% \subsection{Additional Material}

% Important stuff not done in the lecture.

\subsection{Notions of boundedness}
The following is just a short overview of all the notions of
boundedness we used in the lecture.

\begin{definition}+[Boundedness]
    Let $\cX$ be a set of random variables.
    We say that $\cX$ is
    \begin{itemize}
        \item \vocab{uniformly bounded} iff
            \[\sup_{X \in \cX} \sup_{\omega \in \Omega} |X(\omega)| < \infty,\]
        \item \vocab{dominated by $f \in L^p$} for $p \ge 1$ iff
            \[
            \forall X \in \cX .~ |X| \le f,
            \]
        \item \vocab{bounded in $L^p$} for $p \ge 1$ iff
            \[
            \sup_{X \in \cX} \|X\|_{L^p} < \infty,
            \]
        \item \vocab{uniformly integrable} iff
            \[
            \forall \epsilon > 0 .~\exists K .~ \forall X \in \cX.~
            \bE[|X| \One_{|X| > K}] < \epsilon.
            \]
    \end{itemize}
\end{definition}

\begin{fact}+
    Let $\cX$ be a set of random variables.
    Let $1 < p \le q < \infty$
    Then the following implications hold:
    \begin{figure}[H]
        \centering
        \begin{tikzpicture}
            \node at (0,2.5) (ub) {$\cX$ is uniformly bounded};
            \node at (-2.5,1.5) (dq) {$\cX$ is dominated by $f \in L^q$};
            \node at (-2.5,0.5) (dp) {$\cX$ is dominated by $f \in L^p$};
            \node at (2.5,1.0) (bq) {$\cX$ is bounded in $L^q$};
            \node at (2.5,0) (bp) {$\cX$ is bounded in $L^p$};
            \node at (-2.5,-0.5) (d1) {$\cX$ is dominated by $f \in L^1$};
            \node at (0,-1.5) (ui) {$\cX$ is uniformly integrable};
            \node at (2.5,-2.5) (b1) {$\cX$ is bounded in $L^1$};
            \draw[double equal sign distance, -implies] (ub) -- (dq);
            % \draw[double equal sign distance, -implies] (ub) -- (bq);
            \draw[double equal sign distance, -implies] (bq) -- (bp);
            \draw[double equal sign distance, -implies] (dq) -- (dp);
            \draw[double equal sign distance, -implies] (dq) -- (bq);
            \draw[double equal sign distance, -implies] (dp) -- (bp);
            \draw[double equal sign distance, -implies] (bp) -- (ui);
            \draw[double equal sign distance, -implies] (dp) -- (d1);
            \draw[double equal sign distance, -implies] (d1) -- (ui);
            \draw[double equal sign distance, -implies] (ui) -- (b1);
        \end{tikzpicture}
    \end{figure}
\end{fact}





\subsection{Laplace Transforms}
\todo{Write something about Laplace Transforms}