Cylinder set measure
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. Definition. Let formula_1 be a separable real topological vector space. Let formula_2 denote  the collection of all surjective continuous linear maps formula_3 defined on formula_1 whose image is some finite-dimensional real vector space formula_5: formula_6 A cylinder set measure on formula_1 is a collection of probability measures formula_8 where formula_9 is a probability measure on formula_10 These measures are required to satisfy the following consistency condition: if formula_11 is a surjective projection, then the push forward of the measure is as follows: formula_12 Remarks. The consistency condition formula_13 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 formula_14 The cylinder sets are the pre-images in formula_1 of measurable sets in formula_5: if formula_17 denotes the formula_18-algebra on formula_5 on which formula_9 is defined, then formula_21 In practice, one often takes formula_17 to be the Borel formula_18-algebra on formula_10 In this case, one can show that when formula_1 is a  separable Banach space, the σ-algebra generated by the cylinder sets is precisely the Borel formula_18-algebra of formula_1: formula_28 Cylinder set measures versus measures. A cylinder set measure on formula_1 is not actually a measure on formula_1: it is a collection of measures defined on all finite-dimensional images of formula_14 If formula_1 has a probability measure formula_33 already defined on it, then formula_33 gives rise to a cylinder set measure on formula_1 using the push forward: set formula_36on formula_10 When there is a measure formula_33 on formula_1 such that formula_36 in this way, it is customary to abuse  notation slightly and say that the cylinder set measure formula_41 "is" the measure formula_42 Cylinder set measures on Hilbert spaces. When the Banach space formula_1 is actually a Hilbert space formula_44 there is a canonical Gaussian cylinder set measure formula_45 arising from the inner product structure on formula_46 Specifically, if formula_47 denotes the inner product on formula_44 let formula_49 denote the quotient inner product on formula_10 The measure formula_51 on formula_5 is then defined to be the canonical Gaussian measure on formula_5: formula_54 where formula_55 is an isometry of Hilbert spaces taking the Euclidean inner product on formula_56 to the  inner product formula_49 on formula_58 and formula_59 is the standard Gaussian measure on formula_60 The canonical Gaussian cylinder set measure on an infinite-dimensional separable Hilbert space formula_61 does not correspond to a true measure on formula_46 The proof is quite simple: the ball of radius formula_63 (and center 0) has measure at most equal to that of the ball of radius formula_63 in an formula_65-dimensional Hilbert space, and this tends to 0 as formula_65 tends to infinity. So the ball of radius formula_63 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 formula_68 really were a measure, then the identity function on formula_61 would radonify that measure, thus making formula_70 into an abstract Wiener space. By the Cameron–Martin theorem, formula_71 would then be quasi-invariant under translation by any element of formula_44 which implies that either formula_61 is finite-dimensional or that formula_71 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 Fréchet space automatically extends to a measure if its Fourier transform is continuous. Example: Let formula_75 be the space of Schwartz functions on a finite dimensional vector space; it is nuclear. It is contained in the Hilbert space formula_61 of formula_77 functions, which is in turn contained in the space of tempered distributions formula_78 the dual of the nuclear Fréchet space  formula_75: formula_80 The Gaussian cylinder set measure on formula_61 gives a cylinder set measure on the space of tempered distributions, which extends to a measure on the space of tempered distributions, formula_82 The Hilbert space formula_61 has measure 0 in formula_78 by the first argument used above to show that the canonical Gaussian cylinder set measure on formula_61 does not extend to a measure on formula_46