measurable family of Hilbert spaces. A family of von Neumann algebras {Ax}x ∈ X with is measurable if and only if there is a countable set D of measurable operator families that pointwise generate {Ax} x ∈ X as a von Neumann algebra in the following sense: For almost all x ∈ X, where W*(S) denotes the von Neumann algebra generated by the set S. If {Ax}x ∈ X is a measurable family of von Neumann algebras, the direct integral of von Neumann algebras consists of all operators of the form for Tx ∈ Ax. One of the main theorems of von Neumann and Murray in their original series of papers is a proof of the decomposition theorem: Any von Neumann algebra is a direct integral of factors. We state this precisely below. Theorem. If {Ax}x ∈ X is a measurable family of von Neumann algebras and μ is standard, then the family of operator commutants is also measurable and Central decomposition Suppose A is a von Neumann algebra. let Z(A) be the center of A, that is the set of operators in A that commute with all operators A, that is Z(A) is an Abelian von Neumann algebra. Example. The center of L(H) is 1-dimensional. In general, if A is a von Neumann algebra, if the center is 1-dimensional we say A is a factor. Now suppose A is a von Neumann algebra whose center contains a sequence of minimal pairwise orthogonal non-zero projections {Ei}i ∈ N such that Then A Ei is a von Neumann algebra on the range Hi of Ei. It is easy to see A Ei is a factor. Thus in this special case represents A as a direct sum of factors. This is a special case of the central decomposition theorem of von Neumann. In general, we can apply the structure theorem of Abelian von Neumann algebras that represents Z(A) as an algebra of scalar diagonal operators. In any such representation, all the operators in A are decomposable operators. In fact, we can use this to prove the basic result of von Neumann that any von Neumann algebra admits a decomposition into factors. Theorem. Suppose is a direct integral decomposition of H and A is a von Neumann algebra on H so that Z(A) is represented by the algebra of scalar diagonal operators L∞μ(X) where X is a standard Borel space. Then where for