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{{Short description|Concept in mathematics}}

In [[mathematics]], the '''support''' (sometimes '''topological support''' or '''spectrum''') of a [[Measure_spaceMeasure space|measure]] <math>\mu</math> on a [[measuremeasurable space|measurable]] [[topological space]] <math>(X, \mathcaloperatorname{ABorel}, \mu(X))</math> is a precise notion of where in the space <math>X</math> the measure "lives". It is defined to be the largest ([[closed set|closed]]) [[subset]] of <math>X</math> onfor which theevery measure[[open isset|open]] [[Strictlyneighbourhood positive measure(mathematics)|strictlyneighbourhood]] positiveof every point of the [[Set (mathematics)|set]] has positive measure.

==Motivation==

RecallA that the(non-negative) measure <math>\mu</math> on a measurable space <math>(X, \Sigma)</math> is really a function <math>\mu : \mathcal{A}Sigma \to [0, + \infty].</math>. Therefore, in terms of the usual [[definition]] of [[Support_Support (mathematics)|support]], the support of <math>\mu</math> is a subset of the [[sigma algebra|&sigma;-algebra]] <math>\mathcal{A}Sigma:</math>:

:<math display=block>\mathrmoperatorname{supp} (\mu) := \overline{\{ A \in \mathcal{A}Sigma |\,\vert\, \mu (A) >\neq 0 \}}.,</math>

where the overbar denotes [[Closure (topology)|set closure]]. However, this definition is somewhat unsatisfactory: we use the notion of closure, but we do not even have a topology on <math>\mathcal{A}Sigma.</math>! What we really want to know is where in the space <math>X</math> the measure <math>\mu</math> is non-zero. Consider two examples:

# [[Lebesgue measure]] <math>\lambda</math> on the [[real line]] <math>\mathbb{R}Reals.</math>. It seems clear that <math>\lambda</math> "lives on" the whole of the real line.

# A [[Dirac measure]] <math>\delta_{p}delta_p</math> at some point <math>p \in \mathbb{R}Reals.</math>. Again, intuition suggests that the measure <math>\delta_{p}delta_p</math> "lives at" the point <math>p,</math>, and nowhere else.

In light of these two examples, we can reject the following candidate definitions in favour of the one in the next section:

# We could remove the points where <math>\mu</math> is zero, and take the support to be the remainder <math>X \setminus \{ x \in X |\mid \mu ( \{ x \} ) = 0 \}.</math>. This might work for the Dirac measure <math>\delta_{p}delta_p,</math>, but it would definitely not work for <math>\lambda:</math>: since the Lebesgue measure of any pointsingleton is zero, this definition would give <math>\lambda</math> empty support.

# By comparison with the notion of [[Strictly positive measure|strict positivity]] of measures, we could take the support to be the set of all points with a neighbourhood of positive measure: <math display=block>\{x \in X \mid \exists N_x \text{ open} \text{ such that } (x \in N_x \text{ and } \mu(N_x) > 0)\}</math> (or the [[Closure (topology)|closure]] of this). It is also too simplistic: by taking <math>N_x = X</math> for all points <math> x \in X,</math> this would make the support of every measure except the zero measure the whole of <math>X.</math>

:<math>\{ x \in X | \exists \mathrm{open\,} N_{x} \ni x \mathrm{\,s.t\,} \mu(N_{x}) > 0 \}</math>

(or the [[Closure_(topology)|closure]] of this). This is also too simplistic: by taking <math>N_{x} = X</math> for all points <math>x \in X</math>, this would make the support of every measure except the zero measure the whole of <math>X</math>.

TheHowever, the idea of "local strict positivity" is not too far from a workable definition:.

==Definition==

Let <math>(X, \mathcal{T})</math> be a [[topological space]]; let <math>B(X, \mathcal{A}, \muT)</math> alsodenote be athe [[measureBorel spacealgebra|Borel &sigma;-algebra]] suchon that<math>X,</math> i.e. the [[smallest sigma algebra]] on <math>\mathcal{A}X</math> that contains all open sets <math>U \in T.</math> Let <math>\mathcal{mu</math> be a measure on <math>(X, B(T}))</math>. Then the '''support''' of(or the'''spectrum''') measureof <math>\mu</math> is defined to beas the set of all points <math>x</math> \in <math>X</math> for which every [[Open_setOpen set|open]] [[Neighbourhood_Neighbourhood (mathematics)|neighbourhood]] <math>N_x</math> of <math>x</math> has [[positive number|positive]] measure:

:<math display=block>\mathrmoperatorname{supp} (\mu) := \{ x \in X |\mid \forall xN_x \in N_{T \colon (x} \in N_x \mathcal{T},Rightarrow \mu (N_{x}N_x) > 0 )\}.</math>

In other words, the support is the largest subset of <math>X</math> (with respect to inclusion, <math>\subseteq</math>) on which the measure is [[Strictly positive measure|strictly positive]].

Some authors prefer to take the closure of the above set. However, this is not necessary: see "Properties" below.

An equivalent definition of support is as the largest <math>C \in B(T)</math> (with respect to inclusion) such that every open set which has non-empty intersection with <math>C</math> has positive measure, i.e. the largest <math>C</math> such that:

<math display="block">(\forall U \in T)(U \cap C \neq \varnothing \implies \mu (U \cap C) > 0).</math>

===Signed and complex measures===

This definition can be extended to signed and complex measures.

Suppose that <math>\mu : \Sigma \to [-\infty, +\infty]</math> is a [[signed measure]]. Use the [[Hahn decomposition theorem]] to write

<math display=block>\mu = \mu^+ - \mu^-,</math>

where <math>\mu^\pm</math> are both non-negative measures. Then the '''support''' of <math>\mu</math> is defined to be

<math display=block>\operatorname{supp} (\mu) := \operatorname{supp} (\mu^+) \cup \operatorname{supp} (\mu^-).</math>

Similarly, if <math>\mu : \Sigma \to \Complex</math> is a [[complex measure]], the '''support''' of <math>\mu</math> is defined to be the [[Union (set theory)|union]] of the supports of its real and imaginary parts.

==Properties==

<math>\operatorname{supp} (\mu_1 + \mu_2) = \operatorname{supp} (\mu_1) \cup \operatorname{supp} (\mu_2)</math> holds.

* The support of a measure is [[Closed set|closed]] in <math>X</math>. Suppose that <math>x</math> is a [[limit point]] of <math>\mathrm{supp} (\mu)</math>, and let <math>N_{x}</math> be an open neighbourhood of <math>x</math>. Since <math>x</math> is a limit point of the support, there is some <math>y \in N_{x} \cap \mathrm{supp} (\mu)</math>, <math>y \neq x</math>. But <math>N_{x}</math> is also an open neighbourhood of <math>y</math>, so <math>\mu (N_{x}) > 0</math>, as required. Hence, <math>\mathrm{supp} (\mu)</math> contains all its limit points, i.e. it is closed.

* If <math>A</math> is a [[measurable set]] outside the support, then <math>A</math> has [[measure zero]]:

A measure <math>\mu</math> on <math>X</math> is strictly positive [[if and only if]] it has support <math>\operatorname{supp}(\mu) = X.</math> If <math>\mu</math> is strictly positive and <math>x \in X</math> is arbitrary, then any open neighbourhood of <math>x,</math> since it is an [[open set]], has positive measure; hence, <math>x \in \operatorname{supp}(\mu),</math> so <math>\operatorname{supp}(\mu) = X.</math> Conversely, if <math>\operatorname{supp}(\mu) = X,</math> then every non-empty open set (being an open neighbourhood of some point in its interior, which is also a point of the support) has positive measure; hence, <math>\mu</math> is strictly positive.

:<math>A \subseteq X \setminus \mathrm{supp} (\mu) \implies \mu (A) = 0.</math>

The support of a measure is [[Closed set|closed]] in <math>X,</math>as its complement is the union of the open sets of measure <math>0.</math>

The converse is not true in general: it fails if there exists <math>x \in \mathrm{supp} (\mu)</math> such that <math>\mu \left( \{ x \} \right) = 0</math> (e.g. Lebesgue measure).

* One does not need to "integrate outside the support": for any [[measurable function]] <math>f : X \to \mathbb{R}</math> or <math>\mathbb{C}</math>,

In general the support of a nonzero measure may be empty: see the examples below. However, if <math>X</math> is a [[Hausdorff space|Hausdorff]] topological space and <math>\mu</math> is a [[Radon measure]], a Borel set <math>A</math> outside the support has [[measure zero]]:

:<math>\int_{X} f(x) \, \mathrm{d} \mu (x) = \int_{\mathrm{supp} (\mu)} f(x) \, \mathrm{d} \mu (x).</math>

:<math display=block>A \subseteq X \setminus \mathrmoperatorname{supp} (\mu) \implies \mu (A) = 0.</math>

The converse is true if <math>A</math> is open, but it is not true in general: it fails if there exists a point <math>x \in \mathrmoperatorname{supp} (\mu)</math> such that <math>\mu \left( \{ x \} \right) = 0</math> (e.g. Lebesgue measure). Thus, one does not need to "integrate outside the support": for any [[measurable function]] <math>f : X \to \Reals</math> or <math>\Complex,</math>

:<math display=block>\int_{X}int_X f(x) \, \mathrm{d} \mu (x) = \int_{\mathrmoperatorname{supp} (\mu)} f(x) \, \mathrm{d} \mu (x).</math>

The concept of ''support'' of a measure and that of [[spectrum]] of a [[Self-adjoint operator|self-adjoint linear operator]] on a [[Hilbert space]] are closely related. Indeed, if <math>\mu</math> is a [[regular Borel measure]] on the line <math>\mathbb{R},</math> then the multiplication operator <math>(Af)(x) = xf(x)</math> is self-adjoint on its natural ___domain

<math display=block>D(A) = \{f \in L^2(\Reals, d\mu) \mid xf(x) \in L^2(\Reals, d\mu)\}</math>

and its spectrum coincides with the [[essential range]] of the identity function <math>x \mapsto x,</math> which is precisely the support of <math>\mu.</math><ref>Mathematical methods in Quantum Mechanics with applications to Schrödinger Operators</ref>

==Examples==

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===Lebesgue measure===

In the case of Lebesgue measure <math>\lambda</math> on the real line <math>\Reals,</math> consider an arbitrary point <math>x \in \mathbb{R}Reals.</math>. Then any open neighbourhood <math>N_{x}N_x</math> of <math>x</math> must contain some open [[Interval_Interval (mathematics)|interval]] <math>(x - \varepsilonepsilon, x + \varepsilonepsilon)</math> for some <math>\varepsilonepsilon > 0.</math>. This interval has Lebesgue measure <math>2 \varepsilonepsilon > 0,</math>, so <math>\mu lambda(N_{x}N_x) \geq 2 \varepsilonepsilon > 0.</math>. Since <math>x \in \mathbb{R}Reals</math> was arbitrary, <math>\mathrmoperatorname{supp} (\lambda) = \mathbb{R}Reals.</math>.

===Dirac measure===

In the case of [[Dirac measure]] <math>\delta_{p}delta_p,</math>, let <math>x \in \mathbb{R}Reals</math> and consider two cases:

## if <math>x = p,</math>, then every open neighbourhood <math>N_{x}N_x</math> of <math>x</math> contains <math>p,</math>, so <math>\delta_{p} delta_p(N_{x}N_x) = 1 > 0.</math>;

## on the other hand, if <math>x \neq p,</math>, then there exists a sufficiently small open ball <math>B</math> around <math>x</math> that does not contain <math>p,</math>, so <math>\delta_{p} delta_p(B) = 0.</math>.

We conclude that <math>\mathrmoperatorname{supp} (\delta_{p}delta_p)</math> is the closure of the [[Singleton_Singleton (mathematics)|singleton]] set <math>\{ p \},</math>, which is <math>\{ p \}</math> itself.

In fact, a measure <math>\mu</math> on the real line is a Dirac measure <math>\delta_{p}delta_p</math> for some point <math>p</math> [[if and only if]] the support of <math>\mu</math> is the singleton set <math>\{ p \}.</math>. Consequently, Dirac measure on the real line is the unique measure with zero [[variance]] [(provided that the measure has variance at all]).

===A uniform distribution===

Consider the measure <math>\mu</math> on the real line <math>\Reals</math> defined by

:<math display=block>\mu (A) := \lambda (A \cap (0, 1))</math>

i.e. a [[Uniform_distribution_Uniform distribution (continuous)|uniform measure]] on the open interval <math>(0, 1).</math>. A similar argument to the Dirac measure example shows that <math>\mathrmoperatorname{supp} (\mu) = [0, 1].</math>. Note that the boundary points 0 and 1 lie in the support: any open set containing 0 (or 1) contains an open interval about 0 (or 1), which must intersect <math>(0, 1),</math>, and so must have positive <math>\mu</math>-measure.

===A nontrivial measure whose support is empty===

The space of all [[countable ordinal]]s with the topology generated by "open intervals" is a [[locally compact]] [[Hausdorff space]]. The measure ("Dieudonné measure") that assigns measure 1 to Borel sets containing an unbounded closed subset and assigns 0 to other Borel sets is a Borel probability measure whose support is empty. <ref>Measure theory, Example 7.1.3</ref>

===A nontrivial measure whose support has measure zero===

On a compact Hausdorff space the support of a non-zero measure is always non-empty, but may have measure <math>0.</math> An example of this is given by adding the first uncountable ordinal <math>\Omega</math> to the previous example: the support of the measure is the single point <math>\Omega,</math> which has measure <math>0.</math>

==References==

{{reflist}}

* {{cite book | author=Ambrosio, L., Gigli, N. & Savaré, G. | title=Gradient Flows in Metric Spaces and in the Space of Probability Measures | publisher=ETH Zürich, Birkhäuser Verlag, Basel | year=2005 | isbn=3-7643-2428-7}}

* {{cite book | author=Bogachev, V. I. | title=Measure theory | publisher=Springer Berlin Heidelberg | year=2007 | isbn=978-3-540-34514-5 | volume = 2}}

* {{cite book

| last = Parthasarathy

| first = K. R.

| title = Probability measures on metric spaces

|publisher = AMS Chelsea Publishing, Providence, RI

| year = 2005

| isbn = 0-8218-3889-X

| page = xii+276

}} {{MathSciNet|id=2169627}} (See chapter 2, section 2)

* {{cite book

| last = Teschl

| first = Gerald

| title = Mathematical methods in Quantum Mechanics with applications to Schrödinger Operators

|publisher = AMS

| year = 2009

| url = https://www.mat.univie.ac.at/~gerald/ftp/book-schroe/

}}(See chapter 3, section 2)

{{Measure theory}}

[[Category:Measures (measure theory)]]

[[Category:Measure theory]]