, Chapter IX]) but it must not be mistaken with what we defined as "spectral operator measures" for weakly stationary stochastic process. When working with modules, the notion of p.v.m.'s can be extended to gramian-projection-valued measures (g.p.v.m.) which play the same role as p.v.m.'s for the extension of Stone's theorem on modules, H0 a separable Hilbert space. A projection-valued measure (p.v.m.) ? on (X, X , H0) is a p.o.v.m. valued in the space of orthogonal projections on H0, vol.9

, X ) a measurable space. A countably additive orthogonally scattered (c.a.o.s.) measure W on (X, X , H0) is an H0-valued measure which satisfies for all A, B ? X such that A ? B = ?, W (A), W (B) H 0 = 0, Let H0 be a Hilbert space

then ?W : A ? W (A) 2 H 0 is a finite, nonnegative measure on (X, X ) called the intensity measure of W . It satisfies for all ,

X ? H0 is such that there exists a finite, non-negative measure ? on (X, X ) satisfying ?A, B ? X , W (A), W (B) H 0 = ?(A ? B). Then W is a c.a.o.s. measure on ,

, Since we do not assume that a c.a.o.s. measure has finite variation, we cannot use Bochner's integration theory recalled in Section 2.1. However, Assertion (i) implies that we can linearly

, That is, there exists a unique isometric operator IW : L 2 (X, X , ?W ) ? H0 such that ?A ? X , IW (½A) = W (A). Moreover, IW is unitary from L

, and we define integration of L 2 (X, X , ?W ) functions with respect to W by setting, for all f ? L 2 (X, X , ?W ), f dW := IW (f )

non-negative measure on (X, X ) and I is an isometry from L 2 (X, X , ?) to H0, then there exists a unique c.a.o.s. measure W on (X, X , H0) with intensity measure ? such that ,

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