@cref}{{[tcb@cnt@definition][4][1]1.4}{[1][2][]2}{}{}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {1.2}Ergodic theorems}{2}{subsection.1.2}\protected@file@percent }
\newlabel{lem:maximal-ergodic}{{1.5}{2}{Maximal ergodic lemma}{tcb@cnt@definition.1.5}{}}
\newlabel{lem:maximal-ergodic@cref}{{[tcb@cnt@definition][5][1]1.5}{[1][2][]2}{}{}{}}
\newlabel{thm:birkhoff}{{1.6}{3}{Birkhoff's pointwise ergodic theorem}{tcb@cnt@definition.1.6}{}}
\newlabel{thm:birkhoff@cref}{{[tcb@cnt@definition][6][1]1.6}{[1][2][]3}{}{}{}}
\newlabel{thm:von-neumann}{{1.7}{3}{von Neumann's mean ergodic theorem}{tcb@cnt@definition.1.7}{}}
\newlabel{thm:von-neumann@cref}{{[tcb@cnt@definition][7][1]1.7}{[1][3][]3}{}{}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {1.3}Application: the strong law of large numbers, again}{3}{subsection.1.3}\protected@file@percent }
\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces Truncation step in von~Neumann's proof: the unbounded \(f\) (red) is clipped to a bounded \(g=\min \{\max \{-C,f\},C\}\) (blue); the tails are absorbed in \(\|f-g\|_p<\mitvarepsilon /3\), and dominated convergence handles \(g\).}}{4}{figure.caption.2}\protected@file@percent }
\newlabel{fig:truncation}{{2}{4}{Truncation step in von~Neumann's proof: the unbounded \(f\) (red) is clipped to a bounded \(g=\min \{\max \{-C,f\},C\}\) (blue); the tails are absorbed in \(\|f-g\|_p<\varepsilon /3\), and dominated convergence handles \(g\)}{figure.caption.2}{}}
\newlabel{fig:truncation@cref}{{[figure][2][]2}{[1][3][]4}{}{}{}}
\newlabel{def:shift}{{1.8}{4}{Shift map on $\R ^{\N }$}{tcb@cnt@definition.1.8}{}}
\newlabel{def:shift@cref}{{[tcb@cnt@definition][8][1]1.8}{[1][4][]4}{}{}{}}
\newlabel{prop:shift-ergodic}{{1.9}{4}{The shift is measure-preserving and ergodic}{tcb@cnt@definition.1.9}{}}
\newlabel{prop:shift-ergodic@cref}{{[tcb@cnt@definition][9][1]1.9}{[1][4][]4}{}{}{}}
\newlabel{thm:slln-ergodic}{{1.10}{4}{Strong law of large numbers, again}{tcb@cnt@definition.1.10}{}}
\newlabel{thm:slln-ergodic@cref}{{[tcb@cnt@definition][10][1]1.10}{[1][4][]4}{}{}{}}
\gdef \@abspage@last{5}