Random nonsense.

src/measure/weak-integral/definition.tex

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+\section{Weak Integrals*}
+\label{section:weak-integral}
+
+\begin{definition}[Weakly Measurable]
+\label{definition:weakly-measurable}
+    Let $(X, \cm)$ be a measurable space, $E$ be a locally convex space over $K \in \RC$, and $f: X \to E$, then $f$ is \textbf{weakly measurable} if for each $\phi \in E^*$, $\phi \circ f: X \to K$ is Borel measurable. 
+\end{definition}
+
+As I know so little about weak integrals, I will Dunning-Kruger myself right now, give an opinion, and laugh about it later. My gripe with seeing the definition comes from the need to test against \textit{every} continuous linear functional. To me, this seems quite inflexible: consider integrating a distribution-valued function. \textit{Surely} it is wiser to only test this function against test functions rather than \textit{the dual of $\mathcal{D}'$ (dual with respect to $\mathcal{D}'$ with the bounded convergence topology)}. As such, it may be more productive to consider a more flexible form of testing, such as using duality. 
+
+
+\begin{definition}[Weakly Integrable*]
+\label{definition:weakly-integrable}
+    Let $(X, \cm, \mu)$ be a measure space, $\dpn{E, F}{\lambda}$ be a duality over $K \in \RC$, and $f: X \to E$, then $f$ is \textbf{Dunford $\lambda$-integrable*} if:
+    \begin{enumerate}[label=(I\arabic*)]
+        \item $f$ is weakly measurable. 
+        \item For each $\phi \in F$, $\phi \circ f \in L^1(X; K)$.
+        \item For each $A \in \cm$, the mapping
+        \[
+        F \to K \quad \phi \mapsto \int_A \dpn{f(x), \phi}{\lambda} d\mu 
+        \]
+
+        is a continuous linear functional on $F$. 
+    \end{enumerate}
+
+    For each $A \in \cm$, the element $\phi \mapsto \int_A \dpn{f(x), \phi}{\lambda} d\mu$ of $F^*$ is the \textbf{Dunford $\lambda$-integral} of $f$ over $A$, denoted $\int_A^* f d\mu$. 
+
+    The function $f$ is \textbf{Pettis $\lambda$-integrable*} if it satisfies (I1), (I2), and
+    \begin{enumerate}
+        \item[(I3+)] For each $A \in \cm$, the mapping
+        \[
+        F \to K \quad \phi \mapsto \int_X \dpn{f(x), \phi}{\lambda} d\mu 
+        \]
+
+        is a $\sigma(F, E)$-continuous linear functional on $F$. 
+    \end{enumerate}
+
+    In which case, for each $A \in \cm$, $\int_A^* f d\mu$ is the \textbf{Pettis $\lambda$-integral} of $f$ over $A$. 
+\end{definition}
+
+It is at this point that I start to understand why the bidual setup is useful: existence. 
+
+
+

src/measure/weak-integral/index.tex

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+\chapter{Weak Integrals*}
+\label{chap:weak-integral}
+
+\input{./definition.tex}
+