en>Matthew Kastor: adds equivalent (simplified) logical expression and links to "ternary operator" page.

2013-10-19T18:52:12Z

adds equivalent (simplified) logical expression and links to "ternary operator" page.

← Older revision		Revision as of 20:52, 19 October 2013
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	~~Hello buddy~~. Let ~~me introduce myself~~. ~~I am Luther Aubrey~~. ~~Bookkeeping~~ has ~~been his working day occupation for~~ a ~~while~~. The ~~preferred pastime for him~~ and ~~his children~~ is to ~~perform badminton but he~~ is ~~struggling~~ to ~~find time for~~ it. ~~I currently live in Alabama~~.<br><br>~~My page~~ - [~~http~~://~~Brazil~~.~~amor-amore~~.~~com~~/~~nilindon auto warranties~~]		{{mergefrom\|Cylindrical_σ-algebra\|date=April 2012}}
			In [[mathematics]], '''cylinder set measure''' (or '''promeasure''', or '''premeasure''', or '''quasi-measure''', or '''CSM''') is a kind of prototype for a [[Measure (mathematics)\|measure]] on an infinite-dimensional [[vector space]]. An example is the [[Gauss]]ian cylinder set measure on [[Hilbert space]].

			Cylinder set measures are in general '''not''' measures (and in particular need not be [[Sigma additivity\|countably additive]] but only [[Sigma additivity\|finitely additive]]), but can be used to define measures, such as [[Classical Wiener space#Classical Wiener measure\|classical Wiener measure]] on the set of continuous paths starting at the origin in [[Euclidean space]].

			==Definition==

			Let ''E'' be a [[separable space\|separable]], [[real number\|real]], [[topological vector space]]. Let <math>\mathcal{A} (E)</math> denote the collection of all [[surjective]], [[continuous linear map]]s ''T'' : ''E'' → ''F''<sub>''T''</sub> defined on ''E'' whose image is some finite-dimensional real vector space ''F''<sub>''T''</sub>:

			:<math>\mathcal{A} (E) := \{ T \in \mathrm{Lin} (E; F_{T}) \| T \mbox{ surjective and } \dim_{\mathbb{R}} F_{T} < + \infty \}.</math>

			A '''cylinder set measure''' on ''E'' is a collection of [[probability measure]]s

			:<math>\{ \mu_{T} \| T \in \mathcal{A} (E) \}.</math>

			where ''μ''<sub>''T''</sub> is a probability measure on ''F''<sub>''T''</sub>. These measures are required to satisfy the following consistency condition: if ''π''<sub>''ST''</sub> : ''F''<sub>''S''</sub> → ''F''<sub>''T''</sub> is a surjective [[Projection (mathematics)\|projection]], then the measures [[pushforward measure\|push forward]] as follows:

			:<math>\mu_{T} = \left( \pi_{ST} \right)_{*} (\mu_{S}).</math>

			==Remarks==

			The consistency condition

			:<math>\mu_{T} = \left( \pi_{ST} \right)_{*} (\mu_{S})</math>

			is modelled on the way that true measures push forward (see the section [[#Cylinder set measures versus true measures\|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]]s of the topological vector space ''E''. The '''cylinder sets''' are the [[pre-image]]s in ''E'' of measurable sets in ''F''<sub>''T''</sub>: if <math>\mathcal{B}_{T}</math> denotes the [[sigma algebra\|σ-algebra]] on ''F''<sub>''T''</sub> on which ''μ''<sub>''T''</sub> is defined, then

			:<math>\mathrm{Cyl} (E) := \{ T^{-1} (B) \| B \in \mathcal{B}_{T}, T \in \mathcal{A} (E) \}.</math>

			In practice, one often takes <math>\mathcal{B}_{T}</math> to be the [[Borel sigma algebra\|Borel ''σ''-algebra]] on ''F''<sub>''T''</sub>. In this case, one can show that when ''E'' is a [[separable space\|separable]] [[Banach space]], the σ-algebra generated by the cylinder sets is precisely the Borel ''σ''-algebra of ''E'':

			:<math>\mathrm{Borel} (E) = \sigma \left( \mathrm{Cyl} (E) \right).</math>

			==Cylinder set measures versus true measures==

			A cylinder set measure on ''E'' is not actually a measure on ''E'': it is a collection of measures defined on all finite-dimensional images of ''E''. If ''E'' has a probability measure ''μ'' already defined on it, then ''μ'' gives rise to a cylinder set measure on ''E'' using the push forward: set ''μ''<sub>''T''</sub> = ''T''<sub>&lowast;</sub>(''μ'') on ''F''<sub>''T''</sub>.

			When there is a measure ''μ'' on ''E'' such that ''μ''<sub>''T''</sub> = ''T''<sub>&lowast;</sub>(''μ'') in this way, it is customary to [[abuse of notation\|abuse notation]] slightly and say that the cylinder set measure <math>\{ \mu_{T} \| T \in \mathcal{A} (E) \}</math> "is" the measure ''μ''.

			==Cylinder set measures on Hilbert spaces==

			When the Banach space ''E'' is actually a [[Hilbert space]] ''H'', there is a '''[[canon (basic principle)\|canon]]ical Gaussian cylinder set measure''' ''γ''<sup>''H''</sup> arising from the [[inner product]] structure on ''H''. Specifically, if &lang; , &rang; denotes the inner product on ''H'', let &lang; , &rang;<sub>''T''</sub> denote the [[quotient inner product]] on ''F''<sub>''T''</sub>. The measure ''γ''<sub>''T''</sub><sup>''H''</sup> on ''F''<sub>''T''</sub> is then defined to be the canonical [[Gaussian measure]] on ''F''<sub>''T''</sub>:

			:<math>\gamma_{T}^{H} := i_{*} \left( \gamma^{\dim F_{T}} \right),</math>

			where ''i'' : '''R'''<sup>dim(''F''<sub>''T''</sub>)</sup> → ''F''<sub>''T''</sub> is an [[isometry]] of Hilbert spaces taking the [[Euclidean geometry\|Euclidean]] inner product on '''R'''<sup>dim(''F''<sub>''T''</sub>)</sup> to the inner product &lang; , &rang;<sub>''T''</sub> on ''F''<sub>''T''</sub>, and ''γ''<sup>''n''</sup> is the standard [[Gaussian measure]] on '''R'''<sup>''n''</sup>.

			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.

			An alternative proof that the Gaussian cylinder set measure is not a measure uses the [[Cameron–Martin theorem]] and a result on the [[Quasi-invariant measure\|quasi-invariance of measures]]. If ''γ''<sup>''H''</sup> = ''γ'' really were a measure, then the [[identity function]] on ''H'' would [[radonify]] that measure, thus making id : ''H'' → ''H'' into an [[abstract Wiener space]]. By the Cameron–Martin theorem, ''γ'' would then be quasi-invariant under translation by any element of ''H'', which implies that either ''H'' is finite-dimensional or that ''γ'' is the zero measure. In either case, we have a contradiction.

			[[Sazonov's theorem]] gives conditions under which the [[Pushforward measure\|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 space\|nuclear Fréchet space]] automatically extends to a measure if its Fourier transform is continuous.

			'''Example''': Let ''S'' be the space of [[distribution (mathematics)\|Schwartz function]]s on a finite dimensional vector space; it is nuclear. It is contained in the Hilbert space ''H'' of [[Lp space\|''L''<sup>2</sup> functions]], which is in turn contained in the space of [[tempered representation\|tempered distributions]] ''S''′, the dual of the [[Nuclear space\|nuclear]] [[Fréchet space]] ''S'':

			:<math>S \subseteq H \subseteq S'.</math>

			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''′.

			The Hilbert space ''H'' has measure 0 in ''S''′, 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''.

			==References==
			*I.M. Gel'fand, N.Ya. Vilenkin, ''Generalized functions. Applications of harmonic analysis'', Vol 4, Acad. Press (1968)
			*{{springer\|id=C/c027640\|title=cylindrical measure\|author=R.A. Minlos\|authorlink=Robert Adol'fovich Minlos}}
			*{{springer\|id=C/c027620\|title=cylinder set\|author=R.A. Minlos}}
			* L. Schwartz, ''Radon measures''.

			{{Functional Analysis}}

			[[Category:Measures (measure theory)]]
			[[Category:Topological vector spaces]]

en>HRoestBot: r2.6.5) (Robot: Adding ru:Условная дизъюнкция

2012-07-07T10:43:11Z

r2.6.5) (Robot: Adding ru:Условная дизъюнкция

New page

Hello buddy. Let me introduce myself. I am Luther Aubrey. Bookkeeping has been his working day occupation for a while. The preferred pastime for him and his children is to perform badminton but he is struggling to find time for it. I currently live in Alabama.<br><br>My page - [http://Brazil.amor-amore.com/nilindon auto warranties]

Conditioned disjunction - Revision history

en>Matthew Kastor: adds equivalent (simplified) logical expression and links to "ternary operator" page.

en>HRoestBot: r2.6.5) (Robot: Adding ru:Условная дизъюнкция