Support (measure theory)

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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 [math]\displaystyle{ \mu }[/math] on a measurable space [math]\displaystyle{ (X, \Sigma) }[/math] is really a function [math]\displaystyle{ \mu : \Sigma \to [0, +\infty] }[/math]. Therefore, in terms of the usual definition of support, the support of [math]\displaystyle{ \mu }[/math] is a subset of the σ-algebra [math]\displaystyle{ \Sigma }[/math]:

[math]\displaystyle{ \operatorname{supp} (\mu) := \overline{\{A \in \Sigma \,\vert\, \mu(A) \neq 0\}}, }[/math]

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 [math]\displaystyle{ \Sigma }[/math]. What we really want to know is where in the space [math]\displaystyle{ X }[/math] the measure [math]\displaystyle{ \mu }[/math] is non-zero. Consider two examples:

  1. Lebesgue measure [math]\displaystyle{ \lambda }[/math] on the real line [math]\displaystyle{ \mathbb{R} }[/math]. It seems clear that [math]\displaystyle{ \lambda }[/math] "lives on" the whole of the real line.
  2. A Dirac measure [math]\displaystyle{ \delta_p }[/math] at some point [math]\displaystyle{ p \in \mathbb{R} }[/math]. Again, intuition suggests that the measure [math]\displaystyle{ \delta_p }[/math] "lives at" the point [math]\displaystyle{ 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:

  1. We could remove the points where [math]\displaystyle{ \mu }[/math] is zero, and take the support to be the remainder [math]\displaystyle{ X\setminus \{ x \in X \mid \mu (\{x\}) = 0 \} }[/math]. This might work for the Dirac measure [math]\displaystyle{ \delta_p }[/math], but it would definitely not work for [math]\displaystyle{ \lambda }[/math]: since the Lebesgue measure of any singleton is zero, this definition would give [math]\displaystyle{ \lambda }[/math] 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:
[math]\displaystyle{ \{ x \in X \mid \exists N_x \text{ open} \colon (x \in N_x \wedge \mu(N_x) \gt 0) \} }[/math]
(or the closure of this). It is also too simplistic: by taking [math]\displaystyle{ N_x = X }[/math] for all points [math]\displaystyle{ x \in X }[/math], this would make the support of every measure except the zero measure the whole of [math]\displaystyle{ X }[/math].

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

Definition

Let (XT) 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 Nx of x has positive measure:

[math]\displaystyle{ \operatorname{supp} (\mu) := \{ x \in X \mid \forall N_x \in T \colon (x \in N_x \Rightarrow \mu (N_x) \gt 0) \}. }[/math]

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:

[math]\displaystyle{ (\forall U \in T)(U \cap C \neq \varnothing \implies \mu (U \cap C) \gt 0). }[/math]

Properties

  • [math]\displaystyle{ \operatorname{supp} (\mu_1 + \mu_2) = \operatorname{supp} (\mu_1) \cup \operatorname{supp} (\mu_2) }[/math]
  • 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 Hausdorff topological space and μ is a Radon measure, a measurable set A outside the support has measure zero:
[math]\displaystyle{ A \subseteq X \setminus \operatorname{supp} (\mu) \implies \mu (A) = 0. }[/math]
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,
[math]\displaystyle{ \int_X f(x) \, \mathrm{d} \mu (x) = \int_{\operatorname{supp} (\mu)} f(x) \, \mathrm{d} \mu (x). }[/math]
  • 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 [math]\displaystyle{ \mu }[/math] is a regular Borel measure on the line [math]\displaystyle{ \mathbb{R} }[/math], then the multiplication operator [math]\displaystyle{ (Af)(x)=xf(x) }[/math] is self-adjoint on its natural domain
[math]\displaystyle{ D(A)=\{f \in L^2(\mathbb{R}, d\mu) \mid xf(x)\in L^2(\mathbb{R}, d\mu)\} }[/math]
and its spectrum coincides with the essential range of the identity function [math]\displaystyle{ x \mapsto x }[/math], which is precisely the support of [math]\displaystyle{ \mu }[/math].[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 Nx of x must contain some open interval (x − εx + ε) for some ε > 0. This interval has Lebesgue measure 2ε > 0, so λ(Nx) ≥ 2ε > 0. Since x ∈ R was arbitrary, supp(λ) = R.

Dirac measure

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

  1. if x = p, then every open neighbourhood Nx of x contains p, so δp(Nx) = 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 δp(B) = 0.

We conclude that supp(δp) 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 δp 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

[math]\displaystyle{ \mu (A) := \lambda (A \cap (0, 1)) }[/math]

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

[math]\displaystyle{ \mu = \mu^{+} - \mu^{-}, }[/math]

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

[math]\displaystyle{ \operatorname{supp} (\mu) := \operatorname{supp} (\mu^{+}) \cup \operatorname{supp} (\mu^{-}). }[/math]

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

  1. Mathematical methods in Quantum Mechanics with applications to Schrödinger Operators
  • 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. 
  • 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. https://www.mat.univie.ac.at/~gerald/ftp/book-schroe/. (See chapter 3, section 2)