Locally integrable function
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 infinity: in other words, locally integrable functions can grow arbitrarily fast at infinity, but are still manageable in a way similar to ordinary integrable functions.
. Let Ω be an open set in the Euclidean space ℝn and f : Ω → ℂ be a Lebesgue measurable function. If f on Ω is such that
i.e. its Lebesgue integral is finite on all compact subsets K of Ω, then f is called locally integrable. The set of all such functions is denoted by L1,loc(Ω):
where f |K denotes the restriction of f to the set K.
The classical definition of a locally integrable function involves only measure theoretic and topological concepts and can be carried over abstract to complex-valued functions on a topological measure space (X, Σ, μ): however, since the most common application of such functions is to distribution theory on Euclidean spaces, all the definitions in this and the following sections deal explicitly only with this important case.
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