Updated remark for dominated convergence.

src/measure/lebesgue-integral/complex.tex

@@ -122,9 +122,11 @@
     and $\int f d\mu = \limv{n}\int f_n d\mu$.
 \end{proof}
 
-\begin{remark}[There is no dominated convergence theorem for nets]
+\begin{remark}[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 \hyperref[dominated convergence theorem]{theorem:dct} to nets. This limitation arises from the \hyperref[monotone convergence theorem]{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 $\alpha \in A$.
+
+    At least, that is how I thought back in 2025. While arbitrary pointwise convergence has no reason to cooperate with the structure of a measure space, additionally supplying convergence in measure bridges the above gap. The corresponding Monotone Convergence Theorem is given at \autoref{theorem:mct-measure}, and the Dominated Convergence Theorem is given at \autoref{corollary:dct-filter} following the Vitali Convergence Theorem. 
 \end{remark}