📂Lebesgue Spaces

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.

$$ \href{../592}{L^{1}(\Omega)} \subset L_{\text{loc}}^{1}(\Omega) $$

$$ \href{../1594}{C(\Omega)} \subset L_{\text{loc}}^{1}(\Omega) $$

Properties

• Locally integrable functions can be extended to distributions.