# 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 [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:

- Lebesgue measure [math]\displaystyle{ \lambda }[/math] on the real line [math]\displaystyle{ \mathbb{R} }[/math]. It seems clear that [math]\displaystyle{ \lambda }[/math] "lives on" the whole of the real line.
- A Dirac measure [math]\displaystyle{ \delta_p }[/math] at some point [math]\displaystyle{ p \in \mathbb{R} }[/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:

- 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.
- 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} \colon (x \in N_x \wedge \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 (*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* ∈ *T*. Let *μ* be a measure on (*X*, B(*T*)). Then the **support** (or **spectrum**) of *μ* is defined as the set of all points *x* in *X* for which every open neighbourhood *N*_{x} of *x* 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 *C* ∈ 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:

- [math]\displaystyle{ (\forall U \in T)(U \cap C \neq \varnothing \implies \mu (U \cap C) \gt 0). }[/math]

## Properties

- [math]\displaystyle{ \operatorname{supp} (\mu_1 + \mu_2) = \operatorname{supp} (\mu_1) \cup \operatorname{supp} (\mu_2) }[/math]
- 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 Hausdorff topological space and*μ*is a Radon measure, a measurable set*A*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
*A*is open, but it 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**,- [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(\mathbb{R}, d\mu) \mid xf(x)\in L^2(\mathbb{R}, 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].
^{[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 *N*_{x} of *x* must contain some open interval (*x* − *ε*, *x* + *ε*) for some *ε* > 0. This interval has Lebesgue measure 2*ε* > 0, so *λ*(*N*_{x}) ≥ 2*ε* > 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:

- if
*x*=*p*, then every open neighbourhood*N*_{x}of*x*contains*p*, so*δ*_{p}(*N*_{x}) = 1 > 0; - 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

- [math]\displaystyle{ \mu (A) := \lambda (A \cap (0, 1)) }[/math]

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 ("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 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

- [math]\displaystyle{ \mu = \mu^{+} - \mu^{-}, }[/math]

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

- [math]\displaystyle{ \operatorname{supp} (\mu) := \operatorname{supp} (\mu^{+}) \cup \operatorname{supp} (\mu^{-}). }[/math]

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

- ↑ Mathematical methods in Quantum Mechanics with applications to Schrödinger Operators

- 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)

Original source: https://en.wikipedia.org/wiki/Support (measure theory).
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