Inmathematics, thesupport (sometimestopological support orspectrum) of ameasure${\displaystyle \mu }$ on ameasurabletopological space${\displaystyle (X,\mathrm{Borel}(X))}$ is a precise notion of where in the space${\displaystyle X}$ the measure "lives". It is defined to be the largest (closed)subset of${\displaystyle X}$ for which everyopenneighbourhood of every point of theset has positive measure.

A (non-negative) measure${\displaystyle \mu }$ on a measurable space${\displaystyle (X,\mathrm{\Sigma })}$ is really a function${\displaystyle \mu :\mathrm{\Sigma }\to [0,+\mathrm{\infty }].}$ Therefore, in terms of the usualdefinition ofsupport, the support of${\displaystyle \mu }$ is a subset of theσ-algebra${\displaystyle \mathrm{\Sigma }:}$$${\displaystyle \mathrm{supp}(\mu ):=\overline{\{A\in \mathrm{\Sigma }|\mu (A)\ne 0\}},}$$where the overbar denotesset closure. However, this definition is somewhat unsatisfactory: we use the notion of closure, but we do not even have a topology on${\displaystyle \mathrm{\Sigma }.}$ What we really want to know is where in the space${\displaystyle X}$ the measure${\displaystyle \mu }$ is non-zero. Consider two examples:

1. Lebesgue measure${\displaystyle \lambda }$ on thereal line${\displaystyle \mathbb{R}.}$ It seems clear that${\displaystyle \lambda }$ "lives on" the whole of the real line.
2. ADirac measure${\displaystyle {\delta }_p}$ at some point${\displaystyle p\in \mathbb{R}.}$ Again, intuition suggests that the measure${\displaystyle {\delta }_p}$ "lives at" the point${\displaystyle p,}$ 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:

1. We could remove the points where${\displaystyle \mu }$ is zero, and take the support to be the remainder${\displaystyle X\setminus \{x\in X\mid \mu (\{x\})=0\}.}$ This might work for the Dirac measure${\displaystyle {\delta }_p,}$ but it would definitely not work for${\displaystyle \lambda :}$ since the Lebesgue measure of any singleton is zero, this definition would give${\displaystyle \lambda }$ empty support.
2. By comparison with the notion ofstrict positivity of measures, we could take the support to be the set of all points with a neighbourhood of positive measure:$${\displaystyle \{x\in X\mid \mathrm{\exists }{N}_x\text{ open}\text{ such that }(x\in N_x\text{ and }\mu (N_x)>0)\}}$$ (or theclosure of this). It is also too simplistic: by taking${\displaystyle N_x=X}$ for all points${\displaystyle x\in X,}$ this would make the support of every measure except the zero measure the whole of${\displaystyle X.}$

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

Let${\displaystyle (X,T)}$ be atopological space; let${\displaystyle B(T)}$ denote theBorel σ-algebra on${\displaystyle X,}$ i.e. the smallest sigma algebra on${\displaystyle X}$ that contains all open sets${\displaystyle U\in T.}$ Let${\displaystyle \mu }$ be a measure on${\displaystyle (X,B(T))}$. Then thesupport (orspectrum) of${\displaystyle \mu }$ is defined as the set of all points${\displaystyle x}$ in${\displaystyle X}$ for which everyopenneighbourhood${\displaystyle N_x}$ of${\displaystyle x}$ haspositive measure:$${\displaystyle \mathrm{supp}(\mu ):=\{x\in X\mid \mathrm{\forall }{N}_x\in T:(x\in N_x\Rightarrow \mu (N_x)>0)\}.}$$

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${\displaystyle C\in B(T)}$ (with respect to inclusion) such that every open set which has non-empty intersection with${\displaystyle C}$ has positive measure, i.e. the largest${\displaystyle C}$ such that:$${\displaystyle (\mathrm{\forall }U\in T)(U\cap C\ne \varnothing ⟹\mu (U\cap C)>0).}$$

This definition can be extended to signed and complex measures. Suppose that${\displaystyle \mu :\mathrm{\Sigma }\to [-\mathrm{\infty },+\mathrm{\infty }]}$ is asigned measure. Use theHahn decomposition theorem to write$${\displaystyle \mu ={\mu }^{+}-{\mu }^{-},}$$where${\displaystyle {\mu }^{\pm }}$ are both non-negative measures. Then thesupport of${\displaystyle \mu }$ is defined to be$${\displaystyle \mathrm{supp}(\mu ):=\mathrm{supp}({\mu }^{+})\cup \mathrm{supp}({\mu }^{-}).}$$

Similarly, if${\displaystyle \mu :\mathrm{\Sigma }\to \mathbb{C}}$ is acomplex measure, thesupport of${\displaystyle \mu }$ is defined to be theunion of the supports of its real and imaginary parts.

${\displaystyle \mathrm{supp}({\mu }_1+{\mu }_2)=\mathrm{supp}({\mu }_1)\cup \mathrm{supp}({\mu }_2)}$ holds.

A measure${\displaystyle \mu }$ on${\displaystyle X}$ isstrictly positiveif and only if it has support${\displaystyle \mathrm{supp}(\mu )=X.}$ If${\displaystyle \mu }$ is strictly positive and${\displaystyle x\in X}$ is arbitrary, then any open neighbourhood of${\displaystyle x,}$ since it is anopen set, has positive measure; hence,${\displaystyle x\in \mathrm{supp}(\mu ),}$ so${\displaystyle \mathrm{supp}(\mu )=X.}$ Conversely, if${\displaystyle \mathrm{supp}(\mu )=X,}$ 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,${\displaystyle \mu }$ is strictly positive.The support of a measure isclosed in${\displaystyle X,}$as its complement is the union of the open sets of measure${\displaystyle 0.}$

In general the support of a nonzero measure may be empty: see the examples below. However, if${\displaystyle X}$ is aHausdorff topological space and${\displaystyle \mu }$ is aRadon measure, a Borel set${\displaystyle A}$ outside the support hasmeasure zero:$${\displaystyle A\subseteq X\setminus \mathrm{supp}(\mu )⟹\mu (A)=0.}$$ The converse is true if${\displaystyle A}$ is open, but it is not true in general: it fails if there exists a point${\displaystyle x\in \mathrm{supp}(\mu )}$ such that${\displaystyle \mu (\{x\})=0}$ (e.g. Lebesgue measure). Thus, one does not need to "integrate outside the support": for anymeasurable function${\displaystyle f:X\to \mathbb{R}}$ or${\displaystyle \mathbb{C},}$$${\displaystyle {\int }_{X}f(x)\mathrm{d}\mu (x)={\int }_{\mathrm{supp}(\mu )}f(x)\mathrm{d}\mu (x).}$$

The concept ofsupport of a measure and that ofspectrum of aself-adjoint linear operator on aHilbert space are closely related. Indeed, if${\displaystyle \mu }$ is aregular Borel measure on the line${\displaystyle \mathbb{R},}$ then the multiplication operator${\displaystyle (Af)(x)=xf(x)}$ is self-adjoint on its natural domain$${\displaystyle D(A)=\{f\in L^2(\mathbb{R},d\mu )\mid xf(x)\in L^2(\mathbb{R},d\mu )\}}$$ and its spectrum coincides with theessential range of the identity function${\displaystyle x\mapsto x,}$ which is precisely the support of${\displaystyle \mu .}$^[1]

In the case of Lebesgue measure${\displaystyle \lambda }$ on the real line${\displaystyle \mathbb{R},}$ consider an arbitrary point${\displaystyle x\in \mathbb{R}.}$ Then any open neighbourhood${\displaystyle N_x}$ of${\displaystyle x}$ must contain some openinterval${\displaystyle (x-ϵ,x+ϵ)}$ for some${\displaystyle ϵ>0.}$ This interval has Lebesgue measure${\displaystyle 2ϵ>0,}$ so${\displaystyle \lambda (N_x)\ge 2ϵ>0.}$ Since${\displaystyle x\in \mathbb{R}}$ was arbitrary,${\displaystyle \mathrm{supp}(\lambda )=\mathbb{R}.}$

In the case ofDirac measure${\displaystyle {\delta }_p,}$ let${\displaystyle x\in \mathbb{R}}$ and consider two cases:

1. if${\displaystyle x=p,}$ then every open neighbourhood${\displaystyle N_x}$ of${\displaystyle x}$ contains${\displaystyle p,}$ so${\displaystyle {\delta }_p(N_x)=1>0.}$
2. on the other hand, if${\displaystyle x\ne p,}$ then there exists a sufficiently small open ball${\displaystyle B}$ around${\displaystyle x}$ that does not contain${\displaystyle p,}$ so${\displaystyle {\delta }_p(B)=0.}$

We conclude that${\displaystyle \mathrm{supp}({\delta }_p)}$ is the closure of thesingleton set${\displaystyle \{p\},}$ which is${\displaystyle \{p\}}$ itself.

In fact, a measure${\displaystyle \mu }$ on the real line is a Dirac measure${\displaystyle {\delta }_p}$ for some point${\displaystyle p}$if and only if the support of${\displaystyle \mu }$ is the singleton set${\displaystyle \{p\}.}$ Consequently, Dirac measure on the real line is the unique measure with zerovariance (provided that the measure has variance at all).

Consider the measure${\displaystyle \mu }$ on the real line${\displaystyle \mathbb{R}}$ defined by$${\displaystyle \mu (A):=\lambda (A\cap (0,1))}$$i.e. auniform measure on the open interval${\displaystyle (0,1).}$ A similar argument to the Dirac measure example shows that${\displaystyle \mathrm{supp}(\mu )=[0,1].}$ 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${\displaystyle (0,1),}$ and so must have positive${\displaystyle \mu }$-measure.

The space of allcountable ordinals with the topology generated by "open intervals" is alocally compactHausdorff 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.^[2]

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

• Ambrosio, L., Gigli, N. & Savaré, G. (2005).Gradient Flows in Metric Spaces and in the Space of Probability Measures. ETH Zürich, Birkhäuser Verlag, Basel.ISBN 3-7643-2428-7.{{cite book}}: CS1 maint: multiple names: authors list (link)
• Bogachev, V. I. (2007).Measure theory. Vol. 2. Springer Berlin Heidelberg.ISBN 978-3-540-34514-5.
• Parthasarathy, K. R. (2005).Probability measures on metric spaces. AMS Chelsea Publishing, Providence, RI. p. xii+276.ISBN 0-8218-3889-X.MR 2169627 (See chapter 2, section 2)
• Teschl, Gerald (2009).Mathematical methods in Quantum Mechanics with applications to Schrödinger Operators. AMS.(See chapter 3, section 2)

• Measures (measure theory)
• Measure theory

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