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Support (measure theory)

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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 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 \mid \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 δp at some point p ∈ R. Again, intuition suggests that the measure δp "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 δp, 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 \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 Nx = 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 (XT) 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 Nx of x has positive measure:

\mathrm{supp} (\mu) := \{ x \in X \mid 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 \text{ 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 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:
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).
Thus, 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).
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 x \mapsto x, which is precisely the support of \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 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

\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 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

\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

  1. ^ Mathematical methods in Quantum Mechanics with applications to Schrödinger Operators
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