\relax \providecommand\zref@newlabel[2]{} \providecommand\hyper@newdestlabel[2]{} \@writefile{toc}{\contentsline {section}{\numberline {5}Lecture 5 -- Simple and Measurable Functions}{18}{section.5}\protected@file@percent } \newlabel{sec:lec05}{{5}{18}{Lecture 5 -- Simple and Measurable Functions}{section.5}{}} \newlabel{sec:lec05@cref}{{[section][5][]5}{[1][18][]18}{}{}{}} \@writefile{toc}{\contentsline {subsection}{\numberline {5.1}Simple random variables}{18}{subsection.5.1}\protected@file@percent } \newlabel{def:simple-rv}{{5.1}{18}{Simple random variable}{tcb@cnt@definition.5.1}{}} \newlabel{def:simple-rv@cref}{{[tcb@cnt@definition][1][5]5.1}{[1][18][]18}{}{}{}} \@writefile{toc}{\contentsline {subsection}{\numberline {5.2}Simple functions on a measure space}{19}{subsection.5.2}\protected@file@percent } \newlabel{def:simple-function}{{5.2}{19}{Simple function}{tcb@cnt@definition.5.2}{}} \newlabel{def:simple-function@cref}{{[tcb@cnt@definition][2][5]5.2}{[1][19][]19}{}{}{}} \newlabel{prop:simple-algebra}{{5.3}{20}{Algebra of simple functions}{tcb@cnt@definition.5.3}{}} \newlabel{prop:simple-algebra@cref}{{[tcb@cnt@definition][3][5]5.3}{[1][19][]20}{}{}{}} \newlabel{def:simple-integral}{{5.4}{20}{Integral of a simple function}{tcb@cnt@definition.5.4}{}} \newlabel{def:simple-integral@cref}{{[tcb@cnt@definition][4][5]5.4}{[1][20][]20}{}{}{}} \@writefile{toc}{\contentsline {subsection}{\numberline {5.3}Measurable functions}{20}{subsection.5.3}\protected@file@percent } \newlabel{def:measurable-function}{{5.5}{20}{Measurable function}{tcb@cnt@definition.5.5}{}} \newlabel{def:measurable-function@cref}{{[tcb@cnt@definition][5][5]5.5}{[1][20][]20}{}{}{}} \@writefile{toc}{\contentsline {subsection}{\numberline {5.4}Useful facts about measurability}{20}{subsection.5.4}\protected@file@percent } \newlabel{prop:generators-suffice}{{5.6}{21}{Generators suffice}{tcb@cnt@definition.5.6}{}} \newlabel{prop:generators-suffice@cref}{{[tcb@cnt@definition][6][5]5.6}{[1][20][]21}{}{}{}} \newlabel{prop:indicator-measurable}{{5.7}{21}{Indicator functions}{tcb@cnt@definition.5.7}{}} \newlabel{prop:indicator-measurable@cref}{{[tcb@cnt@definition][7][5]5.7}{[1][21][]21}{}{}{}} \newlabel{prop:measurable-algebra}{{5.8}{21}{Algebra of measurable functions}{tcb@cnt@definition.5.8}{}} \newlabel{prop:measurable-algebra@cref}{{[tcb@cnt@definition][8][5]5.8}{[1][21][]21}{}{}{}} \newlabel{prop:measurable-limits}{{5.9}{21}{Pointwise limits of measurable functions}{tcb@cnt@definition.5.9}{}} \newlabel{prop:measurable-limits@cref}{{[tcb@cnt@definition][9][5]5.9}{[1][21][]21}{}{}{}} \newlabel{prop:continuous-measurable}{{5.10}{21}{Continuous functions are Borel measurable}{tcb@cnt@definition.5.10}{}} \newlabel{prop:continuous-measurable@cref}{{[tcb@cnt@definition][10][5]5.10}{[1][21][]21}{}{}{}} \@writefile{toc}{\contentsline {subsection}{\numberline {5.5}Almost everywhere}{21}{subsection.5.5}\protected@file@percent } \newlabel{def:ae}{{5.11}{22}{Almost everywhere / almost surely}{tcb@cnt@definition.5.11}{}} \newlabel{def:ae@cref}{{[tcb@cnt@definition][11][5]5.11}{[1][21][]22}{}{}{}} \@setckpt{lectures/lecture05}{ \setcounter{page}{23} \setcounter{equation}{0} \setcounter{enumi}{5} \setcounter{enumii}{0} \setcounter{enumiii}{0} \setcounter{enumiv}{0} \setcounter{footnote}{0} \setcounter{mpfootnote}{0} \setcounter{part}{0} \setcounter{section}{5} \setcounter{subsection}{5} \setcounter{subsubsection}{0} \setcounter{paragraph}{0} \setcounter{subparagraph}{0} \setcounter{figure}{3} \setcounter{table}{0} \setcounter{parentequation}{0} \setcounter{caption@flags}{2} \setcounter{continuedfloat}{0} \setcounter{section@level}{0} \setcounter{Item}{10} \setcounter{Hfootnote}{0} \setcounter{bookmark@seq@number}{27} \setcounter{tcbbreakpart}{1} \setcounter{tcblayer}{0} \setcounter{tcolorbox@number}{39} \setcounter{tcbrastercolumn}{1} \setcounter{tcbrasterrow}{1} \setcounter{tcbrasternum}{1} \setcounter{tcbraster}{0} \setcounter{lstnumber}{1} \setcounter{tcblisting}{0} \setcounter{tcb@cnt@definition}{11} \setcounter{mdf@globalstyle@cnt}{0} \setcounter{mdfcountframes}{0} \setcounter{mdf@env@i}{0} \setcounter{mdf@env@ii}{0} \setcounter{mdf@zref@counter}{0} \setcounter{example}{3} \setcounter{remark}{5} \setcounter{note}{0} \setcounter{lstlisting}{0} }