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. 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. Let E be a separable, real, topological vector space. Let A ( E ) {displaystyle {mathcal {A}}(E)} denote the collection of all surjective, continuous linear maps T : E → FT defined on E whose image is some finite-dimensional real vector space FT: A cylinder set measure on E is a collection of probability measures where μT is a probability measure on FT. These measures are required to satisfy the following consistency condition: if πST : FS → FT is a surjective projection, then the push forward of the measure is as follows: