construct a test function which majorises the indicator function of . The usual set distance between and the boundary is strictly greater than zero, i.e. hence it is possible to choose a real number such that (if is the empty set, take ). Let and denote the closed -neighborhood and -neighborhood of , respectively. They are likewise compact and satisfy Now use convolution to define the function by where is a mollifier constructed by using the standard positive symmetric one. Obviously is non-negative in the sense that , infinitely differentiable, and its support is contained in , in particular it is a test function. Since for all , we have that . Let be a locally integrable function according to . Then Since this holds for every compact subset of , the function is locally integrable according to . □ Generalization: locally p-integrable functions . Let be an open set in the Euclidean space and be a Lebesgue measurable function. If, for a given with , satisfies i.e., it belongs to for all compact subsets of , then is called locally -integrable or also -locally integrable. The set of all such functions is denoted by : An alternative definition, completely analogous to the one given for locally integrable functions, can also be given for locally -integrable functions: it can also be and proven equivalent to the one in this section. Despite their apparent higher generality, locally -integrable functions form a subset of locally integrable functions for every such that . Notation Apart from the different glyphs which may be used for the uppercase "L", there are few variants for the notation of the set of locally integrable functions adopted by , and . adopted by and . adopted by and . Properties Lp,loc is a complete metric space for all p ≥ 1 . is a complete metrizable space: its topology can be generated by the following metric: where is a family of non empty open sets such that , meaning that is compactly included in i.e. it is a set having compact closure strictly included in the set of higher index. . , k ∈ is an indexed family of seminorms, defined as In references , , and , this theorem is stated but not proved on a formal basis: a complete proof of a more general result, which includes it, is found in . Lp is a subspace