# Probability Measures on a Hilbert Space

• A. V. Balakrishnan
Part of the Lecture Notes in Operations Research and Mathematical Systems book series (LNE, volume 42)

## Abstract

We assume throughout that ℋ is a separable, real Hilbert space. Let x1,...xn be n elements (distinct or not) in ℋ and let B be a Borel set in Euclidean n-space En. Then by a “cylinder” set we mean the set of all y such that the n-tuple f {[y, xi]} is in B:
$$\left\{ {y\left| {\left\{ {\left[ {y,{x_i}} \right]} \right\}} \right.\varepsilon B} \right\}$$
Let ℋn denote the finite dimensional subspace generated by the elements xl,..xn. The dimension of ℋn may well be less than n. Note that if Pn 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 ℋm in ℋ. We know what is meant by a Borel subset of ℋm. 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 ℋm. The Borel set B is then called the “base” of the cylinder, and ℋm the “base” space or “generating” space.

### Keywords

Covariance Cylin