updated definition of convolution

inputs/lecture_09.tex

@@ -86,17 +86,23 @@ characteristic functions of Fourier transforms.
 \subsection{Convolutions${}^\dagger$}
 
 \begin{definition}+[Convolution]
-   Let $\mu$ and $\nu$ be probability measures on $\R^d$
-   with Lebesgue densities $f_\mu$ and $f_\nu$.
+   Let $\mu$ and $\nu$ be probability measures on $\R^d$.
    Then the \vocab{convolution} of $\mu$ and $\nu$,
    $\mu \ast \nu$,
    is the probability measure on $\R^d$
-   with Lebesgue density
+   defined by
    \[
-   f_{\mu \ast \nu}(x) \coloneqq
-       \int_{\R^d} f_\mu(x - t) f_\nu(t) \lambda^d(\dif t).
+       (\mu \ast \nu)(A) = \int_{\R^d} \int_{\R^d} \One_A(x + y) \mu(\dif x) \nu(\dif y).
    \]
 \end{definition}
+\begin{fact}
+    If $\mu$ and $\nu$ have Lebesgue densities $f_\mu$ and $f_\nu$,
+    then the convolution has Lebesgue density
+    \[
+        f_{\mu \ast \nu}(x) \coloneqq
+            \int_{\R^d} f_\mu(x - t) f_\nu(t) \lambda^d(\dif t).
+   \]
+\end{fact}
 
 \begin{fact}+[Exercise 6.1]
     If $X_1,X_2,\ldots$ are independent with