From 6461ef4ea787ca7fa9e7927f1d282e5d22d7578d Mon Sep 17 00:00:00 2001 From: Bokuan Li Date: Mon, 26 Jan 2026 17:58:42 -0500 Subject: [PATCH] Bookkeeping. --- src/measure/bochner-integral/index.tex | 2 +- src/measure/lebesgue-integral/complex.tex | 7 +++++++ 2 files changed, 8 insertions(+), 1 deletion(-) diff --git a/src/measure/bochner-integral/index.tex b/src/measure/bochner-integral/index.tex index adc96a8..aba7179 100644 --- a/src/measure/bochner-integral/index.tex +++ b/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} diff --git a/src/measure/lebesgue-integral/complex.tex b/src/measure/lebesgue-integral/complex.tex index c526846..20ca147 100644 --- a/src/measure/lebesgue-integral/complex.tex +++ b/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}