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Let $E:{\cal B}(X) \to Pr({\cal H})$ be a projection valued measure (PVM), where ${\cal B}(X)$ is the Borel $\sigma$-algebra of a suitable topological space $X$ and $Pr({\cal H})$ is the set of orthogonal projections of a Hilbert space $\cal H$. Let $\cal H'$ be a generic subspace of $\cal H$. Does $E$ induce in a natural way a PVM $E':{\cal B}(X) \to Pr({\cal H'})$?

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What do you mean by "generic"? In particular, is $\mathcal{H}'$ supposed to be closed? –  Nate Eldredge Jul 14 '11 at 17:48
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