The consistency condition is modelled on the way that true measures push forward. 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 spaceE. The cylinder sets are the pre-images in E of measurable sets in FT: if denotes the σ-algebra on FT on which μT is defined, then In practice, one often takes to be the Borel σ-algebra on FT. In this case, one can show that when E is a separable Banach space, the σ-algebra generated by the cylinder sets is precisely the Borel σ-algebra of E:
Cylinder set measures versus 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 μT = T∗ on FT. When there is a measure μ on E such that μT = T∗ in this way, it is customary to abuse notation slightly and say that the cylinder set measure "is" the measure μ.
When the Banach space E is actually a Hilbert space H, there is a canonical Gaussian cylinder set measureγH arising from the inner product structure on H. Specifically, if ⟨ , ⟩ denotes the inner product on H, let ⟨ , ⟩T denote the quotient inner product on FT. The measure γTH on FT is then defined to be the canonical Gaussian measure on FT: where i : Rdim → FT is an isometry of Hilbert spaces taking the Euclidean inner product on Rdim to the inner product ⟨ , ⟩T on FT, and γn is the standard Gaussian measure on Rn. 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 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-invariance of measures. If γH = 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 ofH, 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 push forward of a canonical Gaussian cylinder set measure can be turned into a true measure.
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 be the space of Schwartz functions on a finite dimensional vector space; it is nuclear. It is contained in the Hilbert space H of L2 functions, which is in turn contained in the space of tempered distributionsS′, the dual of the nuclear Fréchet spaceS: 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.
