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 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.If part: Let φ ∈ C ∞c (Ω) be a test function. It is bounded by its supremum norm ||φ||∞, measurable, and has a compact support, let's call it K. Hence 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 1. 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 { extstyle left.f ight|_{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. Definition 2. Let Ω be an open set in the Euclidean space ℝn. Then a function f : Ω → ℂ such that for each test function φ ∈ C ∞c (Ω) is called locally integrable, and the set of such functions is denoted by L1,loc(Ω). Here C ∞c (Ω) denotes the set of all infinitely differentiable functions φ : Ω → ℝ with compact support contained in Ω. This definition has its roots in the approach to measure and integration theory based on the concept of continuous linear functional on a topological vector space, developed by Nicolas Bourbaki and his school: it is also the one adopted by Strichartz (2003) and by Maz'ya & Shaposhnikova (2009, p. 34). This 'distribution theoretic' definition is equivalent to the standard one, as the following lemma proves: Lemma 1. A given function f : Ω → ℂ is locally integrable according to Definition 1 if and only if it is locally integrable according to Definition 2, i.e.