josia-notes/s23-probability-theory
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Josia Pietsch 2023-07-18 22:59:46 +02:00
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commit 76aeeb8d22
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inputs/a_2_additional_stuff.tex
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@ -15,18 +15,17 @@ boundedness we used in the lecture.
\item \vocab{dominated by $f \in L^p$} for $p \ge 1$ iff \item \vocab{dominated by $f \in L^p$} for $p \ge 1$ iff
\[ \[
\forall X \in \cX .~ |X| \le f, \forall X \in \cX .~ |X| \le f,
\] \]
\item \vocab{bounded in $L^p$} for $p \ge 1$ iff \item \vocab{bounded in $L^p$} for $p \ge 1$ iff
\[ \[
\sup_{X \in \cX} \|X\|_{L^p} < \infty, \sup_{X \in \cX} \|X\|_{L^p} < \infty,
\] \]
\item \vocab{uniformly integrable} iff \item \vocab{uniformly integrable} iff
\[ \[
\forall \epsilon > 0 .~\exists K .~ \forall X \in \cX.~ \forall \epsilon > 0 .~\exists K .~ \forall X \in \cX.~
\bE[|X| \One_{|X| > K}] < \epsilon. \bE[|X| \One_{|X| > K}] < \epsilon.
\] \]
\end{itemize} \end{itemize}
\end{definition} \end{definition}
\begin{fact}+ \begin{fact}+