\section{Locally Integrable Functions}
\label{section:locally-integrable}

\begin{definition}[Locally Integrable*]
\label{definition:locally-integrable}
    Let $(X, \cm, \cf, \mu)$ be a \hyperref[scaffolded]{definition:measure-scaffold} measure space, $E$ be a normed vector space over $K \in \RC$, $f: X \to E$ be strongly measurable, and $p \in [1, \infty]$, then $f$ is \textbf{locally $p$-integrable} if for every $A \in \cf$, $\int_A \norm{f}_A^p d\mu < \infty$. 

    The set $\mathcal{L}^p_\cf(X, \cm, \mu; E) = \mathcal{L}^p_\cf(X; E) = \mathcal{L}^p_\cf(\mu; E)$ is the space of all locally $p$-integrable $E$-valued functions on $X$. 
\end{definition}

\begin{definition}[Locally Bounded*]
\label{definition:locally-bounded-measure}
    Let $(X, \cm, \cf, \mu)$ be a \hyperref[scaffolded]{definition:measure-scaffold} measure space, $E$ be a normed vectorr space over $K \in \RC$, and $f: X \to E$ be strongly measurable, then $f$ is \textbf{essentially bounded} if for every $A \in \cf$, $\norm{\one_A f}_{L^\infty(A; E)} < \infty$. 

    The set $\mathcal{L}^\infty_\cf(X, \cm, \mu; E) = \mathcal{L}^\infty_\cf(X; E) = \mathcal{L}^\infty_\cf(\mu; E)$ is the space of all locally bounded $E$-valued functions on $X$. 
\end{definition}

\begin{definition}[Local $L^p$ Space*]
\label{definition:local-lp-space}
    Let $(X, \cm, \cf, \mu)$ be a \hyperref[scaffolded]{definition:measure-scaffold} measure space, $E$ be a normed vector space over $K \in \RC$, and $p \in [1, \infty]$. For each $A \in \cf$ and $f \in \mathcal{L}^p_\cf(X, \cm, \mu; E)$, let $[f]_{L^p_A(X; \mu)} = \norm{\one_A f}_{L^p(X; \mu)}$, then the $[\cdot]_{L^p_A(X; \mu)}$ is a seminorm on \hyperref[$\mathcal{L}^p_\cf(X; E)$]{definition:locally-integrable}. The set
    \[
    L^p_\cf(X, \cm, \mu; E) = \mathcal{L}^p_\cf(X, \cm, \mu; E)/\bracs{f|f = 0\text{ a.e.}}
    \]

    equipped with the seminorms $\bracsn{[\cdot]_{L^p_A(X;\mu)}|A \in \cf}$ is a separated locally convex space, and the \textbf{local $L^p$ space} of $(X, \cm, \mu)$. 
\end{definition}

\begin{lemma}
\label{lemma:gluing-local-lp}
    Let $(X, \cm, \cf, \mu)$ be a \hyperref[scaffolded]{definition:measure-scaffold} localisable measure space, $E$ be a normed vector space over $K \in \RC$, $p \in [1, \infty]$, and $\bracsn{f_A}_{A \in \cf}$ such that:
    \begin{enumerate}[label=(\alph*)]
        \item For each $A \in \cf$, $f_A \in \mathcal{L}^p(A; E)$. 
        \item For each $A, B \in \cf$, $f_A|_{A \cap B} = f_B|_{A \cap B}$ almost everywhere.
        \item $\bigcup_{A \in \cf}f_A(A)$ is a separable subset of $E$. 
    \end{enumerate}

    then there exists a unique $f \in L^p_\cf(X; E)$ such that $f|_A = f_A$ for all $A \in \cf$. 
\end{lemma}
\begin{proof}
    By the \hyperref[gluing lemma for measurable functions]{lemma:gluing-measurable}.
\end{proof}