The simplest example of a direct integral is given by 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 L^2_\mu(X, H).
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 on (
X, μ) is a family {
Hx:
x∈
X} that is locally equivalent to a trivial family in the following sense: There is a countable partition \{X_n: 1 \leq n \leq \omega\} of
X by measurable subsets X_n such that H_x = \mathbf{H}_n, \quad \forall x \in X_n, where
Hn is the canonical
n-dimensional Hilbert spacethat is, \mathbf{H}_n = \begin{cases} \mathbb{C}^n & \text{if } n In the above, \ell^2 is the space of
square summable sequences. All infinite-dimensional
separable Hilbert spaces are isomorphic to \ell^2. 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 that are equal
almost everywhere. Given a measurable family of Hilbert spaces, the direct integral : \int^\oplus_X H_x \, \mathrm{d} \mu(x) 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 : \langle s | t \rangle = \int_X \langle s(x) | t(x) \rangle \, \mathrm{d} \mu(x). 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 is 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 that 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 and ν are σ-finite countably additive measures on
X that have the same sets of measure 0. Then the mapping : s \mapsto \left(\frac{\mathrm{d} \mu}{\mathrm{d} \nu}\right)^{1/2} s is a unitary operator : \int^\oplus_X H_x \, \mathrm{d} \mu(x) \rightarrow \int^\oplus_X H_x \, \mathrm{d} \nu(x).
Example The simplest example occurs when
X is a
countable set and μ is a
discrete measure. Thus, when
X =
N and μ is the counting measure on
N, any sequence {
Hn}
n∈
N of separable Hilbert spaces can be considered as a measurable family. Moreover, : \int^\oplus_X H_x \, \mathrm{d} \mu(x) \cong \bigoplus_{n \in \mathbb{N}} H_n == Decomposable operators ==