Definition:Integrable Function/Locally Integrable Function

Definition

Let $f : \R^d \to \C$ be a function.

Let $K \subseteq \R^d$ be a compact subset.

Suppose:

$\ds \forall K \subseteq \R^d : \int_K \size {\map f {\mathbf x} } \rd {\mathbf x} < \infty$

where $\mathbf x \in \R^d$ and $\rd {\mathbf x}$ and $\rd {\mathbf x} = \rd x_1 \rd x_2 \ldots \rd x_d$ is the volume element in $\R^d$.

Then $f$ is called the locally integrable function.

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Also see

• Results about locally integrable functions can be found here.

Sources

• 2017: Amol Sasane: A Friendly Approach to Functional Analysis ... (previous) ... (next): Chapter $\S 6.1$: A glimpse of distribution theory. Test functions, distributions, and examples