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 E. 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:
Read more about this topic: Cylinder Set Measure
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