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

Definition

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-finite measure. This can also be expressed in terms of a finite-dimensional linear operator. Let X be a topological vector space over \R, denote its algebraic dual as X^* and let be a subspace. Then the set function is a cylinder set measure if for any finite set the restriction to is a σ-finite measure. Notice that is a σ-algebra while is not. is the cylindrical algebra defined for two spaces with dual pairing, i.e. the set of all cylindrical sets for and.

Operatic definition

Let E be a real topological vector space. Let denote the collection of all surjective continuous linear maps defined on E whose image is some finite-dimensional real vector space F_T: A cylinder set measure on E is a collection of probability measures where \mu_T is a probability measure on F_T. 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 E. The cylinder sets are the pre-images in E of measurable sets in F_T: if denotes the \sigma-algebra on F_T on which \mu_T is defined, then In practice, one often takes to be the Borel \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 \sigma-algebra of E:

Cylinder set measures versus true measures

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 \mu already defined on it, then \mu gives rise to a cylinder set measure on E using the push forward: set on F_T. When there is a measure \mu on E such that in this way, it is customary to abuse notation slightly and say that the cylinder set measure "is" the measure \mu.

Cylinder set measures on Hilbert spaces

When**** the Banach**** space E is**** also**** a Hilbert space H,**** there is**** a **** *gam**ma**^H ari**si**ng** fro**m*** the inner product structure on**** H.**** Specifically****,**** if**** denotes the inner product on**** H,**** let **** denote**** the quotient**** inner product on**** F_T.**** The measure \gamma_T^H on F_T is then defined to be the canonical Gaussian measure on F_T: where is an isometry of Hilbert spaces taking the Euclidean inner product on to the inner product on F_T, and \gamma^n is the standard Gaussian measure on \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 really were a measure, then the identity function on H would radonify that measure, thus making into an abstract Wiener space. By the Cameron–Martin theorem, \gamma would then be quasi-invariant under translation by any element of H, which implies that either H is finite-dimensional or that \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

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^\prime, the dual of the nuclear Fréchet space S: 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^\prime. The Hilbert space H has measure 0 in S^\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.