From Wikipedia, the free encyclopedia
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.
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:
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.
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.
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]
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} .}
In the case of Dirac measure δ p , {\displaystyle \delta _{p},} let x ∈ R {\displaystyle x\in \mathbb {R} } and consider two cases:
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.}
{{cite book}}: CS1 maint: multiple names: authors list (link)RetroSearch is an open source project built by @garambo | Open a GitHub Issue
Search and Browse the WWW like it's 1997 | Search results from DuckDuckGo
HTML:
3.2
| Encoding:
UTF-8
| Version:
0.7.4