Editing Support (mathematics) (section)

==Essential support==

If <math>X</math> is a topological [[measure space]] with a [[Borel measure]] <math>\mu</math> (such as <math>\R^n,</math> or a [[Lebesgue measure|Lebesgue measurable]] subset of <math>\R^n,</math> equipped with Lebesgue measure), then one typically identifies functions that are equal <math>\mu</math>-almost everywhere. In that case, the '''{{em|{{visible anchor|essential support}}}}''' of a measurable function <math>f : X \to \R</math> written '''<math>\operatorname{ess\,supp}(f),</math>''' is defined to be the smallest closed subset <math>F</math> of <math>X</math> such that <math>f = 0</math> <math>\mu</math>-almost everywhere outside <math>F.</math> Equivalently, <math>\operatorname{ess\,supp}(f)</math> is the complement of the largest [[open set]] on which <math>f = 0</math> <math>\mu</math>-almost everywhere<ref name=lieb>{{cite book|last1=Lieb|first1=Elliott|author-link1=Elliott H. Lieb|last2=Loss|first2=Michael|author2-link=Michael Loss|title=Analysis|year=2001|edition=2nd|publisher=[[American Mathematical Society]]|series=Graduate Studies in Mathematics|volume=14|isbn=978-0821827833|page=13}}</ref>
<math display="block">\operatorname{ess\,supp}(f) := X \setminus \bigcup \left\{\Omega \subseteq X : \Omega\text{ is open and } f = 0\, \mu\text{-almost everywhere in } \Omega \right\}.</math>

The essential support of a function <math>f</math> depends on the [[Measure (mathematics)|measure]] <math>\mu</math> as well as on <math>f,</math> and it may be strictly smaller than the closed support. For example, if <math>f : [0, 1] \to \R</math> is the [[Dirichlet function]] that is <math>0</math> on irrational numbers and <math>1</math> on rational numbers, and <math>[0, 1]</math> is equipped with Lebesgue measure, then the support of <math>f</math> is the entire interval <math>[0, 1],</math> but the essential support of <math>f</math> is empty, since <math>f</math> is equal almost everywhere to the zero function.

In analysis one nearly always wants to use the essential support of a function, rather than its closed support, when the two sets are different, so <math>\operatorname{ess\,supp}(f)</math> is often written simply as <math>\operatorname{supp}(f)</math> and referred to as the support.<ref name = lieb /><ref>In a similar way, one uses the [[Essential supremum and essential infimum|essential supremum]] of a measurable function instead of its supremum.</ref>