Cylinder set measure

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 Gaussian 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 the classical Wiener measure on the set of continuous paths starting at the origin in Euclidean space.

Definition

There are two equivalent ways to define a cylinder set measure.

One way is to define it directly as a set function on the cylindrical algebra such that certain restrictions on smaller σ-algebras are σ-finite measure. This can also be expressed in terms of a finite-dimensional linear operator.

Let $X$ be a topological vector space over $\mathbb {R}$ , denote its algebraic dual as $X^{*}$ and let $G\subseteq X^{*}$ be a subspace. Then the set function $\mu :{\mathcal {Cyl}}(X,X^{*})\to \mathbb {R} _{+}$ is a cylinder set measure if for any finite set $F=\{f_{1},\dots ,f_{n}\}\subset G$ the restriction to $\mu :\sigma ({\mathcal {Cyl}}(X,F))\to \mathbb {R} _{+}$ is a σ-finite measure. Notice that $\sigma ({\mathcal {Cyl}}(X,F))$ is a σ-algebra while ${\mathcal {Cyl}}(X,X^{*})$ is not.^[1]^[2]

Operatic definition

Let $E$ be a real topological vector space. Let ${\mathcal {A}}(E)$ denote the collection of all surjective continuous linear maps $T:E\to F_{T}$ defined on $E$ whose image is some finite-dimensional real vector space $F_{T}$ : ${\mathcal {A}}(E):=\left\{T\in \mathrm {Lin} (E;F_{T}):T{\mbox{ surjective and }}\dim _{\mathbb {R} }F_{T}<+\infty \right\}.$

A cylinder set measure on $E$ is a collection of probability measures $\left\{\mu _{T}:T\in {\mathcal {A}}(E)\right\}.$

where $\mu _{T}$ is a probability measure on $F_{T}.$ These measures are required to satisfy the following consistency condition: if $\pi _{ST}:F_{S}\to F_{T}$ is a surjective projection, then the push forward of the measure is as follows: $\mu _{T}=\left(\pi _{ST}\right)_{*}\left(\mu _{S}\right).$

Remarks

The consistency condition $\mu _{T}=\left(\pi _{ST}\right)_{*}(\mu _{S})$ 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 sets of the topological vector space $E.$ The cylinder sets are the pre-images in $E$ of measurable sets in $F_{T}$ : if ${\mathcal {B}}_{T}$ denotes the $\sigma$ -algebra on $F_{T}$ on which $\mu _{T}$ is defined, then $\mathrm {Cyl} (E):=\left\{T^{-1}(B):B\in {\mathcal {B}}_{T},T\in {\mathcal {A}}(E)\right\}.$

In practice, one often takes ${\mathcal {B}}_{T}$ to be the Borel $\sigma$ -algebra on $F_{T}.$ 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 $\sigma$ -algebra of $E$ : $\mathrm {Borel} (E)=\sigma \left(\mathrm {Cyl} (E)\right).$

Cylinder set measures versus true measures

A cylinder set measure on $E$ is not actually a true measure on $E$ : it is a collection of measures defined on all finite-dimensional images of $E.$ If $E$ has a probability measure $\mu$ already defined on it, then $\mu$ gives rise to a cylinder set measure on $E$ using the push forward: set $\mu _{T}=T_{*}(\mu )$ on $F_{T}.$

When there is a measure $\mu$ on $E$ such that $\mu _{T}=T_{*}(\mu )$ in this way, it is customary to abuse notation slightly and say that the cylinder set measure $\left\{\mu _{T}:T\in {\mathcal {A}}(E)\right\}$ "is" the measure $\mu .$

Cylinder set measures on Hilbert spaces

When the Banach space $E$ is also a Hilbert space $H,$ there is a canonical Gaussian cylinder set measure $\gamma ^{H}$ arising from the inner product structure on $H.$ Specifically, if $\langle \cdot ,\cdot \rangle$ denotes the inner product on $H,$ let $\langle \cdot ,\cdot \rangle _{T}$ denote the quotient inner product on $F_{T}.$ The measure $\gamma _{T}^{H}$ on $F_{T}$ is then defined to be the canonical Gaussian measure on $F_{T}$ : $\gamma _{T}^{H}:=i_{*}\left(\gamma ^{\dim F_{T}}\right),$ where $i:\mathbb {R} ^{\dim(F_{T})}\to F_{T}$ is an isometry of Hilbert spaces taking the Euclidean inner product on $\mathbb {R} ^{\dim(F_{T})}$ to the inner product $\langle \cdot ,\cdot \rangle _{T}$ on $F_{T},$ and $\gamma ^{n}$ is the standard Gaussian measure on $\mathbb {R} ^{n}.$

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. (See infinite dimensional Lebesgue measure.)

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 $\gamma ^{H}=\gamma$ really were a measure, then the identity function on $H$ would radonify that measure, thus making $\operatorname {id} :H\to H$ into an abstract Wiener space. By the Cameron–Martin theorem, $\gamma$ would then be quasi-invariant under translation by any element of $H,$ which implies that either $H$ is finite-dimensional or that $\gamma$ is the zero measure. In either case, we have a contradiction.

Sazonov's theorem gives conditions under which the push forward of a canonical Gaussian cylinder set measure can be turned into a true measure.

Nuclear spaces and cylinder set measures

A cylinder set measure on the dual of a nuclear Fréchet space automatically extends to a measure if its Fourier transform is continuous.

Example: Let $S$ be the space of Schwartz functions on a finite dimensional vector space; it is nuclear. It is contained in the Hilbert space $H$ of $L^{2}$ functions, which is in turn contained in the space of tempered distributions $S^{\prime },$ the dual of the nuclear Fréchet space $S$ : $S\subseteq H\subseteq S^{\prime }.$

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, $S^{\prime }.$

The Hilbert space $H$ has measure 0 in $S^{\prime },$ by the first argument used above to show that the canonical Gaussian cylinder set measure on $H$ does not extend to a measure on $H.$

References

^ Bogachev, Vladimir (1998). Gaussian Measures. Rhode Island: American Mathematical Society.
^ N. Vakhania, V. Tarieladze and S. Chobanyan (1987). Probability Distributions on Banach Spaces. Mathematics and its Applications. Springer Netherlands. p. 390. ISBN 9789027724960. LCCN 87004931.

I.M. Gel'fand, N.Ya. Vilenkin, Generalized functions. Applications of harmonic analysis, Vol 4, Acad. Press (1968)
R.A. Minlos (2001) [1994], "cylindrical measure", Encyclopedia of Mathematics, EMS Press
R.A. Minlos (2001) [1994], "cylinder set", Encyclopedia of Mathematics, EMS Press
L. Schwartz, Radon measures.

[1] Bogachev, Vladimir (1998). Gaussian Measures. Rhode Island: American Mathematical Society.

[2] N. Vakhania, V. Tarieladze and S. Chobanyan (1987). Probability Distributions on Banach Spaces. Mathematics and its Applications. Springer Netherlands. p. 390. ISBN 9789027724960. LCCN 87004931.

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v t e Analysis in topological vector spaces
Basic concepts	Abstract Wiener space Classical Wiener space Bochner space Convex series Cylinder set measure Infinite-dimensional vector function Matrix calculus Vector calculus
Derivatives	Differentiable vector–valued functions from Euclidean space Differentiation in Fréchet spaces Fréchet derivative Total Functional derivative Gateaux derivative Directional Generalizations of the derivative Hadamard derivative Holomorphic Quasi-derivative
Measurability	Besov measure Cylinder set measure Canonical Gaussian Classical Wiener measure Measure like set functions infinite-dimensional Gaussian measure Projection-valued Vector Bochner / Weakly / Strongly measurable function Radonifying function
Integrals	Bochner Direct integral Dunford Gelfand–Pettis/Weak Regulated Paley–Wiener
Results	Cameron–Martin theorem Inverse function theorem Nash–Moser theorem Feldman–Hájek theorem No infinite-dimensional Lebesgue measure Sazonov's theorem Structure theorem for Gaussian measures
Related	Crinkled arc Covariance operator
Functional calculus	Borel functional calculus Continuous functional calculus Holomorphic functional calculus
Applications	Banach manifold (bundle) Convenient vector space Choquet theory Fréchet manifold Hilbert manifold