15.6 Locally Integrable Functions

Definition 15.6.1 (Locally Integrable*).label Let $(X, \cm, \cf, \mu)$ be a scaffolded 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 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 $X$.

Definition 15.6.2 (Locally Bounded*).label Let $(X, \cm, \cf, \mu)$ be a scaffolded measure space, $E$ be a normed vectorr space over $K \in \RC$, and $f: X \to E$ be strongly measurable, then $f$ is 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 $X$.

Definition 15.6.3 (Local $L^{p}$ Space*).label Let $(X, \cm, \cf, \mu)$ be a scaffolded 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 $\mathcal{L}^{p}_{\cf}(X; E)$. The set

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

Lemma 15.6.4.label Let $(X, \cm, \cf, \mu)$ be a scaffolded 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:

1. (a)

   For each $A \in \cf$, $f_{A} \in \mathcal{L}^{p}(A; E)$.

2. (b)

   For each $A, B \in \cf$, $f_{A}|_{A \cap B}= f_{B}|_{A \cap B}$ almost everywhere.

3. (c)

   $\bigcup_{A \in \cf}f_{A}(A)$ is a separable subset of $E$.

then there exists a unique $f \in L^{p}_{\cf}(X; E)$ such that $f|_{A} = f_{A}$ for all $A \in \cf$.