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.
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 separableBanach 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 nuclearFréchet space automatically extends to a measure if its Fourier transform is continuous.
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
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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 Contents 1 Definition 2 Remarks 3 Cylinder set measures versus measures 4 Cylinder set measures on Hilbert spaces 5 Nuclear spaces and cylinder set measures 6 See also 7 ReferencesDefinition EditLet E displaystyle E be a separable real topological vector space Let A E displaystyle mathcal A E denote the collection of all surjective continuous linear maps T E F T displaystyle T E to F T defined on E displaystyle E whose image is some finite dimensional real vector space F T displaystyle F T A E T L i n E F T T surjective and dim R F T lt displaystyle mathcal A E left T in mathrm Lin E F T T mbox surjective and dim mathbb R F T lt infty right A cylinder set measure on E displaystyle E is a collection of probability measures m T T A E displaystyle left mu T T in mathcal A E right where m T displaystyle mu T is a probability measure on F T displaystyle F T These measures are required to satisfy the following consistency condition if p S T F S F T displaystyle pi ST F S to F T is a surjective projection then the push forward of the measure is as follows m T p S T m S displaystyle mu T left pi ST right left mu S right Remarks EditThe consistency conditionm T p S T m S displaystyle mu T left pi ST right mu S 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 displaystyle E The cylinder sets are the pre images in E displaystyle E of measurable sets in F T displaystyle F T if B T displaystyle mathcal B T denotes the s displaystyle sigma algebra on F T displaystyle F T on which m T displaystyle mu T is defined thenC y l E T 1 B B B T T A E displaystyle mathrm Cyl E left T 1 B B in mathcal B T T in mathcal A E right In practice one often takes B T displaystyle mathcal B T to be the Borel s displaystyle sigma algebra on F T displaystyle F T In this case one can show that when E displaystyle E is a separable Banach space the s algebra generated by the cylinder sets is precisely the Borel s displaystyle sigma algebra of E displaystyle E B o r e l E s C y l E displaystyle mathrm Borel E sigma left mathrm Cyl E right Cylinder set measures versus measures EditA cylinder set measure on E displaystyle E is not actually a measure on E displaystyle E it is a collection of measures defined on all finite dimensional images of E displaystyle E If E displaystyle E has a probability measure m displaystyle mu already defined on it then m displaystyle mu gives rise to a cylinder set measure on E displaystyle E using the push forward set m T T m displaystyle mu T T mu on F T displaystyle F T When there is a measure m displaystyle mu on E displaystyle E such that m T T m displaystyle mu T T mu in this way it is customary to abuse notation slightly and say that the cylinder set measure m T T A E displaystyle left mu T T in mathcal A E right is the measure m displaystyle mu Cylinder set measures on Hilbert spaces EditWhen the Banach space E displaystyle E is actually a Hilbert space H displaystyle H there is a canonical Gaussian cylinder set measure g H displaystyle gamma H arising from the inner product structure on H displaystyle H Specifically if displaystyle langle cdot cdot rangle denotes the inner product on H displaystyle H let T displaystyle langle cdot cdot rangle T denote the quotient inner product on F T displaystyle F T The measure g T H displaystyle gamma T H on F T displaystyle F T is then defined to be the canonical Gaussian measure on F T displaystyle F T g T H i g dim F T displaystyle gamma T H i left gamma dim F T right where i R dim F T F T displaystyle i mathbb R dim F T to F T is an isometry of Hilbert spaces taking the Euclidean inner product on R dim F T displaystyle mathbb R dim F T to the inner product T displaystyle langle cdot cdot rangle T on F T displaystyle F T and g n displaystyle gamma n is the standard Gaussian measure on R n displaystyle mathbb R n The canonical Gaussian cylinder set measure on an infinite dimensional separable Hilbert space H displaystyle H does not correspond to a true measure on H displaystyle H The proof is quite simple the ball of radius r displaystyle r and center 0 has measure at most equal to that of the ball of radius r displaystyle r in an n displaystyle n dimensional Hilbert space and this tends to 0 as n displaystyle n tends to infinity So the ball of radius r displaystyle r 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 g H g displaystyle gamma H gamma really were a measure then the identity function on H displaystyle H would radonify that measure thus making id H H displaystyle operatorname id H to H into an abstract Wiener space By the Cameron Martin theorem g displaystyle gamma would then be quasi invariant under translation by any element of H displaystyle H which implies that either H displaystyle H is finite dimensional or that g displaystyle 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 EditA cylinder set measure on the dual of a nuclear Frechet space automatically extends to a measure if its Fourier transform is continuous Example Let S displaystyle S be the space of Schwartz functions on a finite dimensional vector space it is nuclear It is contained in the Hilbert space H displaystyle H of L 2 displaystyle L 2 functions which is in turn contained in the space of tempered distributions S displaystyle S prime the dual of the nuclear Frechet space S displaystyle S S H S displaystyle S subseteq H subseteq S prime The Gaussian cylinder set measure on H displaystyle H gives a cylinder set measure on the space of tempered distributions which extends to a measure on the space of tempered distributions S displaystyle S prime The Hilbert space H displaystyle H has measure 0 in S displaystyle S prime by the first argument used above to show that the canonical Gaussian cylinder set measure on H displaystyle H does not extend to a measure on H displaystyle H See also EditCylindrical s algebraReferences EditI M Gel fand N Ya Vilenkin Generalized functions Applications of harmonic analysis Vol 4 Acad Press 1968 R A Minlos 2001 1994 cylindrical measure Encyclopedia of Mathematics EMS Press R A Minlos 2001 1994 cylinder set Encyclopedia of Mathematics EMS Press L Schwartz Radon measures Retrieved from https en wikipedia org w index php title Cylinder set measure amp oldid 1136002546, wikipedia, wiki, book, books, library,