Let $M$ be a type I von Neumann algebra whose predual $M_*$ is separable. Then there exist a standard $\sigma$-finite [measure space](/page/Measure%20Space) $(X,\mathcal A,\mu)$, a measurable field of separable Hilbert spaces $(K_x)_{x\in X}$, and a normal $*$-isomorphism
The corresponding direct integral is the von Neumann algebra of essentially bounded measurable operator fields $x\mapsto T_x\in\mathcal L(K_x)$ modulo equality $\mu$-almost everywhere. The isomorphism may be chosen so that, under the induced identification of the center,
is determined uniquely up to equality almost everywhere after identifying the center, equivalently up to measure-class isomorphism of the underlying standard measure spaces.