of bounded operators {Tx}x∈ X with Tx ∈ L(Hx) is said to be strongly measurable if and only if its restriction to each Xn is strongly measurable. This makes sense because Hx is constant on Xn. Measurable families of operators with an essentially bounded norm, that is define bounded linear operators acting in a pointwise fashion, that is Such operators are said to be decomposable. Examples of decomposable operators are those defined by scalar-valued (i.e. C-valued) measurable functions λ on X. In fact, Theorem. The mapping given by is an involutive algebraic isomorphism onto its image. For this reason we will identify L∞μ(X) with the image of φ. Theorem Decomposable operators are precisely those that are in the operator commutant of the abelian algebra L∞μ(X). Decomposition of Abelian von Neumann algebras The spectral theorem has many variants. A particularly powerful version is as follows: Theorem. For any Abelian von Neumann algebra A on a separable Hilbert space H, there is a standard Borel space X and a measure μ on X such that it is unitarily equivalent as an operator algebra to L∞μ(X) acting on a direct integral of Hilbert spaces To assert A is unitarily equivalent to L∞μ(X) as an operator algebra means that there is a unitary such that U A U* is the algebra of diagonal operators L∞μ(X). Note that this asserts more than just the algebraic equivalence of A with the algebra of diagonal operators. This version however does not explicitly state how the underlying standard Borel space X is obtained. There is a uniqueness result for the above decomposition. Theorem. If the Abelian von Neumann algebra A is unitarily equivalent to both L∞μ(X) and L∞ν(Y) acting on the direct integral spaces and μ, ν are standard measures, then there is a Borel isomorphism where E, F are null sets such that φ is a measure class isomorphism, that is φ and its inverse preserve sets of measure 0. This previous two theorems provide the complete classification of Abelian von Neumann algebras on separable Hilbert spaces. Note that this classification actually takes into account the realization of the von Neumann algebra as an algebra of operators. If we only consider the underlying von Neumann algebra independently of its realization as a von Neumann algebra, then its structure is determined by very simple measure-theoretic invariants. Direct integrals of von Neumann algebras Let {Hx}x ∈ X be a