In mathematics, the support of a real-valued function f on a set X is sometimes defined as the subset of X on which f is nonzero. The most common situation occurs when X is a topological space (such as the real line) and f is a continuous function. In this case, the support of f is defined as the smallest closed subset of X outside of which f is zero. The topological support is the closure of the set-theoretic support.

In particular, in probability theory, the support of a probability distribution is the closure of the set of possible values of a random variable having that distribution.

Compact support

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Functions with compact support in X are those with support that is a compact subset of X. For example, if X is the real line, they are examples of functions that vanish at infinity (and negative infinity). Indeed, they are special cases of such functions that must vanish at finite bounds. 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 ε > 0, any function f on the real line X that vanishes at infinity can be approximated by choosing an appropriate compact subset C of X such that

for all x ∈ X, where $I\_C$ is the indicator function of C. Note that every function on a compact space has compact support since every closed subset of a compact space is compact.

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.

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 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 this is clearly a special point, it is more precise to say that the transform qua 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

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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 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 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. If X is a locally compact space, assumed Hausdorff the family of all compact subsets satisfies the further conditions, making it paracompactifying.

Real analysis | Topology | Topology of function spaces

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