on (X, μ) is a family {Hx}x∈ X, which is locally equivalent to a trivial family in the following sense: There is a countable partition by measurable subsets of X such that where Hn is the canonical n-dimensional Hilbert space, that is A cross-section of {Hx}x∈ X is a family {sx}x ∈ X such that sx ∈ Hx for all x ∈ X. A cross-section is measurable if and only if its restriction to each partition element Xn is measurable. We will identify measurable cross-sections s, t that are equal almost everywhere. Given a measurable family of Hilbert spaces, the direct integral consists of equivalence classes (with respect to almost everywhere equality) of measurable square integrable cross-sections of {Hx}x∈ X. This is a Hilbert space under the inner product Given the local nature of our definition, many definitions applicable to single Hilbert spaces apply to measurable families of Hilbert spaces as well. Remark. This definition is apparently more restrictive than the one given by von Neumann and discussed in Dixmier's classic treatise on von Neumann algebras. In the more general definition, the Hilbert space fibers Hx are allowed to vary from point to point without having a local triviality requirement (local in a measure-theoretic sense). One of the main theorems of the von Neumann theory is to show that in fact the more general definition is equivalent to the simpler one given here. Note that the direct integral of a measurable family of Hilbert spaces depends only on the measure class of the measure μ; more precisely: Theorem. Suppose μ, ν are σ-finite countably additive measures on X that have the same sets of measure 0. Then the mapping is a unitary operator Example Technically the simplest examples are when X is a countable set and μ is a discrete measure. Throughout the article, we will consider the following running example in which X = N and μ is counting measure on N. In this case any sequence {Hk} of separable Hilbert spaces can be considered as a measurable family. Moreover, Decomposable operators In our running example, any bounded linear operator T on is given by an infinite matrix Consider operators that are block diagonal, that is all entries off the diagonal are zero. We call these operators decomposable. These operators can be characterized as those that commute with diagonal matrices: We now proceed to the general definition: A family