Definition of a Measure in Hubert Space

Abstract

Let X be a real separable Hilbert space with elements x, y, z, etc. Real numbers will be designated by small Greek letters; α x + β y and (x, y) will denote, as usual, the operations of multiplication of a vector (element of X) by a scalar, vector addition and the scalar product of vectors The norm of a vector will be designated by

$$ \left| x \right| = \sqrt {{(x,x)}} $$

Subsets of X will be denoted by large Latin letters, classes of subsets by large Gothic letters. A class of sets A in which we allow the operations of set difference, union and intersection is called a ring. A ring of sets A containing X as an element is called an algebra. An algebra of sets in which the union operation can be applied countably many times is called a σ-algebra.