Added Fubini's theorem.

src/measure/bochner-integral/bochner.tex

@@ -78,7 +78,7 @@
         \item For any $f \in L^1(X, |\mu|; E)$, $\normn{I_\lambda f}_{G} \le \norm{\lambda}_{L^2(E, F; G)} \cdot \norm{f}_{L^1(X, |\mu|; E)}$.
     \end{enumerate}
     
-    For any $f \in L^1(X; E)$, $I_\lambda f = \int f d\lambda\mu$ is the \textbf{Bochner integral} of $f$ with respect to $\mu$ and $\lambda$. 
+    For any $f \in L^1(X; E)$, $I_\lambda f = \int \lambda(f, d\mu)$ is the \textbf{Bochner integral} of $f$ with respect to $\mu$ and $\lambda$. 
 \end{definition}
 \begin{proof}
     Same as \autoref{definition:bochner-integral}.
@@ -98,10 +98,10 @@
         \item[(b)] There exists $g \in L^1(X) \cap L^+(X)$ such that $\norm{f_n}_E \le g$ for all $n \in \natp$. 
     \end{enumerate}
     
-    then $\int f d\lambda\mu = \limv{n}\int f_n d\lambda\mu$. 
+    then $\int \lambda(f , d\mu) = \limv{n}\int \lambda(f_n, d\mu)$. 
 \end{theorem}
 \begin{proof}
-    By the classical \hyperref[Dominated Convergence Theorem]{proposition:dct-lp}, $f_n \to f$ in $L^1(X, |\mu|; E)$. Since $h \mapsto \int h d\lambda\mu$ is a bounded linear operator, $\int f d\lambda\mu = \limv{n}\int f_n d\lambda\mu$. 
+    By the classical \hyperref[Dominated Convergence Theorem]{proposition:dct-lp}, $f_n \to f$ in $L^1(X, |\mu|; E)$. Since $h \mapsto \int \lambda(f , d\mu)$ is a bounded linear operator, $\int \lambda(f , d\mu) = \limv{n}\int \lambda(f_n, d\mu)$. 
 \end{proof}