Support (measure theory)

In mathematics, the support (sometimes topological support or spectrum) of a measure [math]\displaystyle{ \mu }[/math] on a measurable topological space [math]\displaystyle{ (X, \operatorname{Borel}(X)) }[/math] is a precise notion of where in the space [math]\displaystyle{ X }[/math] the measure "lives". It is defined to be the largest (closed) subset of [math]\displaystyle{ X }[/math] 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{ \Reals. }[/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 \Reals. }[/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} \text{ such that } (x \in N_x \text{ and } \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 [math]\displaystyle{ (X, T) }[/math] be a topological space; let [math]\displaystyle{ B(T) }[/math] denote the Borel σ-algebra on [math]\displaystyle{ X, }[/math] i.e. the smallest sigma algebra on [math]\displaystyle{ X }[/math] that contains all open sets [math]\displaystyle{ U \in T. }[/math] Let [math]\displaystyle{ \mu }[/math] be a measure on [math]\displaystyle{ (X, B(T)) }[/math] Then the support (or spectrum) of [math]\displaystyle{ \mu }[/math] is defined as the set of all points [math]\displaystyle{ x }[/math] in [math]\displaystyle{ X }[/math] for which every open neighbourhood [math]\displaystyle{ N_x }[/math] of [math]\displaystyle{ x }[/math] 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 [math]\displaystyle{ C \in B(T) }[/math] (with respect to inclusion) such that every open set which has non-empty intersection with [math]\displaystyle{ C }[/math] has positive measure, i.e. the largest [math]\displaystyle{ C }[/math] such that: [math]\displaystyle{ (\forall U \in T)(U \cap C \neq \varnothing \implies \mu (U \cap C) \gt 0). }[/math]

Signed and complex measures

This definition can be extended to signed and complex measures. Suppose that [math]\displaystyle{ \mu : \Sigma \to [-\infty, +\infty] }[/math] is a signed measure. Use the Hahn decomposition theorem to write [math]\displaystyle{ \mu = \mu^+ - \mu^-, }[/math] where [math]\displaystyle{ \mu^\pm }[/math] are both non-negative measures. Then the support of [math]\displaystyle{ \mu }[/math] is defined to be [math]\displaystyle{ \operatorname{supp} (\mu) := \operatorname{supp} (\mu^+) \cup \operatorname{supp} (\mu^-). }[/math]

Similarly, if [math]\displaystyle{ \mu : \Sigma \to \Complex }[/math] is a complex measure, the support of [math]\displaystyle{ \mu }[/math] is defined to be the union of the supports of its real and imaginary parts.

Properties

[math]\displaystyle{ \operatorname{supp} (\mu_1 + \mu_2) = \operatorname{supp} (\mu_1) \cup \operatorname{supp} (\mu_2) }[/math] holds.

A measure [math]\displaystyle{ \mu }[/math] on [math]\displaystyle{ X }[/math] is strictly positive if and only if it has support [math]\displaystyle{ \operatorname{supp}(\mu) = X. }[/math] If [math]\displaystyle{ \mu }[/math] is strictly positive and [math]\displaystyle{ x \in X }[/math] is arbitrary, then any open neighbourhood of [math]\displaystyle{ x, }[/math] since it is an open set, has positive measure; hence, [math]\displaystyle{ x \in \operatorname{supp}(\mu), }[/math] so [math]\displaystyle{ \operatorname{supp}(\mu) = X. }[/math] Conversely, if [math]\displaystyle{ \operatorname{supp}(\mu) = X, }[/math] 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, [math]\displaystyle{ \mu }[/math] is strictly positive. The support of a measure is closed in [math]\displaystyle{ X, }[/math]as its complement is the union of the open sets of measure [math]\displaystyle{ 0. }[/math]

In general the support of a nonzero measure may be empty: see the examples below. However, if [math]\displaystyle{ X }[/math] is a Hausdorff topological space and [math]\displaystyle{ \mu }[/math] is a Radon measure, a Borel set [math]\displaystyle{ A }[/math] 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 [math]\displaystyle{ A }[/math] is open, but it is not true in general: it fails if there exists a point [math]\displaystyle{ x \in \operatorname{supp}(\mu) }[/math] such that [math]\displaystyle{ \mu(\{x\}) = 0 }[/math] (e.g. Lebesgue measure). Thus, one does not need to "integrate outside the support": for any measurable function [math]\displaystyle{ f : X \to \Reals }[/math] or [math]\displaystyle{ \Complex, }[/math] [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(\Reals, d\mu) \mid xf(x) \in L^2(\Reals, 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]

Examples

Lebesgue measure

In the case of Lebesgue measure [math]\displaystyle{ \lambda }[/math] on the real line [math]\displaystyle{ \Reals, }[/math] consider an arbitrary point [math]\displaystyle{ x \in \Reals. }[/math] Then any open neighbourhood [math]\displaystyle{ N_x }[/math] of [math]\displaystyle{ x }[/math] must contain some open interval [math]\displaystyle{ (x - \epsilon, x + \epsilon) }[/math] for some [math]\displaystyle{ \epsilon \gt 0. }[/math] This interval has Lebesgue measure [math]\displaystyle{ 2 \epsilon \gt 0, }[/math] so [math]\displaystyle{ \lambda(N_x) \geq 2 \epsilon \gt 0. }[/math] Since [math]\displaystyle{ x \in \Reals }[/math] was arbitrary, [math]\displaystyle{ \operatorname{supp}(\lambda) = \Reals. }[/math]

Dirac measure

In the case of Dirac measure [math]\displaystyle{ \delta_p, }[/math] let [math]\displaystyle{ x \in \Reals }[/math] and consider two cases:

1. if [math]\displaystyle{ x = p, }[/math] then every open neighbourhood [math]\displaystyle{ N_x }[/math] of [math]\displaystyle{ x }[/math] contains [math]\displaystyle{ p, }[/math] so [math]\displaystyle{ \delta_p(N_x) = 1 \gt 0. }[/math]
2. on the other hand, if [math]\displaystyle{ x \neq p, }[/math] then there exists a sufficiently small open ball [math]\displaystyle{ B }[/math] around [math]\displaystyle{ x }[/math] that does not contain [math]\displaystyle{ p, }[/math] so [math]\displaystyle{ \delta_p(B) = 0. }[/math]

We conclude that [math]\displaystyle{ \operatorname{supp}(\delta_p) }[/math] is the closure of the singleton set [math]\displaystyle{ \{p\}, }[/math] which is [math]\displaystyle{ \{p\} }[/math] itself.

In fact, a measure [math]\displaystyle{ \mu }[/math] on the real line is a Dirac measure [math]\displaystyle{ \delta_p }[/math] for some point [math]\displaystyle{ p }[/math] if and only if the support of [math]\displaystyle{ \mu }[/math] is the singleton set [math]\displaystyle{ \{p\}. }[/math] 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 [math]\displaystyle{ \mu }[/math] on the real line [math]\displaystyle{ \Reals }[/math] defined by [math]\displaystyle{ \mu(A) := \lambda(A \cap (0, 1)) }[/math] i.e. a uniform measure on the open interval [math]\displaystyle{ (0, 1). }[/math] A similar argument to the Dirac measure example shows that [math]\displaystyle{ \operatorname{supp}(\mu) = [0, 1]. }[/math] 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 [math]\displaystyle{ (0, 1), }[/math] and so must have positive [math]\displaystyle{ \mu }[/math]-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 [math]\displaystyle{ 0. }[/math] An example of this is given by adding the first uncountable ordinal [math]\displaystyle{ \Omega }[/math] to the previous example: the support of the measure is the single point [math]\displaystyle{ \Omega, }[/math] which has measure [math]\displaystyle{ 0. }[/math]

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. 
• 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)