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The theory of direct integral decompositions of both bounded and unboundedoperators is further developed; in particular, results about spectralprojections, functional calculus and affiliation to von Neumann algebras areproved. For operators belonging to or affiliated to a tracial von Neumannalgebra that is a direct integral von Neumann algebra, the Brown measure isshown to be given by the corresponding integral of Brown measures.

On reduction theory and Brown measure for closed unbounded operators