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Concept
Locally integrable function
Summary
In mathematics, a locally integrable function (sometimes also called locally summable function) is a function which is integrable (so its integral is finite) on every compact subset of its domain of definition. The importance of such functions lies in the fact that their function space is similar to Lp spaces, but its members are not required to satisfy any growth restriction on their behavior at the boundary of their domain (at infinity if the domain is unbounded): in other words, locally integrable functions can grow arbitrarily fast at the domain boundary, but are still manageable in a way similar to ordinary integrable functions.
Definition
Standard definition
. Let Ω be an open set in the Euclidean space \mathbb{R}^n and f : Ω → \mathbb{C} be a Lebesgue measurable function. If f on Ω is such that
: \int_K | f |, \mathrm{d}x
Official source
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