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| In [[mathematics]], the '''support''' of a [[function (mathematics)|function]] is the set of points where the function is not zero-valued or, in the case of functions defined on a topological space, the [[closure (topology)|closure]] of that set. This concept is used very widely in [[mathematical analysis]]. In the form of functions with support that is bounded, it also plays a major part in various types of [[mathematical duality]] theories.
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| ==Formulation==
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| Suppose that ''f'' : ''X'' → '''R''' is a real-valued function whose [[Domain of a function|domain]] is an arbitrary set ''X''. The '''set-theoretic support''' of ''f'', written '''supp(''f'')''', is the set of points in ''X'' where ''f'' is non-zero
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| :<math>\operatorname{supp}(f) = \{x\in X \,|\, f(x)\ne 0\}</math>.
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| The support of ''f'' is the smallest subset of ''X'' with the property that ''f'' is zero on its complement, meaning that the non-zero values of ''f'' "live" on supp(f). If ''f''(''x'') = 0 for all but a finite number of points ''x'' in ''X'', then ''f'' is said to have '''finite support'''. | |
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| If the set ''X'' has an additional structure (for example, a topology), then the support of ''f'' is defined in an analogous way as the smallest subset of ''X'' of an appropriate type such that ''f'' 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 '''R''' and to other objects, such as measures or distributions.
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| ==Closed support==
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| The most common situation occurs when ''X'' is a [[topological space]] (such as the [[real line]] or ''n''-dimensional Euclidean space) and ''f'' : ''X''→'''R''' is a continuous real (or complex)-valued function. In this case, the support of ''f'' is defined topologically as the [[closure (topology)| closure]] of the subset of of ''X'' where ''f'' 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}}</ref> i.e.,
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| :<math>\operatorname{supp}(f) := \overline{\{x \in X \,|\, f(x) \neq 0 \}}</math>.
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| Since the intersection of closed sets is closed, supp(''f'') is the intersection of all closed sets that contain the set-theoretic support of ''f''.
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| For example, if ''f'' : '''R'''→'''R''' is the function defined by
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| :<math> f(x) = \begin{cases} 1-x^2 & \text{if}\ |x| < 1 \\ 0 & \text{if}\ |x| \ge 1 \end{cases}</math>
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| then the support of ''f'' is the closed interval [-1,1], since ''f'' is non-zero on the open interval (-1,1) and the closure of this set is [-1,1].
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| 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 ''f'' : ''X''→'''R''' (or '''C''') 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>
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| ==Compact support==
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| Functions with '''compact support''' on a topological space ''X'' are those whose support is a [[compact space|compact]] subset of ''X''. If ''X'' is the real line, or ''n''-dimensional Euclidean space, then a function has compact support if and only if has '''bounded support''', since the support is closed by definition and a subset of '''R'''<sup>n</sup> is compact if and only if it is closed and bounded.
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| For example, the function ''f'' : '''R'''→'''R''' defined above is a continuous function with compact support [-1,1].
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| The condition of compact support is stronger than the condition of [[vanish at infinity|vanishing at infinity]]. For example, the function ''f'' : '''R'''→'''R''' defined by
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| :<math> f(x) = \frac{x}{1+x^2}</math>
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| vanishes at infinity, since ''f(x)'' → 0 as ''|x|'' → ∞, but its support '''R''' is not compact.
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| 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]].
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| 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 ε > 0, any function ''f'' on the real line '''R''' that vanishes at infinity can be approximated by choosing an appropriate compact subset ''C'' of '''R''' such that
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| :<math>|f(x) - I_C(x)f(x)| < \varepsilon</math> | |
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| for all ''x'' ∈ ''X'', where <math>I_C</math> is the [[indicator function]] of ''C''. Every continuous function on a compact topological space has compact support since every closed subset of a compact space is indeed compact.
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| ==Essential support==
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| If ''X'' is a topological measure space with a Borel measure μ (such as '''R'''<sup>n</sup>, or a Lebesgue measurable subset of '''R'''<sup>n</sup>, equipped with Lebesgue measure), then one typically identifies functions that are equal μ-almost everywhere. In that case, the '''essential support''' of a measurable function ''f'' : ''X''→'''R''', written '''ess supp(''f'')''', is defined to be the smallest closed subset ''F'' of ''X'' such that ''f=0'' μ-almost everywhere outside ''F''. Equivalently, ess supp(f) is the complement of the largest open set on which ''f''=0 μ-almost everywhere<ref name=lieb>{{cite book|last1=Lieb|first1=Elliott|last2 =Loss | first2=Michael|year=2001|title=Analysis, 2nd ed.|page=13|location= Providence|publisher=AMS}}</ref>
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| :<math>\operatorname{ess\,supp}(f) := X \setminus\bigcup \left\{\Omega\subset X \,|\, \Omega\,\text{is open and}\, f = 0\, \mu\text{-almost everywhere in}\, \Omega \right\}</math>.
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| The essential support of a function ''f'' depends on the measure μ as well as on ''f'', and it may be strictly smaller than the closed support. For example, if ''f'' : [0,1] → '''R''' is the [[Dirichlet function]] that is 0 on irrational numbers and 1 on rational numbers, and [0,1] is equipped with Lebesgue measure, then the support of ''f'' is the entire interval [0,1], but the essential support of ''f'' is empty, since ''f'' is equal almost everywhere to the zero function.
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| 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 ess supp(''f'') is often written simply as supp(''f'') 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>
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| ==Generalization==
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| If ''M'' is an arbitrary set containing zero, the concept of support is immediately generalizable to functions ''f'' : ''X''→''M''. ''M'' may also be 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 '''Z'''<sup>'''N'''</sup> of functions from the [[natural numbers]] to the [[integers]] is the [[uncountable]] set of integer sequences. The subfamily { ''f'' in '''Z'''<sup>'''N'''</sup> :''f'' has finite support } is the countable set of all integer sequences that have only finitely many nonzero entries.
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| ==In probability and measure theory==
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| {{details|support (measure theory)}}
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| In [[probability theory]], the support of a [[probability distribution]] can be loosely thought of as the 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.
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| Note that the word ''support'' can refer to the [[logarithm]] of the [[likelihood function|likelihood]] of a [[probability density function]].
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| ==Support of a distribution==
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| It is possible also to talk about the support of a [[distribution (mathematics)|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 function]]s 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 [[measure (mathematics)|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.
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| Suppose that ''f'' is a distribution, and that ''U'' 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 ''U'', <math>f(\phi) = 0</math>. Then ''f'' is said to vanish on ''U''. Now, if ''f'' vanishes on an arbitrary family <math>U_{\alpha}</math> of open sets, then for any test function <math>\phi</math> supported in <math>\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 ''support'' of ''f'' as the complement of the largest open set on which ''f'' vanishes. For example, the support of the Dirac delta is <math>\{0\}</math>.
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| ==Singular support==
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| In [[Fourier analysis]] in particular, it is interesting to study the '''singular support''' of a distribution. This has the intuitive interpretation as the set of points at which a distribution ''fails to be a smooth function''.
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| For example, the [[Fourier transform]] of the [[Heaviside step function]] can, up to constant factors, be considered to be 1/''x'' (a function) ''except'' at ''x'' = 0. While ''x'' = 0 is clearly a special point, it is more precise to say that the transform of the distribution has singular support {0}: it cannot accurately be expressed as a function in relation to test functions with support including 0. It ''can'' be expressed as an application of a [[Cauchy principal value]] ''improper'' integral.
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| For distributions in several variables, singular supports allow one to define ''[[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).
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| ==Family of supports==
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| An abstract notion of '''family of supports''' on a [[topological space]] ''X'', 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]].
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| Bredon, ''Sheaf Theory'' (2nd edition, 1997) gives these definitions. A family Φ of closed subsets of ''X'' is a ''family of supports'', if it is [[down-closed]] and closed under [[finite union]]. Its ''extent'' is the union over Φ. A ''paracompactifying'' family of supports satisfies further than any ''Y'' in Φ is, with the [[subspace topology]], a [[paracompact space]]; and has some ''Z'' in Φ which is a [[neighbourhood (topology)|neighbourhood]]. If ''X'' is a [[locally compact space]], assumed [[Hausdorff space|Hausdorff]] the family of all [[compact subset]]s satisfies the further conditions, making it paracompactifying.
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| ==See also==
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| *[[Titchmarsh convolution theorem]]
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| *[[Support of a module]]
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| ==References==
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| {{reflist}}
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| [[Category:Set theory]]
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| [[Category:Real analysis]]
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| [[Category:Topology]]
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| [[Category:Topology of function spaces]]
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