Support (measure theory)

In mathematics, the support (sometimes topological support or spectrum) of a measure μ on a measurable topological space (X, Borel(X)) is a precise notion of where in the space X the measure "lives". It is defined to be the largest (closed) subset of X for which every open neighbourhood of every point of the set has positive measure.

Motivation[]

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 ${\displaystyle \Sigma }$:

${\displaystyle \operatorname {supp} (\mu ):=\bigcap \{A={\overline {A}}\in \Sigma \mid \mu (A^{c})=0\},}$

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:

${\displaystyle \{x\in X\mid {\text{for some open }}N_{x}\ni x,\mu (N_{x})>0\}}$

(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:

Definition[]

Let (X, T) be a topological space; let B(T) denote the Borel σ-algebra on X, i.e. the smallest sigma algebra on X that contains all open sets U ∈ T. Let μ be a measure on (X, B(T)). Then the support (or spectrum) of μ is defined as the set of all points x in X for which every open neighbourhood N_x of x has positive measure:

${\displaystyle \operatorname {supp} (\mu ):=\{x\in X\mid \forall N_{x}\in T\,\mu (N_{x})>0\}.}$

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 C ∈ B(T) (with respect to inclusion) such that every open set which has non-empty intersection with C has positive measure, i.e. the largest C such that:

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

Properties[]

• ${\displaystyle \operatorname {supp} (\mu _{1}+\mu _{2})=\operatorname {supp} (\mu _{1})\cup \operatorname {supp} (\mu _{2})}$
• A measure μ on X is strictly positive if and only if it has support supp(μ) = X. If μ is strictly positive and x ∈ X is arbitrary, then any open neighbourhood of x, since it is an open set, has positive measure; hence, x ∈ supp(μ), so supp(μ) = X. Conversely, if supp(μ) = 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, μ is strictly positive.
• The support of a measure is closed in X as its complement is the union of the open sets of measure 0.
• In general the support of a nonzero measure may be empty: see the examples below. However, if X is a topological Hausdorff space and μ is a Radon measure, a measurable set A outside the support has measure zero:

${\displaystyle A\subseteq X\setminus \operatorname {supp} (\mu )\implies \mu (A)=0.}$

The converse is true if A is open, but it is not true in general: it fails if there exists a point x ∈ supp(μ) such that μ({x}) = 0 (e.g. Lebesgue measure).

Thus, one does not need to "integrate outside the support": for any measurable function f : X → R or 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]

Examples[]

Lebesgue measure[]

In the case of Lebesgue measure λ on the real line R, consider an arbitrary point x ∈ R. Then any open neighbourhood N_x of x must contain some open interval (x − ε, x + ε) for some ε > 0. This interval has Lebesgue measure 2ε > 0, so λ(N_x) ≥ 2ε > 0. Since x ∈ R was arbitrary, supp(λ) = R.

Dirac measure[]

In the case of Dirac measure δ_p, let x ∈ R and consider two cases:

1. if x = p, then every open neighbourhood N_x of x contains p, so δ_p(N_x) = 1 > 0;
2. on the other hand, if x ≠ p, then there exists a sufficiently small open ball B around x that does not contain p, so δ_p(B) = 0.

We conclude that supp(δ_p) is the closure of the singleton set {p}, which is {p} itself.

In fact, a measure μ on the real line is a Dirac measure δ_p for some point p if and only if the support of μ is the singleton set {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 μ on the real line R defined by

${\displaystyle \mu (A):=\lambda (A\cap (0,1))}$

i.e. a uniform measure on the open interval (0, 1). A similar argument to the Dirac measure example shows that supp(μ) = [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 (0, 1), and so must have positive μ-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 0. An example of this is given by adding the first uncountable ordinal Ω to the previous example: the support of the measure is the single point Ω, which has measure 0.

Signed and complex measures[]

Suppose that μ : Σ → [−∞, +∞] is a signed measure. Use the Hahn decomposition theorem to write

${\displaystyle \mu =\mu ^{+}-\mu ^{-},}$

where μ^± are both non-negative measures. Then the support of μ is defined to be

${\displaystyle \operatorname {supp} (\mu ):=\operatorname {supp} (\mu ^{+})\cup \operatorname {supp} (\mu ^{-}).}$

Similarly, if μ : Σ → C is a complex measure, the support of μ is defined to be the union of the supports of its real and imaginary parts.

References[]

• 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.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)