Support (mathematics) - Support of A Distribution

Support of A Distribution

It is possible also to talk about the support of a distribution, such as the Dirac delta function δ(x) on the real line. In that example, we can consider test functions F, which are smooth functions with support not including the point 0. Since δ(F) (the distribution δ applied as linear functional to F) is 0 for such functions, we can say that the support of δ is {0} only. Since measures (including probability measures) on the real line are special cases of distributions, we can also speak of the support of a measure in the same way.

Suppose that f is a distribution, and that U is an open set in Euclidean space such that, for all test functions such that the support of is contained in U, . Then f is said to vanish on U. Now, if f vanishes on an arbitrary family of open sets, then for any test function supported in, a simple argument based on the compactness of the support of and a partition of unity shows that as well. Hence we can define the support of f as the complement of the largest open set on which f vanishes. For example, the support of the Dirac delta is .