Remark 26.3.2 (Dominated Convergence Theorem for Nets?).label 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 to nets. This limitation arises from the monotone convergence theorem, 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 Unknown (theorem:mct-measure), and the Dominated Convergence Theorem is given at Corollary 15.4.4 following the Vitali Convergence Theorem.