In mathematics, the support of a realvalued function f is the subset of the domain containing those elements which are not mapped to zero. If the domain of f is a topological space, the support of f is instead defined as the smallest closed set containing all points not mapped to zero. This concept is used very widely in mathematical analysis.
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Transcription
Contents
Formulation
Suppose that f : X → R is a realvalued function whose domain is an arbitrary set X. The settheoretic support of f, written supp(f), is the set of points in X where f is nonzero
The support of f is the smallest subset of X with the property that f is zero on the subset's complement. If f(x) = 0 for all but a finite number of points x in X, then f is said to have finite support.
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.
Closed support
The most common situation occurs when X is a topological space (such as the real line or ndimensional 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 of the subset of X where f is nonzero^{[1]}^{[2]}^{[3]} i.e.,
Since the intersection of closed sets is closed, supp(f) is the intersection of all closed sets that contain the settheoretic support of f.
For example, if f : R → R is the function defined by
then the support of f is the closed interval [−1,1], since f is nonzero on the open interval (−1,1) and the closure of this set is [−1,1].
The notion of closed support is usually applied to continuous functions, but the definition makes sense for arbitrary real or complexvalued functions on a topological space, and some authors do not require that f : X → R (or C) be continuous.^{[4]}
Compact support
Functions with compact support on a topological space are those whose closed support is a compact subset of . If is the real line, or dimensional Euclidean space, then a function has compact support if and only if it has bounded support, since a subset of is compact if and only if it is closed and bounded.
For example, the function defined above is a continuous function with compact support [−1, 1].
The condition of compact support is stronger than the condition of vanishing at infinity. For example, the function defined by
vanishes at infinity, since as , but its support is not compact.
Realvalued compactly supported smooth functions on a Euclidean space are called bump functions. Mollifiers are an important special case of bump functions as they can be used in distribution theory to create sequences of smooth functions approximating nonsmooth (generalized) functions, via convolution.
In good cases, functions with compact support are 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 limits, for any , any function on the real line that vanishes at infinity can be approximated by choosing an appropriate compact subset of such that
for all , where is the indicator function of . 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 X is a topological measure space with a Borel measure μ (such as R^{n}, or a Lebesgue measurable subset of R^{n}, equipped with Lebesgue measure), then one typically identifies functions that are equal μalmost 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 μalmost everywhere outside F. Equivalently, ess supp(f) is the complement of the largest open set on which f = 0 μalmost everywhere^{[5]}
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.
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.^{[5]}^{[6]}
Generalization
If M is an arbitrary set containing zero, the concept of support is immediately generalizable to functions f : X→M. Support may also be defined for any algebraic structure with identity (such as a group, monoid, or composition algebra), in which the identity element assumes the role of zero. For instance, the family Z^{N} of functions from the natural numbers to the integers is the uncountable set of integer sequences. The subfamily { f in Z^{N} :f has finite support } 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 rings and free abelian groups.^{[7]}
In probability and measure theory
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.
More formally, if is a random variable on then the support of is the smallest closed set such that .
In practice however, the support of a discrete random variable is often defined as the set and the support of a continuous random variable is defined as the set where is a probability density function of (the settheoretic support).^{[8]}
Note that the word support can refer to the logarithm of the likelihood of a probability density function.^{[9]}
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 .
Singular support
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.
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.
For distributions in several variables, singular supports allow one to define wave front sets 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 family of supports on a topological space X, suitable for sheaf theory, was defined by Henri Cartan. In extending Poincaré duality to manifolds 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 Φ of closed subsets of X is a family of supports, if it is downclosed and closed under finite union. Its extent is the union over Φ. A paracompactifying family of supports that satisfies further that any Y in Φ is, with the subspace topology, a paracompact space; and has some Z in Φ which is a neighbourhood. If X is a locally compact space, assumed Hausdorff the family of all compact subsets satisfies the further conditions, making it paracompactifying.
See also
References
 ^ Folland, Gerald B. (1999). Real Analysis, 2nd ed. New York: John Wiley. p. 132.
 ^ Hörmander, Lars (1990). Linear Partial Differential Equations I, 2nd ed. Berlin: SpringerVerlag. p. 14.
 ^ Pascucci, Andrea (2011). PDE and Martingale Methods in Option Pricing. Bocconi & Springer Series. Berlin: SpringerVerlag. p. 678. doi:10.1007/9788847017818. ISBN 9788847017801.
 ^ Rudin, Walter (1987). Real and Complex Analysis, 3rd ed. New York: McGrawHill. p. 38.
 ^ ^{a} ^{b} Lieb, Elliott; Loss, Michael (2001). Analysis. Graduate Studies in Mathematics. 14 (2nd ed.). American Mathematical Society. p. 13. ISBN 9780821827833.
 ^ In a similar way, one uses the essential supremum of a measurable function instead of its supremum.
 ^ Tomasz, Kaczynski (2004). Computational homology. Mischaikow, Konstantin Michael,, Mrozek, Marian. New York: Springer. p. 445. ISBN 9780387215976. OCLC 55897585.
 ^ Taboga, Marco. "Support of a random variable". statlect.com. Retrieved 29 November 2017.
 ^ Edwards, A. W. F. (1992). Likelihood (Expanded ed.). Baltimore: Johns Hopkins University Press. pp. 31–34. ISBN 0801844436.