Let be a collection of Hilbert spaces such that is a measure space. Now define
where (the th component of ). Then
defines a pre inner product on . Now let . Then is an inner product space, and its completion is a Hilbert space called the direct integral of . We denote the direct integral by
Now suppose is a collection of linear operators where such that is uniformly bounded. Then there is an operator where and where we can define
This gives a representation defined by
which is essentially just a component-wise left action of on . is called the algebra of diagonalizable operators of , which we will denote .
Theorem 1. Let be a representation of von Neumann algebra on a separable Hilbert space such that is a von Neumann subalgebra of . Then there exists a measure space , a collection of Hilbert spaces , and a unitary map such that for all and corresponding and
for all .
Thus if we let from above, then is a factor and we also write
This is called the central decomposition of . This also gives a representation of on defined by .
 Blackadar, Bruce. Operator Algebras. Encyclopedia of Mathematical Sciences. Springer-Verlag. 2006.