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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 denote by c y l n m displaystyle mathcal cyl n m the cylindrical algebra defined for two spaces with dual pairing n m displaystyle langle rangle langle rangle _ n m i e the set of all cylindrical sets c f 1 f m b x n x f 1 x f m b displaystyle c_ f_ 1 dots f_ m b x in n colon langle x f_ 1 rangle dots langle x f_ m rangle in b for some f 1 f m m displaystyle f_ 1 dots f_ m in m and b b r m displaystyle b in mathcal b mathbb r m 4 5 this is an algebra which can also be written as the union of smaller σ algebras definition on the cylindrical algebra edit let x displaystyle x be a topological vector space over r displaystyle mathbb r denote its algebraic dual as x displaystyle x and let g x displaystyle g subseteq x be a subspace then the set function μ c y l x g r displaystyle mu mathcal cyl x g to mathbb r _ is a cylinder set measure or cylinderical measure if for any finite set f f 1 f n g displaystyle f f_ 1 dots f_ n subset g the restriction to μ σ c y l x f r displaystyle mu sigma mathcal cyl x f to mathbb r _ is a σ additive measure notice that σ c y l x f displaystyle sigma mathcal cyl x f is a σ algebra while c y l x g displaystyle mathcal cyl x g is not 1 2 6 as usual if μ x 1 displaystyle mu x 1 we call it a cylindrical probability measure operatic definition edit let e displaystyle e be a 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 displaystyle mathcal a e left t in mathrm lin e f_ t t mbox surjective and dim _ mathbb r f_ t infty right a cylinder set measure on e displaystyle e is a collection of measures μ t t a e displaystyle left mu _ t t in mathcal a e right where μ t displaystyle mu _ t is a measure on f t displaystyle f_ t these measures are required to satisfy the following consistency condition if π 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 μ t π s t μ s displaystyle mu _ t left pi _ st right _ left mu _ s right if μ e 1 displaystyle mu e 1 then it s a cylindrical probability measure some authors define cylindrical measures explicitly as probability measures however they don t need to be connection to the abstract wiener spaces edit main article abstract wiener space let h b i displaystyle h b i be an abstract wiener space in its classical definition by leonard gross this is a separable hilbert space h displaystyle h a separable banach space b displaystyle b that is the completion under a measurable norm or gross measurable norm 1 displaystyle cdot _ 1 and a continuous linear embedding i h b displaystyle i h to b with dense range gross then showed that this construction allows to continue a cylindrical gaussian measure as a σ additive measure on the banach space more precisely let h displaystyle h be the topological dual space of h displaystyle h he showed that a cylindrical gaussian measure on h displaystyle h defined on the cylindrical algebra c y l h h displaystyle mathcal cyl h h will be σ additive on the cylindrical algebra c y l b b displaystyle mathcal cyl b b of the banach space hence the measure is also σ additive on the cylindrical σ algebra e b b σ c y l b b displaystyle mathcal e b b sigma mathcal cyl b b this follows from the carathéodory s extension theorem and is therefore also a measure in the classical sense 7 remarks edit the consistency condition μ t π s t μ 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 σ displaystyle sigma algebra on f t displaystyle f_ t on which μ t displaystyle mu _ t is defined then c 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 σ 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 σ algebra generated by the cylinder sets is precisely the borel σ displaystyle sigma algebra of e displaystyle e b o r e l e σ c y l e displaystyle mathrm borel e sigma left mathrm cyl e right cylinder set measures versus true measures edit a cylinder set measure on e displaystyle e is not actually a true 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 μ displaystyle mu already defined on it then μ displaystyle mu gives rise to a cylinder set measure on e displaystyle e using the push forward set μ t t μ displaystyle mu _ t t_ mu on f t displaystyle f_ t when there is a measure μ displaystyle mu on e displaystyle e such that μ t t μ displaystyle mu _ t t_ mu in this way it is customary to abuse notation slightly and say that the cylinder set measure μ t t a e displaystyle left mu _ t t in mathcal a e right is the measure μ displaystyle mu cylinder set measures on hilbert spaces edit when the banach space e displaystyle e is also a hilbert space h displaystyle h there is a canonical gaussian cylinder set measure γ 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 γ 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 γ t h i γ 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 γ 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 see infinite dimensional lebesgue measure 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 γ h γ 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 γ 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 γ 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 edit a cylinder set measure on the dual of a nuclear fréchet 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 fréchet 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 edit abstract wiener space mathematical construction relating to infinite dimensional spaces cylindrical σ algebra radonifying function structure theorem for gaussian measures mathematical theorem measure theory in topological vector spaces subject in mathematics references edit 1 2 bogachev vladimir 1998 gaussian measures rhode island american mathematical society 1 2 n vakhania v tarieladze and s chobanyan 1987 probability distributions on banach spaces mathematics and its applications springer netherlands p 390 isbn 9789027724960 lccn 87004931 xia dao xing brody elmer j 1972 measure and integration theory on infinite dimensional spaces abstract harmonic analysis ukraine academic press n vakhania v tarieladze and s chobanyan 1987 probability distributions on banach spaces mathematics and its applications springer netherlands p 4 isbn 9789027724960 lccn 87004931 bogachev vladimir igorevich smolyanov oleg georgievich 2017 topological vector spaces and their applications springer monographs in mathematics springer cham p 327 333 doi 10 1007 978 3 319 57117 1 isbn 978 3 319 57116 4 smolyanov oleg georgievich fomin sergei vasilyevich 1976 measures on linear topological spaces russian math surveys 31 4 12 bibcode 1976rumas 31 1s doi 10 1070 rm1976v031n04abeh001553 gross leonard 1967 abstract wiener spaces proceedings of the fifth berkeley symposium on mathematical statistics and probability band 2 contributions to probability theory part 1 university of california press 35 i 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 elworthy david 2008 ma482 stochastic analysis pdf lecture notes university of warwick v t e measure theory basic concepts absolute continuity of measures lebesgue integration l p spaces measure measure space probability space measurable space function sets almost everywhere atom baire set borel set equivalence relation borel space carathéodory s criterion cylindrical σ algebra cylinder set 𝜆 system essential range infimum supremum locally measurable π system σ algebra non measurable set vitali set null set support transverse measure universally measurable types of measures atomic baire banach besov borel brown complex complete content logarithmically convex decomposable discrete equivalent finite inner quasi invariant locally finite maximising metric outer outer perfect pre measure sub probability projection valued radon random regular borel regular inner regular outer regular saturated set function σ finite s finite signed singular spectral strictly positive tight vector particular measures counting dirac euler gaussian haar harmonic hausdorff intensity lebesgue infinite dimensional logarithmic product projections pushforward spherical tangent trivial young maps measurable function bochner strongly weakly convergence almost everywhere of measures in measure of random variables in distribution in probability cylinder set measure random compact set element measure process variable vector projection valued measure main results carathéodory s extension theorem convergence theorems dominated monotone vitali decomposition theorems hahn jordan maharam s egorov s fatou s lemma fubini s fubini tonelli hölder s inequality minkowski inequality radon nikodym riesz markov kakutani representation theorem other results disintegration theorem lifting theory lebesgue s density theorem lebesgue differentiation theorem sard s theorem vitali hahn saks theorem for lebesgue measure isoperimetric inequality brunn minkowski theorem milman s reverse minkowski steiner formula prékopa leindler inequality vitale s random brunn minkowski inequality applications related convex analysis descriptive set theory probability theory real analysis spectral theory v t e analysis in topological vector spaces basic concepts abstract wiener space classical wiener space bochner space convex series cylinder set measure infinite dimensional vector function matrix calculus vector calculus derivatives differentiable vector valued functions from euclidean space differentiation in fréchet spaces fréchet derivative total functional derivative gateaux derivative directional generalizations of the derivative hadamard derivative holomorphic quasi derivative measurability besov measure cylinder set measure canonical gaussian classical wiener measure measure like set functions infinite dimensional gaussian measure projection valued vector bochner weakly strongly measurable function radonifying function measure theory in topological vector spaces integrals bochner direct integral dunford gelfand pettis weak regulated paley wiener results cameron martin theorem inverse function theorem nash moser theorem feldman hájek theorem no infinite dimensional lebesgue measure sazonov s theorem structure theorem for gaussian measures related crinkled arc covariance operator functional calculus borel functional calculus continuous functional calculus holomorphic functional calculus applications banach manifold bundle convenient vector space choquet theory fréchet manifold hilbert manifold v t e functional analysis topics glossary spaces banach besov fréchet hilbert inner product normed nuclear schwartz sobolev topological vector properties barrelled quasi barrelled complete dual algebraic topological locally convex reflexive separable theorems hahn banach riesz representation closed graph uniform boundedness principle kakutani fixed point krein milman min max gelfand nai...