Inmathematics,cylinder set measure (orpromeasure, orpremeasure, orquasi-measure, orCSM) is a kind of prototype for ameasure on an infinite-dimensionalvector space. An example is theGaussian cylinder set measure onHilbert space.

Cylinder set measures are in generalnot measures (and in particular need not becountably additive but onlyfinitely additive), but can be used to define measures, such as theclassical Wiener measure on the set of continuous paths starting at the origin inEuclidean space. This is done in the construction of theabstract Wiener space where one defines a cylinder set Gaussian measure on a separable Hilbert space and chooses a Banach space in such a way that the cylindrical measure becomes σ-additive on the cylindrical algebra.

The terminology is not always consistent in the literature. Some authors call cylinder set measures justcylinder measure orcylindrical measures (see e.g.^[1][2][3]), while some reserve this word only for σ-additive measures.

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

One way is to define it directly as a set function on thecylindrical 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${\displaystyle \mathcal{C}\mathcal{y}\mathcal{l}(N,M)}$ the cylindrical algebra defined for two spaces withdual pairing${\displaystyle ⟨,⟩:=⟨,{⟩}_{N,M}}$, i.e. the set of all cylindrical sets

${\displaystyle C_{f_1,\dots ,f_m,B}=\{x\in N:(⟨x,f_1⟩,\dots ,⟨x,f_m⟩)\in B\}}$

for some${\displaystyle f_1,\dots ,f_m\in M}$ and${\displaystyle B\in \mathcal{B}({\mathbb{R}}^m)}$.^[4][5] This is an algebra which can also be written as the union of smaller σ-algebras.

Let${\displaystyle X}$ be atopological vector space over${\displaystyle \mathbb{R}}$, denote its algebraic dual as${\displaystyle X^{\ast }}$ and let${\displaystyle G\subseteq X^{\ast }}$ be a subspace. Then the set function${\displaystyle \mu :\mathcal{C}\mathcal{y}\mathcal{l}(X,G)\to {\mathbb{R}}_{+}}$is acylinder set measure (orcylinderical measure) if for any finite set${\displaystyle F=\{f_1,\dots ,f_n\}\subset G}$ the restriction to

${\displaystyle \mu :\sigma (\mathcal{C}\mathcal{y}\mathcal{l}(X,F))\to {\mathbb{R}}_{+}}$

is a σ-additive measure. Notice that${\displaystyle \sigma (\mathcal{C}\mathcal{y}\mathcal{l}(X,F))}$ is a σ-algebra while${\displaystyle \mathcal{C}\mathcal{y}\mathcal{l}(X,G)}$ is not.^[1][2][6]

As usual if${\displaystyle \mu (X)=1}$ we call it acylindrical probability measure.

Let${\displaystyle E}$ be arealtopological vector space. Let${\displaystyle \mathcal{A}(E)}$ denote the collection of allsurjectivecontinuous linear maps${\displaystyle T:E\to F_T}$ defined on${\displaystyle E}$ whose image is some finite-dimensional real vector space${\displaystyle F_T}$:

Acylinder set measure on${\displaystyle E}$ is a collection ofmeasures$${\displaystyle \left\{{\mu }_T:T\in \mathcal{A}(E)\right\}.}$$

where${\displaystyle {\mu }_T}$ is a 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 surjectiveprojection, then thepush forward of the measure is as follows:$${\displaystyle {\mu }_T={\left({\pi }_{ST}\right)}_{\ast }\left({\mu }_S\right).}$$

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

Let${\displaystyle (H,B,i)}$ be anabstract Wiener space in its classical definition byLeonard Gross. This is a separableHilbert space${\displaystyle H}$, a separable Banach space${\displaystyle B}$ that is the completion under ameasurable norm orGross-measurable norm${\displaystyle \Vert \cdot {\Vert }_1}$ and a continuous linear embedding${\displaystyle i:H\to B}$ 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${\displaystyle H^{\prime }}$ be the topological dual space of${\displaystyle H}$, he showed that a cylindrical Gaussian measure on${\displaystyle H}$ defined on the cylindrical algebra${\displaystyle \mathcal{C}\mathcal{y}\mathcal{l}(H,H^{\prime })}$ will be σ-additive on the cylindrical algebra${\displaystyle \mathcal{C}\mathcal{y}\mathcal{l}(B,B^{\prime })}$ of the Banach space. Hence the measure is also σ-additive on the cylindrical σ-algebra${\displaystyle \mathcal{E}(B,B^{\prime }):=\sigma (\mathcal{C}\mathcal{y}\mathcal{l}(B,B^{\prime })).}$ This follows from theCarathéodory's extension theorem, and is therefore also a measure in the classical sense.^[7]

The consistency condition$${\displaystyle {\mu }_T={\left({\pi }_{ST}\right)}_{\ast }({\mu }_S)}$$is modelled on the way that true measures push forward (see the sectioncylinder 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 thecylinder sets of the topological vector space${\displaystyle E.}$ Thecylinder sets are thepre-images in${\displaystyle E}$ of measurable sets in${\displaystyle F_T}$: if${\displaystyle {\mathcal{B}}_T}$ denotes the${\displaystyle \sigma }$-algebra on${\displaystyle F_T}$ on which${\displaystyle {\mu }_T}$ is defined, then$${\displaystyle \mathrm{C}\mathrm{y}\mathrm{l}(E):=\left\{T^{-1}(B):B\in {\mathcal{B}}_T,T\in \mathcal{A}(E)\right\}.}$$

In practice, one often takes${\displaystyle {\mathcal{B}}_T}$ to be theBorel${\displaystyle \sigma }$-algebra on${\displaystyle F_T.}$ In this case, one can show that when${\displaystyle E}$ is aseparableBanach space, the σ-algebra generated by the cylinder sets is precisely the Borel${\displaystyle \sigma }$-algebra of${\displaystyle E}$:$${\displaystyle \mathrm{B}\mathrm{o}\mathrm{r}\mathrm{e}\mathrm{l}(E)=\sigma \left(\mathrm{C}\mathrm{y}\mathrm{l}(E)\right).}$$

A cylinder set measure on${\displaystyle E}$ is not actually a true 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${\displaystyle {\mu }_T=T_{\ast }(\mu )}$on${\displaystyle F_T.}$

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

When the Banach space${\displaystyle E}$ is also aHilbert space${\displaystyle H,}$ there is acanonical Gaussian cylinder set measure${\displaystyle {\gamma }^H}$ arising from theinner product structure on${\displaystyle H.}$ Specifically, if${\displaystyle ⟨\cdot ,\cdot ⟩}$ denotes the inner product on${\displaystyle H,}$ let${\displaystyle ⟨\cdot ,\cdot {⟩}_T}$ denote thequotient inner product on${\displaystyle F_T.}$ The measure${\displaystyle {\gamma }_T^H}$ on${\displaystyle F_T}$ is then defined to be the canonicalGaussian measure on${\displaystyle F_T}$:$${\displaystyle {\gamma }_T^H:=i_{\ast }\left({\gamma }^{\dim F_T}\right),}$$where${\displaystyle i:{\mathbb{R}}^{\dim (F_T)}\to F_T}$ is anisometry of Hilbert spaces taking theEuclidean inner product on${\displaystyle {\mathbb{R}}^{\dim (F_T)}}$ to the inner product${\displaystyle ⟨\cdot ,\cdot {⟩}_T}$ on${\displaystyle F_T,}$ and${\displaystyle {\gamma }^n}$ is the standardGaussian measure on${\displaystyle {\mathbb{R}}^n.}$

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. (Seeinfinite dimensional Lebesgue measure.)

An alternative proof that the Gaussian cylinder set measure is not a measure uses theCameron–Martin theorem and a result on thequasi-invariance of measures. If${\displaystyle {\gamma }^H=\gamma }$ really were a measure, then theidentity function on${\displaystyle H}$ wouldradonify that measure, thus making${\displaystyle \mathrm{id}:H\to H}$ into anabstract 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 thepush forward of a canonical Gaussian cylinder set measure can be turned into a true measure.

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

Example: Let${\displaystyle S}$ be the space ofSchwartz 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 oftempered distributions${\displaystyle S^{\mathrm{\prime }},}$ the dual of thenuclearFréchet space${\displaystyle S}$:$${\displaystyle S\subseteq H\subseteq S^{\mathrm{\prime }}.}$$

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,${\displaystyle S^{\mathrm{\prime }}.}$

The Hilbert space${\displaystyle H}$ has measure 0 in${\displaystyle S^{\mathrm{\prime }},}$ 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${\displaystyle H.}$

• Abstract Wiener space – Mathematical construction relating to infinite-dimensional spaces
• Cylindrical σ-algebra
• Radonifying function
• Structure theorem for Gaussian measures – Mathematical theorem
• Measure theory in topological vector spaces – Subject in mathematics

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