-complete Boolean algebra, that is a pointfree -algebra. In the special case one recovers the abstract -algebra . This pointfree approach can be turned into a duality theorem analogue to Gelfand-duality between the category of abelian von Neumann algebras and the category of abstract -algebras. Let μ and ν be non-atomic probability measures on standard Borel spaces X and Y respectively. Then there is a μ null subset N of X, a ν null subset M of Y and a Borel isomorphism which carries μ into ν. Notice that in the above result, it is necessary to clip away sets of measure zero to make the result work. In the above theorem, the isomorphism is required to preserve the weak operator topology. As it turns out (and follows easily from the definitions), for algebras L∞(X, μ), the following topologies agree on norm bounded sets: The weak operator topology on L∞(X, μ); The ultraweak operator topology on L∞(X, μ); The topology of weak* convergence on L∞(X, μ) considered as the dual space of L1(X, μ). However, for an abelian von Neumann algebra A the realization of A as an algebra of operators on a separable Hilbert space is highly non-unique. The complete classification of the operator algebra realizations of A is given by spectral multiplicity theory and requires the use of direct integrals. Spatial isomorphism Using direct integral theory, it can be shown that the abelian von Neumann algebras of the form L∞(X, μ) acting as operators on L2(X, μ) are all maximal abelian. This means that they cannot be extended to properly larger abelian algebras. They are also referred to as Maximal abelian self-adjoint algebras (or M.A.S.A.s). Another phrase used to describe them is abelian von Neumann algebras of uniform multiplicity 1; this description makes sense only in relation to multiplicity theory described below. Von Neumann algebras A on H, B on K are spatially isomorphic (or unitarily isomorphic) if and only if there is a unitary operator U: H → K such that In particular spatially isomorphic von Neumann algebras are algebraically isomorphic. To describe the most general abelian von Neumann algebra on a separable Hilbert space H up to spatial isomorphism, we need to refer the direct integral decomposition of H. The details of this decomposition are discussed in decomposition of abelian von Neumann algebras. In particular: Theorem Any abelian von Neumann algebra on a separable Hilbert space