In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, cylinder set measure (or promeasure, or premeasure, or quasi-measure, or CSM) is a kind of prototype for a
measure
Measure may refer to:
* Measurement, the assignment of a number to a characteristic of an object or event
Law
* Ballot measure, proposed legislation in the United States
* Church of England Measure, legislation of the Church of England
* Mea ...
on an infinite-dimensional
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
. An example is the
Gauss
Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...
ian cylinder set measure on
Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
.
Cylinder set measures are in general not measures (and in particular need not be
countably additive
In mathematics, an additive set function is a function mapping sets to numbers, with the property that its value on a union of two disjoint sets equals the sum of its values on these sets, namely, \mu(A \cup B) = \mu(A) + \mu(B). If this additivit ...
but only
finitely additive
In mathematics, an additive set function is a function mapping sets to numbers, with the property that its value on a union of two disjoint sets equals the sum of its values on these sets, namely, \mu(A \cup B) = \mu(A) + \mu(B). If this additivity ...
), 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
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
.
Definition
Let
be a
separable real
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010)
...
topological vector space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is als ...
. Let
denote the collection of all
surjective
In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of i ...
continuous linear map In functional analysis and related areas of mathematics, a continuous linear operator or continuous linear mapping is a continuous linear transformation between topological vector spaces.
An operator between two normed spaces is a bounded linear o ...
s
defined on
whose image is some finite-dimensional real vector space
:
A cylinder set measure on
is a collection of
probability measure
In mathematics, a probability measure is a real-valued function defined on a set of events in a probability space that satisfies measure properties such as ''countable additivity''. The difference between a probability measure and the more gener ...
s
where
is a probability measure on
These measures are required to satisfy the following consistency condition: if
is a surjective
projection
Projection, projections or projective may refer to:
Physics
* Projection (physics), the action/process of light, heat, or sound reflecting from a surface to another in a different direction
* The display of images by a projector
Optics, graphic ...
, 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 set In mathematics, the cylinder sets form a basis of the product topology on a product of sets; they are also a generating family of the cylinder σ-algebra.
General definition
Given a collection S of sets, consider the Cartesian product X = \prod ...
s of the topological vector space
The cylinder sets are the
pre-image
In mathematics, the image of a function is the set of all output values it may produce.
More generally, evaluating a given function f at each element of a given subset A of its domain produces a set, called the "image of A under (or through) ...
s 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
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
, 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
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
there is a
arising from the
inner product
In mathematics, an inner product space (or, rarely, a Hausdorff space, Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation (mathematics), operation called an inner product. The inner product of two ve ...
structure on
Specifically, if
denotes the inner product on
let
denote the
quotient inner product
In arithmetic, a quotient (from lat, quotiens 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to as the integer part of a ...
on
The measure
on
is then defined to be the canonical
Gaussian measure
In mathematics, Gaussian measure is a Borel measure on finite-dimensional Euclidean space R''n'', closely related to the normal distribution in statistics. There is also a generalization to infinite-dimensional spaces. Gaussian measures are named ...
on
:
where
is an
isometry
In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' mea ...
of Hilbert spaces taking the
Euclidean inner product on
to the inner product
on
and
is the standard
Gaussian measure
In mathematics, Gaussian measure is a Borel measure on finite-dimensional Euclidean space R''n'', closely related to the normal distribution in statistics. There is also a generalization to infinite-dimensional spaces. Gaussian measures are named ...
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
In mathematics, the Cameron–Martin theorem or Cameron–Martin formula (named after Robert Horton Cameron and W. T. Martin) is a theorem of measure theory that describes how abstract Wiener measure changes under translation by certain element ...
and a result on the
quasi-invariance of measures. If
really were a measure, then the
identity function
Graph of the identity function on the real numbers
In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, un ...
on
would
radonify
In measure theory, a radonifying function (ultimately named after Johann Radon) between measurable spaces is one that takes a cylinder set measure (CSM) on the first space to a true measure on the second space. It acquired its name because the pus ...
that measure, thus making
into an
abstract Wiener space The concept of an abstract Wiener space is a mathematical construction developed by Leonard Gross to understand the structure of Gaussian measures on infinite-dimensional spaces. The construction emphasizes the fundamental role played by the Camer ...
. 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
Nuclear may refer to:
Physics
Relating to the nucleus of the atom:
* Nuclear engineering
*Nuclear physics
*Nuclear power
*Nuclear reactor
*Nuclear weapon
*Nuclear medicine
*Radiation therapy
*Nuclear warfare
Mathematics
*Nuclear space
*Nuclear ...
Fréchet space
In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces.
They are generalizations of Banach spaces (normed vector spaces that are complete with respect to the ...
automatically extends to a measure if its Fourier transform is continuous.
Example: Let
be the space of
Schwartz function
In mathematics, Schwartz space \mathcal is the function space of all functions whose derivatives are rapidly decreasing. This space has the important property that the Fourier transform is an automorphism on this space. This property enables on ...
s 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
Distributions, also known as Schwartz distributions or generalized functions, are objects that generalize the classical notion of functions in mathematical analysis. Distributions make it possible to derivative, differentiate functions whose de ...
the dual of the
nuclear
Nuclear may refer to:
Physics
Relating to the nucleus of the atom:
* Nuclear engineering
*Nuclear physics
*Nuclear power
*Nuclear reactor
*Nuclear weapon
*Nuclear medicine
*Radiation therapy
*Nuclear warfare
Mathematics
*Nuclear space
*Nuclear ...
Fréchet space
In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces.
They are generalizations of Banach spaces (normed vector spaces that are complete with respect to the ...
:
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
* I.M. Gel'fand, N.Ya. Vilenkin, ''Generalized functions. Applications of harmonic analysis'', Vol 4, Acad. Press (1968)
*
*
* L. Schwartz, ''Radon measures''.
{{Functional Analysis
Measures (measure theory)
Topological vector spaces