In mathematics and functional analysis a direct integral or Hilbert integral is a generalization of the concept of direct sum. The theory is most developed for direct integrals of Hilbert spaces and direct integrals of von Neumann algebras. The concept was introduced in 1949 by John von Neumann in one of the papers in the series On Rings of Operators. One of von Neumann's goals in this paper was to reduce the classification of (what are now called) von Neumann algebras on separable Hilbert spaces to the classification of so-called factors. Factors are analogous to full matrix algebras over a field, and von Neumann wanted to prove a continuous analogue of the Artin–Wedderburn theorem classifying semi-simple rings. Results on direct integrals can be viewed as generalizations of results about finite-dimensional C*-algebras of matrices; in this case the results are easy to prove directly. The infinite-dimensional case is complicated by measure-theoretic technicalities. Direct integral theory was also used by George Mackey in his analysis of systems of imprimitivity and his general theory of induced representations of locally compact separable groups. Direct integrals of Hilbert spaces The simplest example of a direct integral are the L2 spaces associated to a (σ-finite) countably additive measure μ on a measurable space X. Somewhat more generally one can consider a separable Hilbert space H and the space of square-integrable H-valued functions Terminological note: The terminology adopted by the literature on the subject is followed here, according to which a measurable space X is referred to as a Borel space and the elements of the distinguished σ-algebra of X as Borel sets, regardless of whether or not the underlying σ-algebra comes from a topological space (in most examples it does). A Borel space is standard if and only if it is isomorphic to the underlying Borel space of a Polish space; all Polish spaces of a given cardinality are isomorphic to each other (as Borel spaces). Given a countably additive measure μ on X, a measurable set is one that differs from a Borel set by a null set. The measure μ on X is a standard measure if and only if there is a null set E such that its complement X − E is a standard Borel space. All measures considered here are σ-finite. Definition. Let X be a Borel space equipped with a countably additive measure μ. A measurable family of Hilbert spaces