Locally Integrable Function
Definition
Let $\Omega \subset \mathbb{R}^{n}$ be called an open set.
Definition 11
For every bounded measurable set $K \subset \Omega$,
$$ \int_{K} \left| u(x) \right| dx \lt \infty $$
a function $u : \Omega \to \mathbb{C}$ satisfying this is said to be locally integrable with respect to (the Lebesgue measure).
Definition 22
Let the function $u$ be defined almost everywhere on $\Omega$. For every open set $U \Subset \Omega$ when $u \in L^{1}(U)$, then $u$ is said to be locally integrable on $\Omega$.
Notation
The set of locally integrable functions is denoted as follows.
$$ L_{\text{loc}}^{1}(\Omega) := \left\{ u : \Omega \to \mathbb{C} \Big| u \text{ is locally integrable.}\right\} $$
Explanation
By the definition, the following inclusion relationships are trivially established.
$$ \href{../592}{L^{1}(\Omega)} \subset L_{\text{loc}}^{1}(\Omega) $$
$$ \href{../1594}{C(\Omega)} \subset L_{\text{loc}}^{1}(\Omega) $$