Cylinder set measure#Cylinder set measures on Hilbert spaces

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-additive measure. This can also be expressed in terms of a finite-dimensional linear operator.

Denote by $\backslash mathcal\{Cyl\}(N,M)$ the cylindrical algebra defined for two spaces with dual pairing $\backslash langle,\backslash rangle:=\backslash langle,\backslash rangle\_\{N,M\}$, i.e. the set of all cylindrical sets

:$C\_\{f\_1,\backslash dots,f\_m,B\}=\backslash \{x\backslash in\; N\backslash colon\; (\backslash langle\; x,f\_1\backslash rangle,\backslash dots,\backslash langle\; x,f\_m\backslash rangle)\backslash in\; B\backslash \}$

for some $f\_1,\backslash dots,f\_m\backslash in\; M$ and $B\backslash in\; \backslash mathcal\{B\}(\backslash mathbb\{R\}^m)$.{{cite book |title=Probability Distributions on Banach Spaces |author=N. Vakhania, V. Tarieladze and S. Chobanyan|isbn=9789027724960|lccn=87004931 |series=Mathematics and its Applications| year=1987 | publisher=Springer Netherlands |page=4}}{{cite book |title=Topological Vector Spaces and Their Applications |first1=Vladimir Igorevich|last1=Bogachev |first2=Oleg Georgievich|last2=Smolyanov|DOI=10.1007/978-3-319-57117|lccn=87004931 |series=Springer Monographs in Mathematics| year=2017| publisher=Springer Cham |page=327-333}} This is an algebra which can also be written as the union of smaller σ-algebras.

Let $X$ be a topological vector space over $\backslash R$, denote its algebraic dual as $X^*$ and let $G\backslash subseteq\; X^*$ be a subspace. Then the set function $\backslash mu:\backslash mathcal\{Cyl\}(X,G)\backslash to\; \backslash R\_\{+\}$

is a cylinder set measure (or cylinderical measure) if for any finite set $F=\backslash \{f\_1,\backslash dots,f\_n\backslash \}\backslash subset\; G$ the restriction to

:$\backslash mu:\backslash sigma(\backslash mathcal\{Cyl\}(X,F))\backslash to\; \backslash R\_\{+\}$

is a σ-additive measure. Notice that $\backslash sigma(\backslash mathcal\{Cyl\}(X,F))$ is a σ-algebra while $\backslash mathcal\{Cyl\}(X,G)$ is not.{{cite journal |title=Measures on linear topological spaces |first1=Oleg Georgievich|last1=Smolyanov |first2=Sergei Vasilyevich|last2=Fomin |journal=Russian Math. Surveys |volume=31 |number=4 |date=1976 |pages=12 |doi=10.1070/RM1976v031n04ABEH001553}}

As usual if $\backslash mu(X)=1$ we call it a cylindrical probability measure.

Let $E$ be a real topological vector space. Let $\backslash mathcal\{A\}\; (E)$ denote the collection of all surjective continuous linear maps $T\; :\; E\; \backslash to\; F\_T$ defined on $E$ whose image is some finite-dimensional real vector space $F\_T$:

A cylinder set measure on $E$ is a collection of measures

$$\backslash left\backslash \{\backslash mu\_\{T\}\; :\; T\; \backslash in\; \backslash mathcal\{A\}\; (E)\backslash right\backslash \}.$$

where $\backslash mu\_T$ is a measure on $F\_T.$ These measures are required to satisfy the following consistency condition: if $\backslash pi\_\{ST\}\; :\; F\_S\; \backslash to\; F\_T$ is a surjective projection, then the push forward of the measure is as follows:

$$\backslash mu\_\{T\}\; =\; \backslash left(\backslash pi\_\{ST\}\backslash right)\_\{*\}\; \backslash left(\backslash mu\_\{S\}\backslash right).$$

If $\backslash mu(E)=1$ then it's a cylindrical probability measure. Some authors define cylindrical measures explicitly as probability measures, however they don't need to be.

{{Main|abstract Wiener space}}

Let $(H,B,i)$ be an abstract Wiener space in its classical definition by Leonard Gross. This is a separable Hilbert space $H$, a separable Banach space $B$ that is the completion under a measurable norm or Gross-measurable norm $\backslash |\backslash cdot\backslash |\_1$ and a continuous linear embedding $i:H\backslash to\; B$ with dense range. Gross then showed that this construction allows to continue a cylindrical Gaussian measure as a σ-additive measure on the Banach space. More precisely let $H\text{'}$ be the topological dual space of $H$, he showed that a cylindrical Gaussian measure on $H$ defined on the cylindrical algebra $\backslash mathcal\{Cyl\}(H,H\text{'})$ will be σ-additive on the cylindrical algebra $\backslash mathcal\{Cyl\}(B,B\text{'})$ of the Banach space. Hence the measure is also σ-additive on the cylindrical σ-algebra $\backslash mathcal\{E\}(B,B\text{'}):=\backslash sigma(\backslash mathcal\{Cyl\}(B,B\text{'})).$ This follows from the Carathéodory's extension theorem, and is therefore also a measure in the classical sense.{{cite journal |first=Leonard |last=Gross |publisher=University of California Press |title=Abstract Wiener spaces |journal=Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability |volume=Band 2: Contributions to Probability Theory, Part 1 |date=1967 |pages=35}}

The consistency condition

$$\backslash mu\_\{T\}\; =\; \backslash left(\backslash pi\_\{ST\}\backslash right)\_\{*\}\; (\backslash mu\_\{S\})$$

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 $\backslash mathcal\{B\}\_\{T\}$ denotes the $\backslash sigma$-algebra on $F\_T$ on which $\backslash mu\_T$ is defined, then

$$\backslash mathrm\{Cyl\}\; (E)\; :=\; \backslash left\backslash \{T^\{-1\}\; (B)\; :\; B\; \backslash in\; \backslash mathcal\{B\}\_\{T\},\; T\; \backslash in\; \backslash mathcal\{A\}\; (E)\backslash right\backslash \}.$$

In practice, one often takes $\backslash mathcal\{B\}\_\{T\}$ to be the Borel $\backslash sigma$-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 $\backslash sigma$-algebra of $E$:

$$\backslash mathrm\{Borel\}\; (E)\; =\; \backslash sigma\; \backslash left(\backslash mathrm\{Cyl\}\; (E)\backslash right).$$

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 $\backslash mu$ already defined on it, then $\backslash mu$ gives rise to a cylinder set measure on $E$ using the push forward: set $\backslash mu\_T\; =\; T\_\{*\}(\backslash mu)$on $F\_T.$

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

When the Banach space $E$ is also a Hilbert space $H,$ there is a {{visible anchor|canonical Gaussian cylinder set measure}} $\backslash gamma^H$ arising from the inner product structure on $H.$ Specifically, if $\backslash langle\; \backslash cdot,\; \backslash cdot\; \backslash rangle$ denotes the inner product on $H,$ let $\backslash langle\; \backslash cdot,\; \backslash cdot\; \backslash rangle\_T$ denote the quotient inner product on $F\_T.$ The measure $\backslash gamma\_T^H$ on $F\_T$ is then defined to be the canonical Gaussian measure on $F\_T$:

$$\backslash gamma\_\{T\}^\{H\}\; :=\; i\_\{*\}\; \backslash left(\backslash gamma^\{\backslash dim\; F\_\{T\}\}\backslash right),$$

where $i\; :\; \backslash R^\{\backslash dim(F\_T)\}\; \backslash to\; F\_T$ is an isometry of Hilbert spaces taking the Euclidean inner product on $\backslash R^\{\backslash dim(F\_T)\}$ to the inner product $\backslash langle\; \backslash cdot,\; \backslash cdot\; \backslash rangle\_T$ on $F\_T,$ and $\backslash gamma^n$ is the standard Gaussian measure on $\backslash R^n.$

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

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 $S^\backslash prime,$ the dual of the nuclear Fréchet space $S$:

$$S\; \backslash subseteq\; H\; \backslash subseteq\; S^\backslash prime.$$

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, $S^\backslash prime.$

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

• {{annotated link|Abstract Wiener space}}
• {{annotated link|Cylindrical σ-algebra}}
• {{annotated link|Radonifying function}}
• {{annotated link|Structure theorem for Gaussian measures}}

{{reflist}}

• I.M. Gel'fand, N.Ya. Vilenkin, Generalized functions. Applications of harmonic analysis, Vol 4, Acad. Press (1968)
• {{springer|author=R.A. Minlos|authorlink=Robert Adol'fovich Minlos|id=C/c027640|title=cylindrical measure}}
• {{springer|id=C/c027620|title=cylinder set|author=R.A. Minlos}}
• L. Schwartz, Radon measures.
• {{citation

| last1=Elworthy | first1=David

| date=2008

| url=http://www.tjsullivan.org.uk/pdf/MA482_Stochastic_Analysis.pdf

| title=MA482 Stochastic Analysis

| publisher=Lecture Notes, University of Warwick}}

{{Measure theory}}

{{Analysis in topological vector spaces}}

{{Functional Analysis}}

Category:Measures (measure theory)

Category:Topological vector spaces