Definition:Integrable Function/Locally Integrable Function

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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 $\R^d$.


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Then $f$ is called the locally integrable function.


This article, or a section of it, needs explaining.
In particular: Why "the"? Is it a unique concept? And what is it "the" locally integrable function of?
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Also see

  • Results about locally integrable functions can be found here.


Sources

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