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.
This is done in the construction of the abstract 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.
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.
the cylindrical algebra defined for two spaces with dual pairing
[4][5] This is an algebra which can also be written as the union of smaller σ-algebras.
denote the collection of all surjective continuous linear maps
whose image is some finite-dimensional real vector space
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:
be an abstract Wiener space in its classical definition by Leonard Gross, this is a separable Hilbert space
Gross then showed that this construction allows to continue a cylindrical Gaussian measure as a σ-additive measure on the Banach space.
, which follows from the Carathéodory's extension theorem, and is therefore also a measure in the classical sense.
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
is a separable Banach space, the σ-algebra generated by the cylinder sets is precisely the Borel
: it is a collection of measures defined on all finite-dimensional images of
in this way, it is customary to abuse notation slightly and say that the cylinder set measure
there is a canonical Gaussian cylinder set measure
is an isometry of Hilbert spaces taking the Euclidean inner product on
The canonical Gaussian cylinder set measure on an infinite-dimensional separable Hilbert space
(and center 0) has measure at most equal to that of the ball of radius
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.
functions, which is in turn contained in the space of tempered distributions
the dual of the nuclear Fréchet space
gives a cylinder set measure on the space of tempered distributions, which extends to a measure on the space of tempered distributions,
by the first argument used above to show that the canonical Gaussian cylinder set measure on