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Cylinder set measure

Definition[edit]

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

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

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

Remarks[edit]

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

In practice, one often takes ${\displaystyle {\mathcal {B}}_{T}}$ to be the Borel ${\displaystyle \sigma }$-algebraon${\displaystyle F_{T}.}$ In this case, one can show that when ${\displaystyle E}$ is a separable Banach space, the σ-algebra generated by the cylinder sets is precisely the Borel ${\displaystyle \sigma }$-algebra of ${\displaystyle E}$: $${\displaystyle \mathrm {Borel} ($$E)=\sigma \left(\mathrm {Cyl} (E)\right).}

Cylinder set measures versus measures[edit]

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

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

Cylinder set measures on Hilbert spaces[edit]

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

The canonical Gaussian cylinder set measure on an infinite-dimensional separable Hilbert space ${\displaystyle H}$ does not correspond to a true measure on ${\displaystyle H.}$ The proof is quite simple: the ball of radius ${\displaystyle r}$ (and center 0) has measure at most equal to that of the ball of radius ${\displaystyle r}$ in an ${\displaystyle n}$-dimensional Hilbert space, and this tends to 0 as ${\displaystyle n}$ tends to infinity. So the ball of radius ${\displaystyle r}$ 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 ${\displaystyle \gamma ^{H}=\gamma }$ really were a measure, then the identity functionon${\displaystyle H}$ would radonify that measure, thus making ${\displaystyle \operatorname {id} :H\to H}$ into an abstract Wiener space. By the Cameron–Martin theorem, ${\displaystyle \gamma }$ would then be quasi-invariant under translation by any element of ${\displaystyle H,}$ which implies that either ${\displaystyle H}$ is finite-dimensional or that ${\displaystyle \gamma }$ 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[edit]

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

The Gaussian cylinder set measure on ${\displaystyle H}$ gives a cylinder set measure on the space of tempered distributions, which extends to a measure on the space of tempered distributions, ${\displaystyle S^{\prime }.}$

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

See also[edit]

• Abstract Wiener space – separable Banach space equipped with a Hilbert subspace such that the standard cylinder set measure on the Hilbert subspace induces a Gaussian measure on the whole Banach spacePages displaying wikidata descriptions as a fallback
• Cylindrical σ-algebra
• Radonifying function
• Structure theorem for Gaussian measures – Mathematical theorem

References[edit]

• 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.