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

Definition

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 ${\displaystyle F_{T}}$: $${\displaystyle {\mathcal {A}}(E):=\left\{T\in \mathrm {Lin} (E;F_{T}):T{\mbox{ surjective and }}\dim _{\mathbb {R} }F_{T}<+\infty \right\}.}$$

A cylinder set measure on ${\displaystyle E}$ 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

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

In practice, one often takes ${\displaystyle {\mathcal {B}}_{T}}$ to be the Borel ${\displaystyle \sigma }$-algebra on ${\displaystyle F_{T}.}$ 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

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

When the Banach space ${\displaystyle E}$ is actually a Hilbert space ${\displaystyle H,}$Zdroj:https://en.wikipedia.org?pojem=Canonical_Gaussian_cylinder_set_measure
Text je dostupný za podmienok Creative Commons Attribution/Share-Alike License 3.0 Unported; prípadne za ďalších podmienok. Podrobnejšie informácie nájdete na stránke Podmienky použitia.