diff --git a/inputs/lecture_09.tex b/inputs/lecture_09.tex
index ddac17e..99f4671 100644
--- a/inputs/lecture_09.tex
+++ b/inputs/lecture_09.tex
@@ -113,7 +113,7 @@ We have
     $X: (\Omega, \cF) \to  (\R, \cB(\R))$ is a random variable.
     Then we can define
     \[
-    \phi_X(t) \coloneqq  \bE[e^{\i t x}]
+    \phi_X(t) \coloneqq  \bE[e^{\i t X}]
       = \int e^{\i t X(\omega)} \bP(\dif \omega)
       = \int_{\R} e^{\i t x} \mu(dx) = \phi_\mu(t),
     \]
diff --git a/inputs/lecture_10.tex b/inputs/lecture_10.tex
index 80bedbe..1ffaadd 100644
--- a/inputs/lecture_10.tex
+++ b/inputs/lecture_10.tex
@@ -30,13 +30,13 @@ where $\mu = \bP X^{-1}$.
         &&\lim_{T \to  \infty} \frac{1}{2 \pi} \int_{-T}^T \int_{\R} \frac{e^{-\i t b}- e^{-\i t a}}{-\i t} e^{\i t x} \dif t \bP(\dif x)\\
         &\overset{\text{Fubini for  $L^1$}}{=}& \lim_{T \to  \infty} \frac{1}{2 \pi} \int_{\R} \int_{-T}^T \frac{e^{-\i t b}- e^{-\i t a}}{-\i t} e^{\i t x} \dif t \bP(\dif x)\\
         &=& \lim_{T \to  \infty} \frac{1}{2 \pi} \int_{\R} \int_{-T}^T \frac{e^{\i t (b-x)}- e^{\i t (x-a)}}{-\i t} \dif t \bP(\dif x)\\
-        &=& \lim_{T \to \infty} \frac{1}{2 \pi} \int_{\R} \underbrace{\int_{-T}^T \left[ \frac{\cos(t (x-b)) - \cos(t(x-a))}{-\i t}\right] \dif t \bP(\dif x)}_{=0 \text{, as the function is odd}}
-      \\&&
-        + \lim_{T \to \infty} \frac{1}{2\pi} \int_{\R}\int_{-T}^T \frac{\sin(t ( x - b)) - \sin(t(x-a))}{-t} \dif t \bP(\dif x)\\
+        &=& \lim_{T \to \infty} \frac{1}{2 \pi} \int_{\R} \underbrace{\int_{-T}^T \left[ \frac{\cos(t (x-b)) - \cos(t(x-a))}{-\i t}\right] \dif t}_{=0 \text{, as the function is odd}} \bP(\dif x) \\
+        &&  + \lim_{T \to \infty} \frac{1}{2\pi} \int_{\R}\int_{-T}^T \frac{\sin(t ( x - b)) - \sin(t(x-a))}{-t} \dif t \bP(\dif x)\\
         &=& \lim_{T \to \infty} \frac{1}{\pi} \int_\R \int_{0}^T \frac{\sin(t(x-a)) - \sin(t(x-b))}{t} \dif t \bP(\dif x)\\
-        &\overset{\substack{\text{\autoref{fact:intsinxx},}\\\text{dominated convergence}}}{=}& \frac{1}{\pi} \int -\frac{\pi}{2} \One_{x < a} + \frac{\pi}{2} \One_{x > a }
-                                                                - (- \frac{\pi}{2} \One_{x < b} + \frac{\pi}{2} \One_{x > b}) \bP(\dif x)\\
-        &=&  \frac{1}{2} \bP(\{a\} ) + \frac{1}{2} \bP(\{b\}) + \bP((a,b))\\
+        &\overset{\substack{\text{\autoref{fact:intsinxx},}\\\text{dominated convergence}}}{=}&
+          \frac{1}{\pi} \int -\frac{\pi}{2} \One_{x < a} + \frac{\pi}{2} \One_{x > a}
+          - (- \frac{\pi}{2} \One_{x < b} + \frac{\pi}{2} \One_{x > b}) \bP(\dif x)\\
+        &=& \frac{1}{2} \bP(\{a\} ) + \frac{1}{2} \bP(\{b\}) + \bP((a,b))\\
         &=& \frac{F(b) + F(b-)}{2} - \frac{F(a) - F(a-)}{2}
     \end{IEEEeqnarray*}
 \end{refproof}