Algebras of unbounded scalar-type spectral operators

If P: Σ → L(X) is a closed spectral measure in the quasicomplete locally convex space X and if T is a densely defined linear operator in X with domain invariant under each operator of the form ∫_Ω fdP, with f a complex bounded Σ-measurable function then T is closable and there exists a complex Σ-measurable function f such that the closure of T is the spectral integral ∫Ω fdP if and only if T leaves invariant each closed subspace of X which is invariant under the range of the spectral measure P.