Editing Support (mathematics)

{{Short description|Inputs for which a function's value is non-zero}}
{{For|other uses in mathematics|Support (disambiguation)#Mathematics{{!}}Support § Mathematics}}
{{Refimprove|date=November 2009}}
In [[mathematics]], the '''support''' of a [[Real number|real-valued]] [[Function (mathematics)|function]] <math>f</math> is the [[subset]] of the function [[Domain of a function|domain]] of elements that are not mapped to zero. If the domain of <math>f</math> is a [[topological space]], then the support of <math>f</math> is instead defined as the smallest [[closed set]] containing all points not mapped to zero. This concept is used widely in [[mathematical analysis]].

==Formulation==

Suppose that <math>f : X \to \R</math> is a real-valued function whose [[Domain of a function|domain]] is an arbitrary set <math>X.</math> The '''{{em|{{visible anchor|set-theoretic support}}}}''' of <math>f,</math> written <math>\operatorname{supp}(f),</math> is the set of points in <math>X</math> where <math>f</math> is non-zero:
<math display="block">\operatorname{supp}(f) = \{ x \in X \,:\, f(x) \neq 0\}.</math>

The support of <math>f</math> is the smallest subset of <math>X</math> with the property that <math>f</math> is zero on the subset's complement. If <math>f(x) = 0</math> for all but a finite number of points <math>x \in X,</math> then <math>f</math> is said to have '''{{em|{{visible anchor|finite support}}}}'''.

If the set <math>X</math> has an additional structure (for example, a [[topology]]), then the support of <math>f</math> is defined in an analogous way as the smallest subset of <math>X</math> of an appropriate type such that <math>f</math> vanishes in an appropriate sense on its complement. The notion of support also extends in a natural way to functions taking values in more general sets than <math>\R</math> and to other objects, such as [[Measure (mathematics)|measures]] or [[Distribution (mathematics)|distributions]].

==Closed support==

The most common situation occurs when <math>X</math> is a [[topological space]] (such as the [[real line]] or <math>n</math>-dimensional [[Euclidean space]]) and <math>f : X \to \R</math> is a [[Continuous function|continuous]] real- (or [[Complex number|complex]]-) valued function. In this case, the '''{{em|{{visible anchor|support}}}} of <math>f</math>''', <math>\operatorname{supp}(f)</math>, or the '''{{em|{{visible anchor|closed support}}}}''' '''of''' '''<math>f</math>''', is defined topologically as the [[Closure (topology)|closure]] (taken in <math>X</math>) of the subset of <math>X</math> where <math>f</math> is non-zero<ref name='folland'>{{cite book|last=Folland|first=Gerald B.|year=1999|title=Real Analysis, 2nd ed.|page=132|location=New York|publisher=John Wiley}}</ref><ref name='hormander'>{{cite book|last=Hörmander|first=Lars|year=1990|title=Linear Partial Differential Equations I, 2nd ed.|page=14|location=Berlin|publisher=Springer-Verlag}}</ref><ref name=Pasc>{{cite book|last=Pascucci|first=Andrea|year=2011|title=PDE and Martingale Methods in Option Pricing|page=678|isbn=978-88-470-1780-1|doi=10.1007/978-88-470-1781-8|location=Berlin|publisher=Springer-Verlag|series=Bocconi & Springer Series}}</ref> that is,
<math display="block">\operatorname{supp}(f) := \operatorname{cl}_X\left(\{x \in X \,:\, f(x) \neq 0 \}\right) = \overline{f^{-1}\left(\{ 0 \}^{\mathrm{c}}\right)}.</math>Since the intersection of closed sets is closed, <math>\operatorname{supp}(f)</math> is the intersection of all closed sets that contain the set-theoretic support of <math>f.</math> Note that if the function <math>f: \mathbb{R}^n \supseteq X \to \mathbb{R}</math> is defined on an open subset <math>X \subseteq \mathbb{R}^n</math>, then the closure is still taken with respect to <math>X</math> and not with respect to the ambient <math>\mathbb{R}^n</math>.

For example, if <math>f : \R \to \R</math> is the function defined by
<math display="block">f(x) = \begin{cases} 1 - x^2 & \text{if } |x| < 1 \\ 0 & \text{if } |x| \geq 1 \end{cases}</math>
then <math>\operatorname{supp}(f)</math>, the support of <math>f</math>, or the closed support of <math>f</math>, is the closed interval <math>[-1, 1],</math> since <math>f</math> is non-zero on the open interval <math>(-1, 1)</math> and the [[Closure (topology)|closure]] of this set is <math>[-1, 1].</math>

The notion of closed support is usually applied to continuous functions, but the definition makes sense for arbitrary real or complex-valued functions on a topological space, and some authors do not require that <math>f : X \to \R</math> (or <math>f : X \to \Complex</math>) be continuous.<ref>{{cite book|last=Rudin|first=Walter|year=1987|title=Real and Complex Analysis, 3rd ed.|page=38|location=New York|publisher=McGraw-Hill}}</ref>

==Compact support==

Functions with '''{{em|{{visible anchor|compact support}}}}''' on a topological space <math>X</math> are those whose closed support is a [[Compact space|compact]] subset of <math>X.</math> If <math>X</math> is the real line, or <math>n</math>-dimensional Euclidean space, then a function has compact support if and only if it has '''{{em|{{visible anchor|bounded support}}}}''', since a subset of <math>\R^n</math> is compact if and only if it is closed and bounded.

For example, the function <math>f : \R \to \R</math> defined above is a continuous function with compact support <math>[-1, 1].</math> If <math>f : \R^n \to \R</math> is a smooth function then because <math>f</math> is identically <math>0</math> on the open subset <math>\R^n \setminus \operatorname{supp}(f),</math> all of <math>f</math>'s partial derivatives of all orders are also identically <math>0</math> on <math>\R^n \setminus \operatorname{supp}(f).</math> 

The condition of compact support is stronger than the condition of [[Vanish at infinity|vanishing at infinity]]. For example, the function <math>f : \R \to \R</math> defined by
<math display="block">f(x) = \frac{1}{1+x^2}</math>
vanishes at infinity, since <math>f(x) \to 0</math> as <math>|x| \to \infty,</math> but its support <math>\R</math> is not compact.

Real-valued compactly supported [[smooth function]]s on a [[Euclidean space]] are called [[bump function]]s. [[Mollifier]]s are an important special case of bump functions as they can be used in [[Distribution (mathematics)|distribution theory]] to create [[sequence]]s of smooth functions approximating nonsmooth (generalized) functions, via [[convolution]].

In [[Well-behaved|good cases]], functions with compact support are [[Dense set|dense]] in the space of functions that vanish at infinity, but this property requires some technical work to justify in a given example. As an intuition for more complex examples, and in the language of [[Limit (mathematics)|limits]], for any <math>\varepsilon > 0,</math> any function <math>f</math> on the real line <math>\R</math> that vanishes at infinity can be approximated by choosing an appropriate compact subset <math>C</math> of <math>\R</math> such that
<math display="block">\left|f(x) - I_C(x) f(x)\right| < \varepsilon</math>
for all <math>x \in X,</math> where <math>I_C</math> is the [[indicator function]] of <math>C.</math> Every continuous function on a compact topological space has compact support since every closed subset of a compact space is indeed compact.

==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>

==Generalization==

If <math>M</math> is an arbitrary set containing zero, the concept of support is immediately generalizable to functions <math>f : X \to M.</math>  Support may also be defined for any [[algebraic structure]] with [[Identity element|identity]] (such as a [[Group (mathematics)|group]], [[monoid]], or [[composition algebra]]), in which the identity element assumes the role of zero. For instance, the family <math>\Z^{\N}</math> of functions from the [[natural numbers]] to the [[integers]] is the [[uncountable]] set of integer sequences.  The subfamily <math>\left\{ f \in \Z^{\N} : f \text{ has finite support } \right\}</math> is the countable set of all integer sequences that have only finitely many nonzero entries.

Functions of finite support are used in defining algebraic structures such as [[Group ring|group rings]] and [[Free abelian group|free abelian groups]].<ref>{{Cite book|title=Computational homology|last=Tomasz|first=Kaczynski|date=2004|publisher=Springer|others=Mischaikow, Konstantin Michael,, Mrozek, Marian|isbn=9780387215976|location=New York|pages=445|oclc=55897585}}</ref>

==In probability and measure theory==
{{details|Support (measure theory)}}

In [[probability theory]], the support of a [[probability distribution]] can be loosely thought of as the [[Closure (mathematics)|closure]] of the set of possible values of a random variable having that distribution. There are, however, some subtleties to consider when dealing with general distributions defined on a [[sigma algebra]], rather than on a topological space.

More formally, if <math>X : \Omega \to \R</math> is a random variable on <math>(\Omega, \mathcal{F}, P)</math> then the support of <math>X</math> is the smallest closed set <math>R_X \subseteq \R</math> such that <math>P\left(X \in R_X\right) = 1.</math>

In practice however, the support of a [[discrete random variable]] <math>X</math> is often defined as the set <math>R_X = \{x \in \R : P(X = x) > 0 \}</math> and the support of a [[continuous random variable]] <math>X</math> is defined as the set <math>R_X = \{x \in \R : f_X(x) > 0 \}</math> where <math>f_X(x)</math> is a [[probability density function]] of <math>X</math> (the [[#set-theoretic support|set-theoretic support]]).<ref>{{cite web|last1=Taboga|first1=Marco|title=Support of a random variable|url=https://www.statlect.com/glossary/support-of-a-random-variable|website=statlect.com|access-date=29 November 2017}}</ref> 

Note that the word {{em|support}} can refer to the [[logarithm]] of the [[likelihood function|likelihood]] of a probability density function.<ref>{{cite book|first=A. W. F.|last=Edwards|title=Likelihood|edition=Expanded|location=Baltimore|publisher=Johns Hopkins University Press|year=1992|isbn=0-8018-4443-6|pages=31–34|url=https://books.google.com/books?id=LL08AAAAIAAJ&pg=PA31 }}</ref>

==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>

==Singular support==

In [[Fourier analysis]] in particular, it is interesting to study the '''{{em|{{visible anchor|singular support}}}}''' of a distribution. This has the intuitive interpretation as the set of points at which a distribution {{em|fails to be a smooth function}}.

For example, the [[Fourier transform]] of the [[Heaviside step function]] can, up to constant factors, be considered to be <math>1/x</math> (a function) {{em|except}} at <math>x = 0.</math> While <math>x = 0</math> is clearly a special point, it is more precise to say that the transform of the distribution has singular support <math>\{ 0 \}</math>: it cannot accurately be expressed as a function in relation to test functions with support including <math>0.</math> It {{em|can}} be expressed as an application of a [[Cauchy principal value]] {{em|improper}} integral.

For distributions in several variables, singular supports allow one to define {{em|[[wave front set]]s}} and understand [[Huygens' principle]] in terms of [[mathematical analysis]]. Singular supports may also be used to understand phenomena special to distribution theory, such as attempts to 'multiply' distributions (squaring the Dirac delta function fails – essentially because the singular supports of the distributions to be multiplied should be disjoint).

==Family of supports==

An abstract notion of '''{{em|{{visible anchor|family of supports}}}}''' on a [[topological space]] <math>X,</math> suitable for [[sheaf theory]], was defined by [[Henri Cartan]]. In extending [[Poincaré duality]] to [[manifold]]s that are not compact, the 'compact support' idea enters naturally on one side of the duality; see for example [[Alexander–Spanier cohomology]].

Bredon, ''Sheaf Theory'' (2nd edition, 1997) gives these definitions. A family <math>\Phi</math> of closed subsets of <math>X</math> is a {{em|family of supports}}, if it is [[down-closed]] and closed under [[finite union]]. Its {{em|extent}} is the union over <math>\Phi.</math> A {{em|paracompactifying}} family of supports that satisfies further that any <math>Y</math> in <math>\Phi</math> is, with the [[subspace topology]], a [[paracompact space]]; and has some <math>Z</math> in <math>\Phi</math> which is a [[Neighbourhood (topology)|neighbourhood]]. If <math>X</math> is a [[locally compact space]], assumed [[Hausdorff space|Hausdorff]], the family of all [[compact subset]]s satisfies the further conditions, making it paracompactifying.

== See also ==

* {{annotated link|Bounded function}}
* {{annotated link|Bump function}}
* {{annotated link|Support of a module}}
* {{annotated link|Titchmarsh convolution theorem}}

== Citations ==

{{reflist|group=note}}
{{reflist}}

== References ==

* {{Rudin Walter Functional Analysis|edition=2}} <!-- {{sfn|Rudin|1991|p=}} -->
* {{Trèves François Topological vector spaces, distributions and kernels}} <!-- {{sfn|Trèves|2006|p=}} -->

[[Category:Set theory]]
[[Category:Real analysis]]
[[Category:Topology]]
[[Category:Topology of function spaces]]
[[Category:Schwartz distributions]]