Support (measure theory)

Concept in mathematics

In mathematics, the support (sometimes topological support or spectrum) of a measure μ {\displaystyle \mu } on a measurable topological space ( X , Borel ⁡ ( X ) ) {\displaystyle (X,\operatorname {Borel} (X))} is a precise notion of where in the space  X {\displaystyle X} the measure "lives". It is defined to be the largest (closed) subset of  X {\displaystyle X} for which every open neighbourhood of every point of the set has positive measure.

## Motivation

[edit]

A (non-negative) measure  μ {\displaystyle \mu } on a measurable space  ( X , Σ ) {\displaystyle (X,\Sigma )} is really a function  μ : Σ -> [ 0 , + ∞ ] . {\displaystyle \mu :\Sigma \to [0,+\infty ].} Therefore, in terms of the usual definition of support, the support of  μ {\displaystyle \mu } is a subset of the σ-algebra Σ : {\displaystyle \Sigma :} supp ⁡ ( μ ) := { A ∈ Σ | μ ( A ) ≠ 0 } ¯ , {\displaystyle \operatorname {supp} (\mu ):={\overline {\\{A\in \Sigma \,\vert \,\mu (A)\neq 0\\}}},} 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  Σ . {\displaystyle \Sigma .} What we really want to know is where in the space  X {\displaystyle X} the measure  μ {\displaystyle \mu } is non-zero. Consider two examples:

1. Lebesgue measure λ {\displaystyle \lambda } on the real line R . {\displaystyle \mathbb {R} .} It seems clear that  λ {\displaystyle \lambda } "lives on" the whole of the real line.
2. A Dirac measure δ p {\displaystyle \delta _{p}} at some point  p ∈ R . {\displaystyle p\in \mathbb {R} .} Again, intuition suggests that the measure  δ p {\displaystyle \delta _{p}} "lives at" the point  p , {\displaystyle 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 {\displaystyle \mu } is zero, and take the support to be the remainder  X ∖ { x ∈ X ∣ μ ( { x } ) = 0 } . {\displaystyle X\setminus \\{x\in X\mid \mu (\\{x\\})=0\\}.} This might work for the Dirac measure  δ p , {\displaystyle \delta _{p},} but it would definitely not work for  λ : {\displaystyle \lambda :} since the Lebesgue measure of any singleton is zero, this definition would give  λ {\displaystyle \lambda } 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 ∈ X ∣ ∃ N x open such that ( x ∈ N x and μ ( N x ) > 0 ) } {\displaystyle \\{x\in X\mid \exists N_{x}{\text{ open}}{\text{ such that }}(x\in N_{x}{\text{ and }}\mu (N_{x})>0)\\}} (or the closure of this). It is also too simplistic: by taking  N x = X {\displaystyle N_{x}=X} for all points  x ∈ X , {\displaystyle x\in X,} this would make the support of every measure except the zero measure the whole of  X . {\displaystyle X.}

However, the idea of "local strict positivity" is not too far from a workable definition.

## Definition

[edit]

Let  ( X , T ) {\displaystyle (X,T)} be a topological space; let  B ( T ) {\displaystyle B(T)} denote the Borel σ-algebra on  X , {\displaystyle X,} i.e. the smallest sigma algebra on  X {\displaystyle X} that contains all open sets  U ∈ T . {\displaystyle U\in T.} Let  μ {\displaystyle \mu } be a measure on  ( X , B ( T ) ) {\displaystyle (X,B(T))} Then the support (or spectrum) of  μ {\displaystyle \mu } is defined as the set of all points  x {\displaystyle x} in  X {\displaystyle X} for which every open neighbourhood N x {\displaystyle N_{x}} of  x {\displaystyle x} has positive measure:  supp ⁡ ( μ ) := { x ∈ X ∣ ∀ N x ∈ T : ( x ∈ N x ⇒ μ ( N x ) > 0 ) } . {\displaystyle \operatorname {supp} (\mu ):=\\{x\in X\mid \forall N_{x}\in T\colon (x\in N_{x}\Rightarrow \mu (N_{x})>0)\\}.}

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 ) {\displaystyle C\in B(T)} (with respect to inclusion) such that every open set which has non-empty intersection with  C {\displaystyle C} has positive measure, i.e. the largest  C {\displaystyle C} such that:  ( ∀ U ∈ T ) ( U ∩ C ≠ ∅ ⟹ μ ( U ∩ C ) > 0 ) . {\displaystyle (\forall U\in T)(U\cap C\neq \varnothing \implies \mu (U\cap C)>0).}

### Signed and complex measures

[edit]

This definition can be extended to signed and complex measures. Suppose that  μ : Σ -> [ − ∞ , + ∞ ] {\displaystyle \mu :\Sigma \to [-\infty ,+\infty ]} is a signed measure. Use the Hahn decomposition theorem to write  μ = μ + − μ − , {\displaystyle \mu =\mu ^{+}-\mu ^{-},} where  μ ± {\displaystyle \mu ^{\pm }} are both non-negative measures. Then the support of  μ {\displaystyle \mu } is defined to be  supp ⁡ ( μ ) := supp ⁡ ( μ + ) ∪ supp ⁡ ( μ − ) . {\displaystyle \operatorname {supp} (\mu ):=\operatorname {supp} (\mu ^{+})\cup \operatorname {supp} (\mu ^{-}).}

Similarly, if  μ : Σ -> C {\displaystyle \mu :\Sigma \to \mathbb {C} } is a complex measure, the support of  μ {\displaystyle \mu } is defined to be the union of the supports of its real and imaginary parts.

## Properties

[edit]

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

A measure  μ {\displaystyle \mu } on  X {\displaystyle X} is strictly positive if and only if it has support  supp ⁡ ( μ ) = X . {\displaystyle \operatorname {supp} (\mu )=X.} If  μ {\displaystyle \mu } is strictly positive and  x ∈ X {\displaystyle x\in X} is arbitrary, then any open neighbourhood of  x , {\displaystyle x,} since it is an open set, has positive measure; hence,  x ∈ supp ⁡ ( μ ) , {\displaystyle x\in \operatorname {supp} (\mu ),} so  supp ⁡ ( μ ) = X . {\displaystyle \operatorname {supp} (\mu )=X.} Conversely, if  supp ⁡ ( μ ) = X , {\displaystyle \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,  μ {\displaystyle \mu } is strictly positive. The support of a measure is closed in  X , {\displaystyle X,} as its complement is the union of the open sets of measure  0. {\displaystyle 0.}

In general the support of a nonzero measure may be empty: see the examples below. However, if  X {\displaystyle X} is a Hausdorff topological space and  μ {\displaystyle \mu } is a Radon measure, a Borel set  A {\displaystyle A} outside the support has measure zero:  A ⊆ X ∖ supp ⁡ ( μ ) ⟹ μ ( A ) = 0. {\displaystyle A\subseteq X\setminus \operatorname {supp} (\mu )\implies \mu (A)=0.} The converse is true if  A {\displaystyle A} is open, but it is not true in general: it fails if there exists a point  x ∈ supp ⁡ ( μ ) {\displaystyle x\in \operatorname {supp} (\mu )} such that  μ ( { x } ) = 0 {\displaystyle \mu (\\{x\\})=0} (e.g. Lebesgue measure). Thus, one does not need to "integrate outside the support": for any measurable function f : X -> R {\displaystyle f:X\to \mathbb {R} } or  C , {\displaystyle \mathbb {C} ,} ∫ X f ( x ) d μ ( x ) = ∫ supp ⁡ ( μ ) f ( x ) d μ ( x ) . {\displaystyle \int _{X}f(x)\,\mathrm {d} \mu (x)=\int _{\operatorname {supp} (\mu )}f(x)\,\mathrm {d} \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  μ {\displaystyle \mu } is a regular Borel measure on the line  R , {\displaystyle \mathbb {R} ,} then the multiplication operator  ( A f ) ( x ) = x f ( x ) {\displaystyle (Af)(x)=xf(x)} is self-adjoint on its natural domain  D ( A ) = { f ∈ L 2 ( R , d μ ) ∣ x f ( x ) ∈ L 2 ( R , d μ ) } {\displaystyle 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 ↦ x , {\displaystyle x\mapsto x,} which is precisely the support of  μ . {\displaystyle \mu .} [1]

## Examples

[edit]

### Lebesgue measure

[edit]

In the case of Lebesgue measure  λ {\displaystyle \lambda } on the real line  R , {\displaystyle \mathbb {R} ,} consider an arbitrary point  x ∈ R . {\displaystyle x\in \mathbb {R} .} Then any open neighbourhood  N x {\displaystyle N_{x}} of  x {\displaystyle x} must contain some open interval ( x − ϵ , x + ϵ ) {\displaystyle (x-\epsilon ,x+\epsilon )} for some  ϵ > 0. {\displaystyle \epsilon >0.} This interval has Lebesgue measure  2 ϵ > 0 , {\displaystyle 2\epsilon >0,} so  λ ( N x ) ≥ 2 ϵ > 0. {\displaystyle \lambda (N_{x})\geq 2\epsilon >0.} Since  x ∈ R {\displaystyle x\in \mathbb {R} } was arbitrary,  supp ⁡ ( λ ) = R . {\displaystyle \operatorname {supp} (\lambda )=\mathbb {R} .}

### Dirac measure

[edit]

In the case of Dirac measure δ p , {\displaystyle \delta _{p},} let  x ∈ R {\displaystyle x\in \mathbb {R} } and consider two cases:

1. if  x = p , {\displaystyle x=p,} then every open neighbourhood  N x {\displaystyle N_{x}} of  x {\displaystyle x} contains  p , {\displaystyle p,} so  δ p ( N x ) = 1 > 0. {\displaystyle \delta _{p}(N_{x})=1>0.}
2. on the other hand, if  x ≠ p , {\displaystyle x\neq p,} then there exists a sufficiently small open ball  B {\displaystyle B} around  x {\displaystyle x} that does not contain  p , {\displaystyle p,} so  δ p ( B ) = 0. {\displaystyle \delta _{p}(B)=0.}

We conclude that  supp ⁡ ( δ p ) {\displaystyle \operatorname {supp} (\delta _{p})} is the closure of the singleton set  { p } , {\displaystyle \\{p\\},} which is  { p } {\displaystyle \\{p\\}} itself.

In fact, a measure  μ {\displaystyle \mu } on the real line is a Dirac measure  δ p {\displaystyle \delta _{p}} for some point  p {\displaystyle p} if and only if the support of  μ {\displaystyle \mu } is the singleton set  { p } . {\displaystyle \\{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

[edit]

Consider the measure  μ {\displaystyle \mu } on the real line  R {\displaystyle \mathbb {R} } defined by  μ ( A ) := λ ( A ∩ ( 0 , 1 ) ) {\displaystyle \mu (A):=\lambda (A\cap (0,1))} i.e. a uniform measure on the open interval  ( 0 , 1 ) . {\displaystyle (0,1).} A similar argument to the Dirac measure example shows that  supp ⁡ ( μ ) = [ 0 , 1 ] . {\displaystyle \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 ) , {\displaystyle (0,1),} and so must have positive  μ {\displaystyle \mu } -measure.

### A nontrivial measure whose support is empty

[edit]

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. [2]

### A nontrivial measure whose support has measure zero

[edit]

On a compact Hausdorff space the support of a non-zero measure is always non-empty, but may have measure  0. {\displaystyle 0.} An example of this is given by adding the first uncountable ordinal  Ω {\displaystyle \Omega } to the previous example: the support of the measure is the single point  Ω , {\displaystyle \Omega ,} which has measure  0. {\displaystyle 0.}

## References

[edit]

1. ^ Mathematical methods in Quantum Mechanics with applications to Schrödinger Operators
2. ^ Measure theory, Example 7.1.3

* 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.`{{cite book}}`: CS1 maint: multiple names: authors list (link)
* Bogachev, V. I. (2007). Measure theory. Vol. 2\. Springer Berlin Heidelberg. ISBN 978-3-540-34514-5.
* Parthasarathy, K. R. (2005). Probability measures on metric spaces. AMS Chelsea Publishing, Providence, RI. p. xii+276. ISBN 0-8218-3889-X. MR 2169627 (See chapter 2, section 2)
* Teschl, Gerald (2009). Mathematical methods in Quantum Mechanics with applications to Schrödinger Operators. AMS.(See chapter 3, section 2)

* v
* t
* e

Measure theory

Basic concepts|

* Absolute continuity of measures
* Lebesgue integration
* Lp spaces
* Measure
* Measure space
* Probability space
* Measurable space/function

Sets|

* Almost everywhere
* Atom
* Baire set
* Borel set
* equivalence relation
* Borel space
* Carathéodory's criterion
* Cylindrical σ-algebra
* Cylinder set
* 𝜆-system
* Essential range
* infimum/supremum
* Locally measurable
* π-system
* σ-algebra
* Non-measurable set
* Vitali set
* Null set
* Support
* Transverse measure
* Universally measurable

Types of measures|

* Atomic
* Baire
* Banach
* Besov
* Borel
* Brown
* Complex
* Complete
* Content
* (Logarithmically) Convex
* Decomposable
* Discrete
* Equivalent
* Finite
* Inner
* (Quasi-) Invariant
* Locally finite
* Maximising
* Metric outer
* Outer
* Perfect
* Pre-measure
* (Sub-) Probability
* Projection-valued
* Radon
* Random
* Regular
* Borel regular
* Inner regular
* Outer regular
* Saturated
* Set function
* σ-finite
* s-finite
* Signed
* Singular
* Spectral
* Strictly positive
* Tight
* Vector

Particular measures|

* Counting
* Dirac
* Euler
* Gaussian
* Haar
* Harmonic
* Hausdorff
* Intensity
* Lebesgue
* Infinite-dimensional
* Logarithmic
* Product
* Projections
* Pushforward
* Spherical measure
* Tangent
* Trivial
* Young

Maps|

* Measurable function
* Bochner
* Strongly
* Weakly
* Convergence: almost everywhere
* of measures
* in measure
* of random variables
* in distribution
* in probability
* Cylinder set measure
* Random: compact set
* element
* measure
* process
* variable
* vector
* Projection-valued measure

Main results|

* Carathéodory's extension theorem
* Convergence theorems
* Dominated
* Monotone
* Vitali
* Decomposition theorems
* Hahn
* Jordan
* Maharam's
* Egorov's
* Fatou's lemma
* Fubini's
* Fubini–Tonelli
* Hölder's inequality
* Minkowski inequality
* Radon–Nikodym
* Riesz–Markov–Kakutani representation theorem

Other results|

* Disintegration theorem
* Lifting theory
* Lebesgue's density theorem
* Lebesgue differentiation theorem
* Sard's theorem
* Vitali–Hahn–Saks theorem

For Lebesgue measure|

* Isoperimetric inequality
* Brunn–Minkowski theorem
* Milman's reverse
* Minkowski–Steiner formula
* Prékopa–Leindler inequality
* Vitale's random Brunn–Minkowski inequality

Applications & related|

* Convex analysis
* Descriptive set theory
* Probability theory
* Real analysis
* Spectral theory

*[v]: View this template
*[t]: Discuss this template
*[e]: Edit this template