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 μ on a measurable space (X, Σ) is really a function μ : Σ → [0, +∞]. Therefore, in terms of the usual definition of support, the support of μ is a subset of the σ-algebra Σ:

$\mathrm {supp} (\mu ):={\overline {\{A\in \Sigma |\mu (A)>0\}}}.$

However, this definition is somewhat unsatisfactory: we do not even have a topology on Σ! What we really want to know is where in the space X the measure μ is non-zero. Consider two examples:

1. Lebesgue measure λ on the real line R. It seems clear that λ "lives on" the whole of the real line.
2. A Dirac measure δ_p at some point p ∈ R. Again, intuition suggests that the measure δ_p "lives at" the point 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 μ is zero, and take the support to be the remainder X \ { x ∈ X | μ({x}) ≠ 0 }. This might work for the Dirac measure δ_p, but it would definitely not work for λ: since the Lebesgue measure of any point is zero, this definition would give λ 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:

$\{x\in X|{\mbox{for some open }}N_{x}\ni x,\mu (N_{x})>0\}$

(or the closure of this). This is also too simplistic: by taking N_{x<- this should be lower case} = X for all points x ∈ X, this would make the support of every measure except the zero measure the whole of 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 Borel(X) 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, Borel(X)). Then the support (or spectrum) of μ is defined to be the set of all points x in X for which every open neighbourhood N_x of x has positive measure:

$\mathrm {supp} (\mu ):=\{x\in X|x\in N_{x}\in T\implies \mu (N_{x})>0\}.$

Some authors prefer to take the closure of the above set. However, this is not necessary: see "Properties" below. As such, an equivalent definition of the support is as the largest closed set C ⊆ X (with respect to inclusion) such that

$U\in T{\mbox{ and }}U\cap C\neq \varnothing \implies \mu (U\cap C)>0,$

i.e. every open set that has non-trivial intersection with the support has positive measure.

Properties

• 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 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. Suppose that x is a limit point of supp(μ), and let N_x be an open neighbourhood of x. Since x is a limit point of the support, there is some y ∈ N_x ∩ supp(μ), y ≠ x. But N_x is also an open neighbourhood of y, so μ(N_x) > 0, as required. Hence, supp(μ) contains all its limit points, i.e. it is closed.
• If A is a measurable set outside the support, then A has measure zero:

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

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

• One does not need to "integrate outside the support": for any measurable function f : X → R or C,

$\int _{X}f(x)\,\mathrm {d} \mu (x)=\int _{\mathrm {supp} (\mu )}f(x)\,\mathrm {d} \mu (x).$

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

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

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.

Signed and complex measures

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

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

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

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

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.{{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.)

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