Probability Measures on a Hilbert Space

Abstract

We assume throughout that ℋ is a separable, real Hilbert space. Let x₁,...x_n be n elements (distinct or not) in ℋ and let B be a Borel set in Euclidean n-space E_n. Then by a “cylinder” set we mean the set of all y such that the n-tuple f {[y, x_i]} is in B:

$$\left\{ {y\left| {\left\{ {\left[ {y,{x_i}} \right]} \right\}} \right.\varepsilon B} \right\}$$

Let ℋ_n denote the finite dimensional subspace generated by the elements x_l,..x_n. The dimension of ℋ_n may well be less than n. Note that if P_n denotes the projection operator projecting ℋ onto ℋ _n, then if y belongs to the cylinder set, so does

$${P_n}y + \left( {I - {P_n}} \right)\mathcal{H}$$

which explains the name “cylinder” set. We can also describe the set in a slightly different (and more general) language. Let us take any finite dimensional subspace ℋ_m in ℋ. We know what is meant by a Borel subset of ℋ_m. By a cylinder set we mean any set of the form

$$B + orthogonal{\text{ complement of }}{{\text{H}}_m}$$

where B is a Borel subset of ℋ_m. The Borel set B is then called the “base” of the cylinder, and ℋ_m the “base” space or “generating” space.