diff --git a/inputs/lecture_10.tex b/inputs/lecture_10.tex
index d032207..3561637 100644
--- a/inputs/lecture_10.tex
+++ b/inputs/lecture_10.tex
@@ -123,7 +123,7 @@ However, Fourier analysis is not only useful for continuous probability density
     Let $\bP \in M_1(\lambda)$.
     Then
     \[
-    \forall x \in  \R ~ \bP\left( \{x\}  \right) = \lim_{T \to \infty} \frac{1}{2 T} \int_{-T}^T e^{-\i t x } \phi(t) \dif t.
+    \forall x \in  \R .~ \bP\left( \{x\}  \right) = \lim_{T \to \infty} \frac{1}{2 T} \int_{-T}^T e^{-\i t x } \phi(t) \dif t.
     \]
 
 \end{theorem}
@@ -175,18 +175,18 @@ However, Fourier analysis is not only useful for continuous probability density
                                                        &=& \int_{\R} \left| \sum_{l}  c_l e^{\i t_l x}\right|^2 \ge  0
     \end{IEEEeqnarray*}
 \end{refproof}
-\begin{theorem}[Bochner's theorem]\label{bochnersthm}
-    The converse to \autoref{thm:lec_10thm5} holds, i.e.~ any
+\begin{theorem}[Bochner's theorem]\label{thm:bochner}
+    The converse to \autoref{thm:lec_10thm5} holds, i.e.~any
     $\phi: \R \to  \C$ satisfying (a) and (b) of \autoref{thm:lec_10thm5}
     must be the Fourier transform of a probability measure $\bP$
     on $(\R, \cB(\R))$.
 \end{theorem}
-Unfortunately, we won't prove \autoref{bochnersthm} in this lecture.
+Unfortunately, we won't prove \autoref{thm:bochner} in this lecture.
 
 
 \begin{definition}[Convergence in distribution / weak convergence]
     \label{def:weakconvergence}
-    We say that $\bP_n \subseteq  M_1(\R)$ \vocab[Convergence!weak]{converges weakly} towards $\bP \in M_1(\R)$ (notation: $\bP_n \implies \bP$), iff
+    We say that $\bP_n \in M_1(\R)$ \vocab[Convergence!weak]{converges weakly} towards $\bP \in M_1(\R)$ (notation: $\bP_n \implies \bP$), iff
     \[
     \forall f \in C_b(\R)~  \int f \dif\bP_n \to \int f \dif\bP.
     \]
@@ -245,6 +245,8 @@ Unfortunately, we won't prove \autoref{bochnersthm} in this lecture.
      $\bP_n \implies \bP$, where $\bP_n$ is the distribution of $X_n$
      and  $\bP$ is the distribution of $X$.
 \end{definition}
+It is easy to see, that this is equivalent to $\bE[f(X_n)] \to \bE[f(X)]$
+for all $f \in  C_b(\R)$.
 \begin{example}
     Let $X_n \coloneqq  \frac{1}{n}$
     and $F_n$ the distribution function, i.e.~$F_n = \One_{[\frac{1}{n},\infty)}$.
diff --git a/inputs/lecture_23.tex b/inputs/lecture_23.tex
index df324e6..41b3763 100644
--- a/inputs/lecture_23.tex
+++ b/inputs/lecture_23.tex
@@ -67,8 +67,8 @@ In this lecture we recall the most important point from the lecture.
         inversion formula (\autoref{inversionformula}), ...
     \item Levy's continuity theorem (\autoref{levycontinuity}),
         (\autoref{genlevycontinuity})
-    \item Bockner's theorem for positive definite function % TODO REF
-    \item Bockner's theorem for the mass at a point (\autoref{bochnersformula})
+    \item Bochner's theorem for positive definite function (\autoref{thm:bochner})
+    \item Bochner's theorem for the mass at a point (\autoref{bochnersformula})
     \item Related notions
         \todo{TODO}
         \begin{itemize}