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If      A \mathcal {A}       is a      W - ∗  W{{\text {-}}^\ast }       algebra on separable Hilbert space  H , and if      A ( λ  ) \mathcal {A}(\lambda )       are the factors in the direct integral decomposition of      A \mathcal {A}      , then      P = { λ  | A ( λ  ) \mathcal {P} = \{ \lambda |\mathcal {A}(\lambda )       has property P} is      μ  \mu      -measurable, and      A \mathcal {A}      , has property P iff      A ( λ  ) \mathcal {A}(\lambda )       has property P      μ  \mu      -a.e.

Property 𝑃 and direct integral decomposition of 𝑊*-algebras