Support (measure theory) explained

\mu on a measurable topological space (X, \operatorname{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

\mu on a measurable space (X, \Sigma) is really a function \mu : \Sigma \to [ 0 , +inf ty]. Therefore, in terms of the usual definition of support, the support of \mu is a subset of the σ-algebra \Sigma: $$\backslash operatorname\; (\backslash mu)\; :=\; \backslash overline,$$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:λ on the real line \Reals. It seems clear that λ "lives on" the whole of the real line.\delta _{p} at some point p \in \Reals. Again, intuition suggests that the measure \delta _{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

\mu is zero, and take the support to be the remainder X \setminus \{x \in X \mid \mu(\{x\}) = 0 \}. This might work for the Dirac measure \delta _{p,} but it would definitely not work for λ : since the Lebesgue measure of any singleton is zero, this definition would give λ empty support.

1. 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: $$\backslash$$ (or the closure of this). It is also too simplistic: by taking

N _{x} = X for all points x \in 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 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:$$\backslash operatorname\; (\backslash mu)\; :=\; \backslash .$$

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:$$(\backslash forall\; U\; \backslash in\; T)(U\; \backslash cap\; C\; \backslash neq\; \backslash varnothing\; \backslash implies\; \backslash mu\; (U\; \backslash cap\; C)\; >\; 0).$$

Signed and complex measures

This definition can be extended to signed and complex measures. Suppose that

\mu : \Sigma \to [ -inf ty, +inf ty] is a signed measure. Use the Hahn decomposition theorem to write$$\backslash mu\; =\; \backslash mu^+\; -\; \backslash mu^-,$$where \mu ^{\pm} are both non-negative measures. Then the support of \mu is defined to be$$\backslash operatorname\; (\backslash mu)\; :=\; \backslash operatorname\; (\backslash mu^+)\; \backslash cup\; \backslash operatorname\; (\backslash mu^-).$$

Similarly, if

\mu : \Sigma \to \Complex is a complex measure, the support of \mu is defined to be the union of the supports of its real and imaginary parts.

Properties

\operatorname{supp} (\mu _{1} + \mu _{2)} = \operatorname{supp} (\mu _{1)} \cup \operatorname{supp} (\mu _{2)} holds.

A measure

\mu on X is strictly positive if and only if it has support \operatorname{supp}(\mu) = X. 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, x \in \operatorname{supp}(\mu), so \operatorname{supp}(\mu) = X. Conversely, if \operatorname{supp}(\mu) = 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, \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: $$A\; \backslash subseteq\; X\; \backslash setminus\; \backslash operatorname\; (\backslash mu)\; \backslash implies\; \backslash mu\; (A)\; =\; 0.$$ The converse is true if A is open, but it is not true in general: it fails if there exists a point x \in \operatorname{supp}(\mu) such that \mu(\{x\}) = 0 (e.g. Lebesgue measure). Thus, one does not need to "integrate outside the support": for any measurable function f : X \to \Reals or \Complex, $$\backslash int\_X\; f(x)\; \backslash ,\; \backslash mathrm\; \backslash mu\; (x)\; =\; \backslash int\_\; f(x)\; \backslash ,\; \backslash mathrm\; \backslash mu\; (x).$$

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 R , then the multiplication operator (Af)(x) = xf(x) is self-adjoint on its natural domain $$D(A)\; =\; \backslash$$ 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 \Reals, consider an arbitrary point x \in \Reals. Then any open neighbourhood N _{x} of x must contain some open interval (x - \epsilon, x + \epsilon) for some \epsilon > 0 . This interval has Lebesgue measure 2 \epsilon > 0 , so λ (N _{x)} \geq 2 \epsilon > 0 . Since x \in \Reals was arbitrary, \operatorname{supp}( λ ) = \Reals.

Dirac measure

\delta _{p,} let x \in \Reals and consider two cases:

1. if

x = p, then every open neighbourhood N _{x} of x contains p, so \delta _{p(N} _{x)} = 1 > 0 .

1. on the other hand, if

x ≠ p, then there exists a sufficiently small open ball B around x that does not contain p, so \delta _{p(B)} = 0 . We conclude that \operatorname{supp}(\delta _{p)} 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$$\backslash mu(A)\; :=\; \backslash lambda(A\; \backslash cap\; (0,\; 1))$$i.e. a uniform measure on the open interval ( 0 , 1 ). A similar argument to the Dirac measure example shows that \operatorname{supp}(\mu) = [ 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 \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 .

References

• Book: Ambrosio, L., Gigli, N. & Savaré, G. . Gradient Flows in Metric Spaces and in the Space of Probability Measures . ETH Zürich, Birkhäuser Verlag, Basel . 2005 . 3-7643-2428-7.
• Book: Parthasarathy , K. R. . Probability measures on metric spaces. AMS Chelsea Publishing, Providence, RI. 2005. 0-8218-3889-X. xii+276. (See chapter 2, section 2.)
• Book: Teschl , Gerald . Mathematical methods in Quantum Mechanics with applications to Schrödinger Operators. AMS . 2009. (See chapter 3, section 2)

Notes and References

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