Cylinder set measure
Encyclopedia
In mathematics
, cylinder set measure (or promeasure, or premeasure, or quasi-measure, or CSM) is a kind of prototype for a measure
on an infinite-dimensional vector space
. An example is the Gauss
ian cylinder set measure on Hilbert space
.
Cylinder set measures are in general not measures (and in particular need not be countably additive
but only finitely additive
), but can be used to define measures, such as classical Wiener measure on the set of continuous paths starting at the origin in Euclidean space
.
, topological vector space
. Let
denote the collection of all surjective, continuous linear maps T : E → FT defined on E whose image is some finite-dimensional real vector space FT:
A cylinder set measure on E is a collection of probability measure
s
where μT is a probability measure on FT. These measures are required to satisfy the following consistency condition: if πST : FS → FT is a surjective projection
, then the measures push forward
as follows:
is modelled on the way that true measures push forward (see the section cylinder set measures versus true measures). However, it is important to understand that in the case of cylinder set measures, this is a requirement that is part of the definition, not a result.
A cylinder set measure can be intuitively understood as defining a finitely additive function on the cylinder set
s of the topological vector space E. The cylinder sets are the pre-images in E of measurable sets in FT: if
denotes the σ-algebra on FT on which μT is defined, then
In practice, one often takes
to be the Borel σ-algebra on FT. In this case, one can show that when E is a separable Banach space
, the σ-algebra generated by the cylinder sets is precisely the Borel σ-algebra of E:
When there is a measure μ on E such that μT = T∗(μ) in this way, it is customary to abuse notation
slightly and say that the cylinder set measure
"is" the measure μ.
H, there is a canonical
Gaussian cylinder set measure γH arising from the inner product structure on H. Specifically, if ⟨ , ⟩ denotes the inner product on H, let ⟨ , ⟩T denote the quotient inner product on FT. The measure γTH on FT is then defined to be the canonical Gaussian measure
on FT:
where i : Rdim(FT) → FT is an isometry
of Hilbert spaces taking the Euclidean
inner product on Rdim(FT) to the inner product ⟨ , ⟩T on FT, and γn is the standard Gaussian measure
on Rn.
The canonical Gaussian cylinder set measure on an infinite-dimensional separable Hilbert space H does not correspond to a true measure on H. The proof is quite simple: the ball of radius r (and center 0) has measure at most equal to that of the ball of radius r in an n-dimensional Hilbert space, and this tends to 0 as n tends to infinity. So the ball of radius r has measure 0; as the Hilbert space is a countable union of such balls it also has measure 0, which is a contradiction.
An alternative proof that the Gaussian cylinder set measure is not a measure uses the Cameron–Martin theorem and a result on the quasi-invariance of measures
. If γH = γ really were a measure, then the identity function
on H would radonify that measure, thus making id : H → H into an abstract Wiener space
. By the Cameron–Martin theorem, γ would then be quasi-invariant under translation by any element of H, which implies that either H is finite-dimensional or that γ is the zero measure. In either case, we have a contradiction.
automatically extends to a measure if its Fourier transform is continuous.
Example: Let S be the space of Schwartz function
s on a finite dimensional vector space; it is nuclear. It is contained in the Hilbert space H of L2 functions
, which is in turn contained in the space of tempered distribution
s S′, the dual of the nuclear
Fréchet space
S:
The Gaussian cylinder set measure on H gives a cylinder set measure on the space of tempered distributions, which extends to a measure on the space of tempered distributions, S′.
The Hilbert space H has measure 0 in S′, by the first argument used above to show that the canonical Gaussian cylinder set measure on H does not extend to a measure on H.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, cylinder set measure (or promeasure, or premeasure, or quasi-measure, or CSM) is a kind of prototype for a measure
Measure (mathematics)
In mathematical analysis, a measure on a set is a systematic way to assign to each suitable subset a number, intuitively interpreted as the size of the subset. In this sense, a measure is a generalization of the concepts of length, area, and volume...
on an infinite-dimensional vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
. An example is the Gauss
Gauss
Gauss may refer to:*Carl Friedrich Gauss, German mathematician and physicist*Gauss , a unit of magnetic flux density or magnetic induction*GAUSS , a software package*Gauss , a crater on the moon...
ian cylinder set measure on Hilbert space
Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
.
Cylinder set measures are in general not measures (and in particular need not be countably additive
Sigma additivity
In mathematics, additivity and sigma additivity of a function defined on subsets of a given set are abstractions of the intuitive properties of size of a set.- Additive set functions :...
but only finitely additive
Sigma additivity
In mathematics, additivity and sigma additivity of a function defined on subsets of a given set are abstractions of the intuitive properties of size of a set.- Additive set functions :...
), but can be used to define measures, such as classical Wiener measure on the set of continuous paths starting at the origin in Euclidean space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
.
Definition
Let E be a separable, realReal number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
, topological vector space
Topological vector space
In mathematics, a topological vector space is one of the basic structures investigated in functional analysis...
. Let
denote the collection of all surjective, continuous linear maps T : E → FT defined on E whose image is some finite-dimensional real vector space FT:A cylinder set measure on E is a collection of probability measure
Probability measure
In mathematics, a probability measure is a real-valued function defined on a set of events in a probability space that satisfies measure properties such as countable additivity...
s
where μT is a probability measure on FT. These measures are required to satisfy the following consistency condition: if πST : FS → FT is a surjective projection
Projection (mathematics)
Generally speaking, in mathematics, a projection is a mapping of a set which is idempotent, which means that a projection is equal to its composition with itself. A projection may also refer to a mapping which has a left inverse. Bot notions are strongly related, as follows...
, then the measures push forward
Pushforward measure
In measure theory, a pushforward measure is obtained by transferring a measure from one measurable space to another using a measurable function.-Definition:...
as follows:
Remarks
The consistency conditionis modelled on the way that true measures push forward (see the section cylinder set measures versus true measures). However, it is important to understand that in the case of cylinder set measures, this is a requirement that is part of the definition, not a result.
A cylinder set measure can be intuitively understood as defining a finitely additive function on the cylinder set
Cylinder set
In mathematics, a cylinder set is the natural open set of a product topology. Cylinder sets are particularly useful in providing the base of the natural topology of the product of a countable number of copies of a set...
s of the topological vector space E. The cylinder sets are the pre-images in E of measurable sets in FT: if
denotes the σ-algebra on FT on which μT is defined, thenIn practice, one often takes
to be the Borel σ-algebra on FT. In this case, one can show that when E is a separable Banach spaceBanach space
In mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...
, the σ-algebra generated by the cylinder sets is precisely the Borel σ-algebra of E:
Cylinder set measures versus true measures
A cylinder set measure on E is not actually a measure on E: it is a collection of measures defined on all finite-dimensional images of E. If E has a probability measure μ already defined on it, then μ gives rise to a cylinder set measure on E using the push forward: set μT = T∗(μ) on FT.When there is a measure μ on E such that μT = T∗(μ) in this way, it is customary to abuse notation
Abuse of notation
In mathematics, abuse of notation occurs when an author uses a mathematical notation in a way that is not formally correct but that seems likely to simplify the exposition or suggest the correct intuition . Abuse of notation should be contrasted with misuse of notation, which should be avoided...
slightly and say that the cylinder set measure
"is" the measure μ.Cylinder set measures on Hilbert spaces
When the Banach space E is actually a Hilbert spaceHilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
H, there is a canonical
Canonical
Canonical is an adjective derived from canon. Canon comes from the greek word κανών kanon, "rule" or "measuring stick" , and is used in various meanings....
Gaussian cylinder set measure γH arising from the inner product structure on H. Specifically, if ⟨ , ⟩ denotes the inner product on H, let ⟨ , ⟩T denote the quotient inner product on FT. The measure γTH on FT is then defined to be the canonical Gaussian measure
Gaussian measure
In mathematics, Gaussian measure is a Borel measure on finite-dimensional Euclidean space Rn, closely related to the normal distribution in statistics. There is also a generalization to infinite-dimensional spaces...
on FT:
where i : Rdim(FT) → FT is an isometry
Isometry
In mathematics, an isometry is a distance-preserving map between metric spaces. Geometric figures which can be related by an isometry are called congruent.Isometries are often used in constructions where one space is embedded in another space...
of Hilbert spaces taking the Euclidean
Euclidean
Euclidean relates to Euclid , a town or others. It may refer to:Geometry...
inner product on Rdim(FT) to the inner product ⟨ , ⟩T on FT, and γn is the standard Gaussian measure
Gaussian measure
In mathematics, Gaussian measure is a Borel measure on finite-dimensional Euclidean space Rn, closely related to the normal distribution in statistics. There is also a generalization to infinite-dimensional spaces...
on Rn.
The canonical Gaussian cylinder set measure on an infinite-dimensional separable Hilbert space H does not correspond to a true measure on H. The proof is quite simple: the ball of radius r (and center 0) has measure at most equal to that of the ball of radius r in an n-dimensional Hilbert space, and this tends to 0 as n tends to infinity. So the ball of radius r has measure 0; as the Hilbert space is a countable union of such balls it also has measure 0, which is a contradiction.
An alternative proof that the Gaussian cylinder set measure is not a measure uses the Cameron–Martin theorem and a result on the quasi-invariance of measures
Quasi-invariant measure
In mathematics, a quasi-invariant measure μ with respect to a transformation T, from a measure space X to itself, is a measure which, roughly speaking, is multiplied by a numerical function by T...
. If γH = γ really were a measure, then the identity function
Identity function
In mathematics, an identity function, also called identity map or identity transformation, is a function that always returns the same value that was used as its argument...
on H would radonify that measure, thus making id : H → H into an abstract Wiener space
Abstract Wiener space
An abstract Wiener space is a mathematical object in measure theory, used to construct a "decent" measure on an infinite-dimensional vector space. It is named after the American mathematician Norbert Wiener...
. By the Cameron–Martin theorem, γ would then be quasi-invariant under translation by any element of H, which implies that either H is finite-dimensional or that γ is the zero measure. In either case, we have a contradiction.
Nuclear spaces and cylinder set measures
A cylinder set measure on the dual of a nuclear Frechet spaceNuclear space
In mathematics, a nuclear space is a topological vector space with many of the good properties of finite-dimensional vector spaces. The topology on them can be defined by a family of seminorms whose unit balls decrease rapidly in size...
automatically extends to a measure if its Fourier transform is continuous.
Example: Let S be the space of Schwartz function
Distribution (mathematics)
In mathematical analysis, distributions are objects that generalize functions. Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense. In particular, any locally integrable function has a distributional derivative...
s on a finite dimensional vector space; it is nuclear. It is contained in the Hilbert space H of L2 functions
Lp space
In mathematics, the Lp spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces...
, which is in turn contained in the space of tempered distribution
Tempered distribution
*Distribution *Tempered representation...
s S′, the dual of the nuclear
Nuclear space
In mathematics, a nuclear space is a topological vector space with many of the good properties of finite-dimensional vector spaces. The topology on them can be defined by a family of seminorms whose unit balls decrease rapidly in size...
Fréchet space
Fréchet space
In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces. They are generalizations of Banach spaces...
S:
The Gaussian cylinder set measure on H gives a cylinder set measure on the space of tempered distributions, which extends to a measure on the space of tempered distributions, S′.
The Hilbert space H has measure 0 in S′, by the first argument used above to show that the canonical Gaussian cylinder set measure on H does not extend to a measure on H.