# 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:

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:

- Lebesgue measure
*λ*on the real line**R**. It seems clear that*λ*"lives on" the whole of the real line. - 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:

- 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. - 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:

- (or the closure of this). It is also too simplistic: by taking
*N*_{x}=*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 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 closed set *C* ⊆ *X* (with respect to inclusion) such that every open set which 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:

- 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**,

- 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 is a regular Borel measure on the line , 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 .
^{[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

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

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

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

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