Added the scaffold.

src/measure/measure/semifinite.tex

@@ -59,3 +59,30 @@
     \]
 
 \end{proof}
+
+
+\begin{definition}[Scaffold*]
+\label{definition:measure-scaffold}
+    Let $(X, \cm, \mu)$ be a measure space and $\cf \subset \bracs{A \in \cm|\mu(A) < \infty}$, then $\cf$ is a \textbf{scaffold} for $\mu$ if for all $E \in \cm$, 
+    \[
+    \mu(E) = \sup\bracs{\mu(E \cap A)|A \in \cf}
+    \]
+
+    and the quadruple $(X, \cm, \cf, \mu)$ is a \textbf{scaffolded measure space}. 
+\end{definition}
+
+\begin{lemma}
+\label{lemma:measure-scaffold-semifinite}
+    Let $(X, \cm, \mu)$ be a measure space:
+    \begin{enumerate}
+        \item If $\cf \subset \cm$ is a \hyperref[scaffold]{definition:measure-scaffold} for $\mu$, then $\mu$ is semifinite.
+        \item If $\mu$ is semifinite, then $\bracs{A \in \cm|\mu(A) < \infty}$ is a scaffold for $\mu$. 
+    \end{enumerate}
+\end{lemma}
+% Omitted
+
+\begin{example}
+    Let $X$ be a LCH space, $\mu$ be a Radon measure, and $\mathcal{K}$ be the collection of compact subsets of $X$, then $\mathcal{K}$ is a scaffold for $\mu$. 
+\end{example}
+% Omitted
+