# Probability Measures on a Hilbert Space

Chapter

## Abstract

We assume throughout that ℋ is a separable, real Hilbert space. Let xLet ℋwhich 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 ℋwhere B is a Borel subset of ℋ

_{1},...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:$$\left\{ {y\left| {\left\{ {\left[ {y,{x_i}} \right]} \right\}} \right.\varepsilon B} \right\}$$

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

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

_{m}. The Borel set B is then called the “base” of the cylinder, and ℋ_{m}the “base” space or “generating” space.## Keywords

Hilbert Space Borel Subset Finite Dimensional Subspace Cylinder Measure Complete Orthonormal System
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© Springer-Verlag Berlin · Heidelberg 1971