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 be a separable real topological vector space. Let denote the collection of all surjective continuous linear maps defined on whose image is some finite-dimensional real vector space :

A cylinder set measure on is a collection of probability measures

where is a probability measure on These measures are required to satisfy the following consistency condition: if is a surjective projection, then the push forward of the measure is as follows:

Remarks

The consistency condition 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 The cylinder sets are the pre-images in of measurable sets in : if denotes the -algebra on on which is defined, then

In practice, one often takes to be the Borel -algebra on In this case, one can show that when is a separable Banach space, the σ-algebra generated by the cylinder sets is precisely the Borel -algebra of :

Cylinder set measures versus measures

A cylinder set measure on is not actually a measure on : it is a collection of measures defined on all finite-dimensional images of If has a probability measure already defined on it, then gives rise to a cylinder set measure on using the push forward: set on

When there is a measure on such that in this way, it is customary to abuse notation slightly and say that the cylinder set measure "is" the measure

Cylinder set measures on Hilbert spaces

When the Banach space is actually a Hilbert space there is a canonical Gaussian cylinder set measure arising from the inner product structure on Specifically, if denotes the inner product on let denote the quotient inner product on The measure on is then defined to be the canonical Gaussian measure on : where is an isometry of Hilbert spaces taking the Euclidean inner product on to the inner product on and is the standard Gaussian measure on

The canonical Gaussian cylinder set measure on an infinite-dimensional separable Hilbert space does not correspond to a true measure on The proof is quite simple: the ball of radius (and center 0) has measure at most equal to that of the ball of radius in an -dimensional Hilbert space, and this tends to 0 as tends to infinity. So the ball of radius 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 really were a measure, then the identity function on would radonify that measure, thus making into an abstract Wiener space. By the Cameron–Martin theorem, would then be quasi-invariant under translation by any element of which implies that either is finite-dimensional or that 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 be the space of Schwartz functions on a finite dimensional vector space; it is nuclear. It is contained in the Hilbert space of functions, which is in turn contained in the space of tempered distributions the dual of the nuclear Fréchet space :

The Gaussian cylinder set measure on gives a cylinder set measure on the space of tempered distributions, which extends to a measure on the space of tempered distributions,

The Hilbert space has measure 0 in by the first argument used above to show that the canonical Gaussian cylinder set measure on does not extend to a measure on

See also

References