\relax \providecommand\zref@newlabel[2]{} \providecommand\hyper@newdestlabel[2]{} \@writefile{toc}{\contentsline {section}{\numberline {7}Lecture 7 -- Lebesgue-Stieltjes Measure; Fubini-Tonelli}{26}{section.7}\protected@file@percent } \newlabel{sec:lec07}{{7}{26}{Lecture 7 -- Lebesgue-Stieltjes Measure; Fubini-Tonelli}{section.7}{}} \newlabel{sec:lec07@cref}{{[section][7][]7}{[1][26][]26}{}{}{}} \@writefile{toc}{\contentsline {subsection}{\numberline {7.1}Image measures and Lebesgue--Stieltjes}{26}{subsection.7.1}\protected@file@percent } \newlabel{def:image-measure}{{7.1}{26}{Image measure}{tcb@cnt@definition.7.1}{}} \newlabel{def:image-measure@cref}{{[tcb@cnt@definition][1][7]7.1}{[1][26][]26}{}{}{}} \newlabel{thm:lebesgue-stieltjes}{{7.2}{26}{Lebesgue--Stieltjes measure}{tcb@cnt@definition.7.2}{}} \newlabel{thm:lebesgue-stieltjes@cref}{{[tcb@cnt@definition][2][7]7.2}{[1][26][]26}{}{}{}} \newlabel{lem:g-properties}{{7.3}{26}{Properties of the left-inverse $g$}{tcb@cnt@definition.7.3}{}} \newlabel{lem:g-properties@cref}{{[tcb@cnt@definition][3][7]7.3}{[1][26][]26}{}{}{}} \@writefile{lof}{\contentsline {figure}{\numberline {5}{\ignorespaces A right-continuous non-decreasing \(F\) with a jump. \(g(y)=\inf \{x:y\le F(x)\}\) reads the picture sideways: a level \(y\) inside the jump still resolves to a single \(x\)-value.}}{27}{figure.caption.5}\protected@file@percent } \newlabel{fig:cdf-leftinverse}{{5}{27}{A right-continuous non-decreasing \(F\) with a jump. \(g(y)=\inf \{x:y\le F(x)\}\) reads the picture sideways: a level \(y\) inside the jump still resolves to a single \(x\)-value}{figure.caption.5}{}} \newlabel{fig:cdf-leftinverse@cref}{{[figure][5][]5}{[1][27][]27}{}{}{}} \newlabel{def:radon}{{7.4}{27}{Radon measure}{tcb@cnt@definition.7.4}{}} \newlabel{def:radon@cref}{{[tcb@cnt@definition][4][7]7.4}{[1][27][]27}{}{}{}} \@writefile{toc}{\contentsline {subsection}{\numberline {7.2}Product $\sigma $-fields and the product measure}{27}{subsection.7.2}\protected@file@percent } \newlabel{def:prod-sigma}{{7.5}{27}{Rectangles and product $\sigma $-field}{tcb@cnt@definition.7.5}{}} \newlabel{def:prod-sigma@cref}{{[tcb@cnt@definition][5][7]7.5}{[1][27][]27}{}{}{}} \newlabel{thm:product-measure}{{7.6}{27}{Existence and uniqueness of the product measure}{tcb@cnt@definition.7.6}{}} \newlabel{thm:product-measure@cref}{{[tcb@cnt@definition][6][7]7.6}{[1][27][]27}{}{}{}} \@writefile{toc}{\contentsline {subsection}{\numberline {7.3}The monotone class theorem}{28}{subsection.7.3}\protected@file@percent } \newlabel{def:monotone-class}{{7.7}{28}{Monotone class}{tcb@cnt@definition.7.7}{}} \newlabel{def:monotone-class@cref}{{[tcb@cnt@definition][7][7]7.7}{[1][28][]28}{}{}{}} \newlabel{thm:monotone-class-thm}{{7.8}{28}{Monotone class theorem}{tcb@cnt@definition.7.8}{}} \newlabel{thm:monotone-class-thm@cref}{{[tcb@cnt@definition][8][7]7.8}{[1][28][]28}{}{}{}} \newlabel{lem:indic-swap}{{7.9}{28}{Order of integration for indicators}{tcb@cnt@definition.7.9}{}} \newlabel{lem:indic-swap@cref}{{[tcb@cnt@definition][9][7]7.9}{[1][28][]28}{}{}{}} \@writefile{toc}{\contentsline {subsection}{\numberline {7.4}The Fubini--Tonelli theorem}{28}{subsection.7.4}\protected@file@percent } \newlabel{thm:fubini-tonelli}{{7.10}{29}{Fubini--Tonelli}{tcb@cnt@definition.7.10}{}} \newlabel{thm:fubini-tonelli@cref}{{[tcb@cnt@definition][10][7]7.10}{[1][28][]29}{}{}{}} \@setckpt{lectures/lecture07}{ \setcounter{page}{30} \setcounter{equation}{0} \setcounter{enumi}{2} \setcounter{enumii}{0} \setcounter{enumiii}{0} \setcounter{enumiv}{0} \setcounter{footnote}{0} \setcounter{mpfootnote}{0} \setcounter{part}{0} \setcounter{section}{7} \setcounter{subsection}{4} \setcounter{subsubsection}{0} \setcounter{paragraph}{0} \setcounter{subparagraph}{0} \setcounter{figure}{5} \setcounter{table}{0} \setcounter{parentequation}{0} \setcounter{caption@flags}{2} \setcounter{continuedfloat}{0} \setcounter{section@level}{0} \setcounter{Item}{14} \setcounter{Hfootnote}{0} \setcounter{bookmark@seq@number}{35} \setcounter{tcbbreakpart}{1} \setcounter{tcblayer}{0} \setcounter{tcolorbox@number}{57} \setcounter{tcbrastercolumn}{1} \setcounter{tcbrasterrow}{1} \setcounter{tcbrasternum}{1} \setcounter{tcbraster}{0} \setcounter{lstnumber}{1} \setcounter{tcblisting}{0} \setcounter{tcb@cnt@definition}{10} \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}{0} \setcounter{remark}{7} \setcounter{note}{0} \setcounter{lstlisting}{0} }