Radonifying function
Encyclopedia
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 pushforward measure
on the second space was historically thought of as a Radon measure
.
s
and 
, a CSM 
on 
and a continuous
linear map
, we say that 
is radonifying if the push forward CSM (see below) 
on 
"is" a measure, i.e. there is a measure 
on 
such that
for each
, where 
is the usual push forward of the measure 
by the linear map 
.
requires that the maps in 
be surjective, the definition of the push forward for a CSM requires careful attention. The CSM
is defined by
if the composition
is surjective. If 
is not surjective, let 
be the image of 
, let 
be the inclusion map
, and define
where
(so 
) is such that 
.
Johann Radon
Johann Karl August Radon was an Austrian mathematician. His doctoral dissertation was on calculus of variations .- Life :...
) between measurable spaces is one that takes a cylinder set measure
Cylinder set measure
In mathematics, cylinder set measure is a kind of prototype for a measure on an infinite-dimensional vector space...
(CSM) on the first space to a true measure on the second space. It acquired its name because the pushforward measure
Pushforward measure
In measure theory, a pushforward measure is obtained by transferring a measure from one measurable space to another using a measurable function.-Definition:...
on the second space was historically thought of as a Radon measure
Radon measure
In mathematics , a Radon measure, named after Johann Radon, is a measure on the σ-algebra of Borel sets of a Hausdorff topological space X that is locally finite and inner regular.-Motivation:...
.
Definition
Given two separable Banach spaceBanach space
In mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...
s
and , a CSM on and a continuousContinuous function
In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...
linear map
, we say that is radonifying if the push forward CSM (see below) on "is" a measure, i.e. there is a measure on such that
for each
, where is the usual push forward of the measure by the linear map .Push forward of a CSM
Because the definition of a CSM on requires that the maps in be surjective, the definition of the push forward for a CSM requires careful attention. The CSM
is defined by
if the composition
Function composition
In mathematics, function composition is the application of one function to the results of another. For instance, the functions and can be composed by computing the output of g when it has an argument of f instead of x...
is surjective. If is not surjective, let be the image of , let be the inclusion mapInclusion map
In mathematics, if A is a subset of B, then the inclusion map is the function i that sends each element, x of A to x, treated as an element of B:i: A\rightarrow B, \qquad i=x....
, and define
-
,
where
(so ) is such that .See also
- Classical Wiener spaceClassical Wiener spaceIn mathematics, classical Wiener space is the collection of all continuous functions on a given domain , taking values in a metric space . Classical Wiener space is useful in the study of stochastic processes whose sample paths are continuous functions...
- Abstract Wiener spaceAbstract Wiener spaceAn abstract Wiener space is a mathematical object in measure theory, used to construct a "decent" measure on an infinite-dimensional vector space. It is named after the American mathematician Norbert Wiener...