Support (measure theory)

Definition

Let ${\displaystyle (X,T)}$ be a topological space; let ${\displaystyle B(T)}$ denote the Borel 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 the support (or spectrum) of ${\displaystyle \mu }$ is defined as the set of all points ${\displaystyle x}$ in ${\displaystyle X}$ for which every open neighbourhood ${\displaystyle N_x}$ of ${\displaystyle x}$ has positive 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: $${\displaystyle (\mathrm{\forall }U\in T)(U\cap C\ne \varnothing ⟹\mu (U\cap C)>0).}$$

Signed and complex measures

This definition can be extended to signed and complex measures. Suppose that ${\displaystyle \mu :\mathrm{\Sigma }\to [-\mathrm{\infty },+\mathrm{\infty }]}$ is a signed measure. Use the Hahn decomposition theorem to write $${\displaystyle \mu ={\mu }^{+}-{\mu }^{-},}$$ where ${\displaystyle {\mu }^{\pm }}$ are both non-negative measures. Then the support 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 a complex measure, the support of ${\displaystyle \mu }$ is defined to be the union of the supports of its real and imaginary parts.

Properties

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

A measure ${\displaystyle \mu }$ on ${\displaystyle X}$ is strictly positive if 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 an open 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 is closed 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 a Hausdorff topological space and ${\displaystyle \mu }$ is a Radon measure, a Borel set ${\displaystyle A}$ outside the support has measure 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 any measurable 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 of support of a measure and that of spectrum of a self-adjoint linear operator on a Hilbert space are closely related. Indeed, if ${\displaystyle \mu }$ is a regular 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 the essential range of the identity function ${\displaystyle x\mapsto x,}$ which is precisely the support of ${\displaystyle \mu .}$

Examples

Lebesgue measure

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 open interval ${\displaystyle (x-ϵ,x+ϵ)}$ 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, ${\displaystyle \mathrm{supp}(\lambda )=\mathbb{R}.}$

Dirac measure

In the case of Dirac 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 the singleton 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 zero variance (provided that the measure has variance at all).

A uniform distribution

Consider the measure ${\displaystyle \mu }$ on the real line ${\displaystyle \mathbb{R}}$ defined by $${\displaystyle \mu (A):=\lambda (A\cap (0,1))}$$ i.e. a uniform 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.

A nontrivial measure whose support is empty

The space of all countable ordinals 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.

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 ${\displaystyle 0.}$ 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 ${\displaystyle 0.}$