Support (measure theory)

In mathematics, the support (sometimes topological support or spectrum) of a measure ${\displaystyle \mu }$ on a measurable topological space ${\displaystyle (X,\operatorname {Borel} (X))}$ 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 every open neighbourhood of every point of the set has positive measure.

A (non-negative) measure ${\displaystyle \mu }$ on a measurable space ${\displaystyle (X,\Sigma )}$ is really a function ${\displaystyle \mu :\Sigma \to [0,+\infty ].}$ Therefore, in terms of the usual definition of support, the support of ${\displaystyle \mu }$ is a subset of the σ-algebra ${\displaystyle \Sigma :}$ $${\displaystyle \operatorname {supp} (\mu ):={\overline {\{A\in \Sigma \,\vert \,\mu (A)\neq 0\}}},}$$ where the overbar denotes set closure. However, this definition is somewhat unsatisfactory: we use the notion of closure, but we do not even have a topology on ${\displaystyle \Sigma .}$ 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 the real line ${\displaystyle \mathbb {R} .}$ It seems clear that ${\displaystyle \lambda }$ "lives on" the whole of the real line.
2. A Dirac 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 of strict positivity of measures, we could take the support to be the set of all points with a neighbourhood of positive measure: $${\displaystyle \{x\in X\mid \exists N_{x}{\text{ open}}{\text{ such that }}(x\in N_{x}{\text{ and }}\mu (N_{x})>0)\}}$$ (or the closure 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 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: $${\displaystyle \operatorname {supp} (\mu ):=\{x\in X\mid \forall N_{x}\in T\colon (x\in N_{x}\Rightarrow \mu (N_{x})>0)\}.}$$

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: $${\displaystyle (\forall U\in T)(U\cap C\neq \varnothing \implies \mu (U\cap C)>0).}$$

This definition can be extended to signed and complex measures. Suppose that ${\displaystyle \mu :\Sigma \to [-\infty ,+\infty ]}$ 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 $${\displaystyle \operatorname {supp} (\mu ):=\operatorname {supp} (\mu ^{+})\cup \operatorname {supp} (\mu ^{-}).}$$

Similarly, if ${\displaystyle \mu :\Sigma \to \mathbb {C} }$ is a complex measure, the support of ${\displaystyle \mu }$ is defined to be the union of the supports of its real and imaginary parts.

${\displaystyle \operatorname {supp} (\mu _{1}+\mu _{2})=\operatorname {supp} (\mu _{1})\cup \operatorname {supp} (\mu _{2})}$ holds.

A measure ${\displaystyle \mu }$ on ${\displaystyle X}$ is strictly positive if and only if it has support ${\displaystyle \operatorname {supp} (\mu )=X.}$ 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, ${\displaystyle x\in \operatorname {supp} (\mu ),}$ so ${\displaystyle \operatorname {supp} (\mu )=X.}$ Conversely, if ${\displaystyle \operatorname {supp} (\mu )=X,}$ 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: $${\displaystyle A\subseteq X\setminus \operatorname {supp} (\mu )\implies \mu (A)=0.}$$ The converse is true if ${\displaystyle A}$ is open, but it is not true in general: it fails if there exists a point ${\displaystyle x\in \operatorname {supp} (\mu )}$ 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 ${\displaystyle \mathbb {C} ,}$ $${\displaystyle \int _{X}f(x)\,\mathrm {d} \mu (x)=\int _{\operatorname {supp} (\mu )}f(x)\,\mathrm {d} \mu (x).}$$

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 ${\displaystyle \mu .}$^[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 open interval ${\displaystyle (x-\epsilon ,x+\epsilon )}$ for some ${\displaystyle \epsilon >0.}$ This interval has Lebesgue measure ${\displaystyle 2\epsilon >0,}$ so ${\displaystyle \lambda (N_{x})\geq 2\epsilon >0.}$ Since ${\displaystyle x\in \mathbb {R} }$ was arbitrary, ${\displaystyle \operatorname {supp} (\lambda )=\mathbb {R} .}$

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 ${\displaystyle x\neq p,}$ 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 ${\displaystyle \operatorname {supp} (\delta _{p})}$ 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).

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 ${\displaystyle \operatorname {supp} (\mu )=[0,1].}$ 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 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.

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 ${\displaystyle \Omega }$ to the previous example: the support of the measure is the single point ${\displaystyle \Omega ,}$ 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)
• Parthasarathy, K. R. (2005). Probability measures on metric spaces. AMS Chelsea Publishing, Providence, RI. p. xii+276. ISBN 0-8218-3889-X. MR2169627 (See chapter 2, section 2.)
• Teschl, Gerald (2009). Mathematical methods in Quantum Mechanics with applications to Schrödinger Operators. AMS.(See chapter 3, section 2)