Added MCT for convergence in measure.

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Bokuan Li
2026-06-19 20:50:42 -04:00
parent 8742c4f7cd
commit 2d75e7c029
2 changed files with 23 additions and 5 deletions

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@@ -87,7 +87,7 @@
(3): By (2), there exists a subsequence $\seq{n_k}$ such that $f_{n_k} \to f$ and $f_{n_k} \to g$ almost everywhere, so $f = g$ almost everywhere.
\end{proof}
\begin{theorem}[Monotone Convergence Theorem (In Measure)]
\begin{theorem}[Monotone Convergence Theorem (in Measure)]
\label{theorem:mct-measure}
Let $(X, \cm, \mu)$ be a semifinite measure space, $\net{f} \subset \mathcal{L}^+(X, \cm)$, and $f \in \mathcal{L}^+(X, \cm)$ such that
\begin{enumerate}[label=(\alph*)]
@@ -117,4 +117,5 @@
\end{align*}
As the above holds for all $\eps > 0$, $\lambda \in (0, 1)$, $\sup_{\alpha \in A}\int f_\alpha d\mu \ge \int \phi d\mu$. Therefore $\sup_{\alpha \in A}\int f_\alpha d\mu \ge \int f d\mu$ by \autoref{lemma:lebesgue-non-negative-strict}.
\end{proof}
\end{proof}