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Support (mathematics)

2025-09-22T05:24:40Z

2601:204:F181:9410:F00A:A5AF:A897:8D69: Improved wording

{{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's [[Domain of a function|domain]] consisting of those 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 (structure)|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}} 
* {{Trèves François Topological vector spaces, distributions and kernels}} 

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

Probability measure

2025-09-21T23:48:04Z

2601:204:F181:9410:F00A:A5AF:A897:8D69: /* Definition */ must return results → must take values

{{Short description|Measure of total value one, generalizing probability distributions}}
{{Use American English|date = March 2019}}
{{Probability fundamentals}}
In [[mathematics]], a '''probability measure''' is a [[real-valued function]] defined on a set of events in a [[σ-algebra]] that satisfies [[Measure (mathematics)|measure]] properties such as ''countable additivity''.<ref>''An introduction to measure-theoretic probability'' by George G. Roussas 2004 {{isbn|0-12-599022-7}} [https://books.google.com/books?id=J8ZRgCNS-wcC&pg=PA47 page 47]</ref> The difference between a probability measure and the more general notion of measure (which includes concepts like [[area]] or [[volume]]) is that a probability measure must assign value 1 to the entire space.

Intuitively, the additivity property says that the probability assigned to the union of two disjoint (mutually exclusive) events by the measure should be the sum of the probabilities of the events; for example, the value assigned to the outcome "1 or 2" in a throw of a die should be the sum of the values assigned to the outcomes "1" and "2".

Probability measures have applications in diverse fields, from physics to finance and biology.

==Definition==
[[File:Probability-measure.svg|thumb|300px|A ''probability measure'' mapping the σ-algebra for <math>2^3</math> events to the [[unit interval]].]]
The requirements for a [[set function]] <math>\mu</math> to be a probability measure on a [[σ-algebra]] are that:

* <math>\mu</math> must take values in the [[unit interval]] <math>[0, 1],</math> including <math>0</math> on the empty set and <math>1</math> on the entire space.
* <math>\mu</math> must satisfy the ''[[Sigma-additive set function|countable additivity]]'' property that for all [[countable]] collections <math>E_1, E_2, \ldots</math> of pairwise [[disjoint sets]]: <math display=block> \mu\left(\bigcup_{i \in \N} E_i\right) = \sum_{i \in \N} \mu(E_i).</math>

For example, given three elements 1, 2 and 3 with probabilities <math>1/4, 1/4</math> and <math>1/2,</math> the value assigned to <math>\{1, 3\}</math> is <math>1/4 + 1/2 = 3/4,</math> as in the diagram on the right.

The [[conditional probability]] based on the intersection of events defined as:
<math display=block>\mu (B \mid A) = \frac{\mu(A \cap B)}{\mu(A)}.</math>
satisfies the probability function requirements so long as <math>\mu(A)</math> is not zero.<ref>{{Cite journal |last1=Dekking |first1=Frederik Michel |last2=Kraaikamp |first2=Cornelis |last3=Lopuhaä |first3=Hendrik Paul |last4=Meester |first4=Ludolf Erwin |date=2005 |title=A Modern Introduction to Probability and Statistics |url=https://link.springer.com/book/10.1007/1-84628-168-7 |journal=Springer Texts in Statistics |language=en |doi=10.1007/1-84628-168-7 |isbn=978-1-85233-896-1 |issn=1431-875X|url-access=subscription }}</ref><ref>''Probability, Random Processes, and Ergodic Properties'' by Robert M. Gray 2009 {{isbn|1-4419-1089-1}} [https://books.google.com/books?id=x-VbL8mZWl8C&pg=PA163 page 163]</ref>

Probability measures are distinct from the more general notion of [[Fuzzy measure theory|fuzzy measures]] in which there is no requirement that the fuzzy values sum up to <math>1,</math> and the additive property is replaced by an order relation based on [[set inclusion]].

==Example applications==

In many cases, [[statistical physics]] uses ''probability measures'', but not all [[measure theory|measures]] it uses are probability measures.{{clarify|reason=this sentence make no sense on its own, see Talk page|date=May 2025}}<ref name="stern">''A course in mathematics for students of physics, Volume 2'' by Paul Bamberg, Shlomo Sternberg 1991 {{isbn|0-521-40650-1}} [https://books.google.com/books?id=eSmC4qQ0SCAC&pg=PA802 page 802]</ref><ref name="gut">''The concept of probability in statistical physics'' by Yair M. Guttmann 1999 {{isbn|0-521-62128-3}} [https://books.google.com/books?id=Q1AUhivGmyUC&pg=PA149 page 149]</ref>

''Market measures'' which assign probabilities to [[financial market]] spaces based on observed market movements are examples of probability measures which are of interest in [[mathematical finance]]; for example, in the pricing of [[financial derivative]]s.<ref>''Quantitative methods in derivatives pricing'' by Domingo Tavella 2002 {{isbn|0-471-39447-5}} [https://books.google.com/books?id=dHIMulKy8dYC&pg=PA11 page 11]</ref> For instance, a [[risk-neutral measure]] is a probability measure which assumes that the current value of assets is the [[expected value]] of the future payoff taken with respect to that same risk neutral measure (i.e. calculated using the corresponding risk neutral density function), and [[discounted]] at the [[risk-free rate]]. If there is a unique probability measure that must be used to price assets in a market, then the market is called a [[complete market]].<ref>''Irreversible decisions under uncertainty'' by Svetlana I. Boyarchenko, Serge Levendorskiĭ 2007 {{isbn|3-540-73745-6}} [https://books.google.com/books?id=lpsrP5mQG_QC&pg=PA11 page 11]</ref>

Not all measures that intuitively represent chance or likelihood are probability measures. For instance, although the fundamental concept of a system in [[statistical mechanics]] is a measure space, such measures are not always probability measures.<ref name=stern/> In statistical physics, for sentences of the form "the probability of a system S assuming state A is p," the geometry of the system does not always lead to the definition of a probability measure [[congruence relation|under congruence]], although it may do so in the case of systems with just one degree of freedom.<ref name=gut/>

Probability measures are also used in [[mathematical biology]].<ref>''Mathematical Methods in Biology'' by J. David Logan, William R. Wolesensky 2009 {{isbn|0-470-52587-8}} [https://books.google.com/books?id=6GGyquH8kLcC&pg=PA195 page 195]</ref> For instance, in comparative [[sequence analysis]] a probability measure may be defined for the likelihood that a variant may be permissible for an [[amino acid]] in a sequence.<ref>''Discovering biomolecular mechanisms with computational biology'' by Frank Eisenhaber 2006 {{isbn|0-387-34527-2}} [https://books.google.com/books?id=Pygg7cIZTwIC&pg=PA127 page 127]</ref>

==See also==

* {{annotated link|Borel measure}}
* {{annotated link|Fuzzy measure}}
* {{annotated link|Haar measure}}
* {{annotated link|Lebesgue measure}}
* {{annotated link|Martingale measure}}
* {{annotated link|Set function}}
* [[Probability distribution]]

==References==
{{reflist}}

==Further reading==
* {{cite book |title=Probability and Measure |first=Patrick |last=Billingsley |author-link=Patrick Billingsley |year=1995 |publisher=John Wiley |isbn=0-471-00710-2 }}
* {{cite book |title=Probability & Measure Theory |first1=Robert B. |last1=Ash |first2=Catherine A. |last2=Doléans-Dade |year=1999 |publisher=Academic Press |isbn= 0-12-065202-1}}
* [https://math.stackexchange.com/q/1073744/29780 Distinguishing probability measure, function and distribution], Math Stack Exchange

==External links==
* {{Commons category-inline}}

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