In mathematics, the support (sometimes topological support or spectrum) of a measure \mu on a measurable topological space 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 \mu on a measurable space (X, \Sigma) is really a function Therefore, in terms of the usual definition of support, the support of \mu is a subset of the σ-algebra \Sigma: 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 \Sigma. What we really want to know is where in the space X the measure \mu is non-zero. Consider two examples: In light of these two examples, we can reject the following candidate definitions in favour of the one in the next section: 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 \in T. Let \mu be a measure on (X, B(T)) Then the support (or spectrum) of \mu is defined as the set of all points x in X for which every open neighbourhood N_x of x has positive measure: 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 \in 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:

Signed and complex measures

This definition can be extended to signed and complex measures. Suppose that is a signed measure. Use the Hahn decomposition theorem to write where \mu^\pm are both non-negative measures. Then the support of \mu is defined to be Similarly, if is a complex measure, the support of \mu is defined to be the union of the supports of its real and imaginary parts.

Properties

holds. A measure \mu on X is strictly positive if and only if it has support If \mu is strictly positive and x \in X is arbitrary, then any open neighbourhood of x, since it is an open set, has positive measure; hence, so Conversely, if 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, \mu 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 \mu is a Radon measure, a Borel set A outside the support has measure zero: The converse is true if A is open, but it is not true in general: it fails if there exists a point such that (e.g. Lebesgue measure). Thus, one does not need to "integrate outside the support": for any measurable function or \Complex, 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 \mu is a regular Borel measure on the line \mathbb{R}, then the multiplication operator is self-adjoint on its natural domain and its spectrum coincides with the essential range of the identity function which is precisely the support of \mu.

Examples

Lebesgue measure

In the case of Lebesgue measure \lambda on the real line \Reals, consider an arbitrary point Then any open neighbourhood N_x of x must contain some open interval for some This interval has Lebesgue measure so Since was arbitrary,

Dirac measure

In the case of Dirac measure \delta_p, let and consider two cases: We conclude that is the closure of the singleton set {p}, which is {p} itself. In fact, a measure \mu on the real line is a Dirac measure \delta_p for some point p if and only if the support of \mu 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 \mu on the real line \Reals defined by i.e. a uniform measure on the open interval (0, 1). A similar argument to the Dirac measure example shows that 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 \mu-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 \Omega to the previous example: the support of the measure is the single point \Omega, which has measure 0.