Bookkeeping.

src/measure/bochner-integral/index.tex

@@ -1,4 +1,4 @@
-\chapter{Bochner Integral}
+\chapter{The Bochner Integral}
 \label{chap:bochner-integral}
 
 \input{./src/measure/bochner-integral/strongly.tex}

src/measure/lebesgue-integral/complex.tex

@@ -124,3 +124,10 @@
     \]
     and $\int f d\mu = \limv{n}\int f_n d\mu$.
 \end{proof}
+
+\begin{remark}[There is no dominated convergence theorem for nets]
+\label{remark:dct-no-net}
+    In analysis, one frequently encounters places where only sequential continuity is provided or required. It is my opinion that a good portion of this comes from the lack of an extension of the dominated convergence theorem (\ref{theorem:dct}) to nets. This limitation arises from the monotone convergence theorem (\ref{theorem:mct}), where continuity from below is used.
+
+    For an example, consider the Lebesgue measure on $[0, 1]$. Let $A$ be the net of all finite subsets of $[0, 1]$, directed by inclusion, then $\lim_{\alpha \in A}\one_\alpha = 1$ pointwise. However, $\int \one_\alpha = 0$ for all $\alpha \in A$.
+\end{remark}