Editing Support (mathematics) (section)

==Support of a distribution{{anchor|Support (statistics)}}==

It is possible also to talk about the support of a [[Distribution (mathematics)|distribution]], such as the [[Dirac delta function]] <math>\delta(x)</math> on the real line. In that example, we can consider test functions <math>F,</math> which are [[smooth function]]s with support not including the point <math>0.</math> Since <math>\delta(F)</math> (the distribution <math>\delta</math> applied as [[linear functional]] to <math>F</math>) is <math>0</math> for such functions, we can say that the support of <math>\delta</math> is <math>\{ 0 \}</math> only. Since measures (including [[probability measure]]s) 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 <math>f</math> is a distribution, and that <math>U</math> is an open set in Euclidean space such that, for all test functions <math>\phi</math> such that the support of <math>\phi</math> is contained in <math>U,</math> <math>f(\phi) = 0.</math> Then <math>f</math> is said to vanish on <math>U.</math> Now, if <math>f</math> vanishes on an arbitrary family <math>U_{\alpha}</math> of open sets, then for any test function <math>\phi</math> supported in <math display="inline">\bigcup U_{\alpha},</math> a simple argument based on the compactness of the support of <math>\phi</math> and a partition of unity shows that <math>f(\phi) = 0</math> as well. Hence we can define the {{em|support}} of <math>f</math> as the complement of the largest open set on which <math>f</math> vanishes. For example, the support of the Dirac delta is <math>\{ 0 \}.</math>