fixed typo

inputs/lecture_09.tex

@@ -141,7 +141,7 @@ We have
     Then
     \begin{itemize}
         \item $\phi_{a X + b}(t) = e^{\i t b} \phi_X(\frac{t}{a})$,
-        \item $\phi_{X + Y}(t) = \phi_X(t) + \phi_Y(t)$.
+        \item $\phi_{X + Y}(t) = \phi_X(t) \cdot \phi_Y(t)$.
     \end{itemize}
 \end{fact}
 \begin{proof}

inputs/prerequisites.tex

@@ -48,6 +48,7 @@ from the lecture on stochastic.
 
 \pagebreak
 \begin{theorem}+
+    % \footnote{see exercise 3.4}
     \label{thm:convergenceimplications}
     \vspace{10pt}
     Let $X$ be a random variable and $X_n, n \in \N$ a sequence of random variables.
@@ -247,7 +248,20 @@ from the lecture on stochastic.
     We have $[-k,k] \uparrow \R$, hence $\mu([-k,k]) \uparrow \mu(\R)$.
 \end{proof}
 
-\begin{theorem}[Riemann-Lebesgue]
+\begin{theorem}+[Change of variables formula]
+    Let $X$ be a random variable with a continuous density $f$,
+    and let $g: \R \to  \R$ be continuous such that
+    $g(X)$ is integrable.
+    Then
+    \[
+        \bE[g(X)] = \int g \circ X \dif \bP
+        = \int_{-\infty}^\infty g(y) f(y) \lambda(\dif y)
+        = \int_{-\infty}^\infty g(y) f(y) \dif y.
+    \]
+\end{theorem}
+
+\begin{theorem}+[Riemann-Lebesgue]
+    %\footnote{see exercise 3.3}
     \label{riemann-lebesgue}
     Let $f: \R \to  \R$ be integrable.
     Then
@@ -256,7 +270,27 @@ from the lecture on stochastic.
     \]
 \end{theorem}
 
-
+\begin{theorem}+[Fubini-Tonelli]
+    %\footnote{exercise sheet 1}
+    Let $(\Omega_{i}, \cF_i, \bP_i), i \in \{0,1\}$
+    be probability spaces and $\Omega \coloneqq \Omega_0 \otimes \Omega_1$,
+    $\cF \coloneqq \cF_1 \otimes\cF_2$,
+    $\bP \coloneqq  \bP_0 \otimes \bP_1$.
+    Let $f \ge 0$ be $(\Omega, \cF)$-measurable,
+    then
+    \[
+        \Omega_0 \ni x \mapsto \int_{\Omega_{2}} f(x,y) \bP_2(\dif y)
+    \]
+    and
+    \[
+    \Omega_1 \ni y \mapsto \int_{\Omega_1} f(x,y) \bP_1(\dif x)
+    \]
+    are measurable, and
+    \[
+    \int f \dif \bP = \int_{\Omega_1} \int_{\Omega_2} f(x,y) \bP_2(\dif y) \bP_1(\dif x)(\dif x)
+    = \int_{\Omega_2} \int_{\Omega_1} f(x,y) \bP_1(\dif x) \bP_2(\dif y).
+    \]
+\end{theorem}
 
 \subsection{Inequalities}
 This is taken from section 6.1 of the notes on Stochastik.