Editing Direct integral (section)

== Direct integrals of Hilbert spaces ==
The simplest example of a direct integral are the ''L''<sup>2</sup> 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

<math display=block> L^2_\mu(X, H). </math>

'''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]].{{clarify|reason=Is this X assumed in advance to be a Borel space, or to be a standard Borel space?|date=September 2015}} 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 {''H''<sub>''x''</sub>}<sub>''x''∈''X''</sub>, which is locally equivalent to a trivial family in the following sense: There is a countable partition

<math display=block> \{X_n\}_{1 \leq n \leq \omega} </math>

of ''X'' by measurable subsets <math>X_n</math> such that

<math display=block> H_x = \mathbf{H}_n, \quad  \forall x \in X_n, </math>

where '''H'''<sub>''n''</sub> is the canonical ''n''-dimensional Hilbert space{{emdash}}that is,

<math display=block>\mathbf{H}_n = \begin{cases} \mathbb{C}^n & \text{if } n < \omega, \\ \ell^2 & \text{if } n = \omega. \end{cases}</math>

In the above, <math>\ell^2</math> is the space of [[Sequence_space#ℓp_spaces|square summable sequences]]; all infinite-dimensional [[separable Hilbert space]]s are isomorphic to <math>\ell^2.</math>

A ''cross-section'' of {''H''<sub>''x''</sub>}<sub>''x''∈ ''X''</sub> is a family {''s''<sub>''x''</sub>}<sub>''x'' ∈ ''X''</sub> such that ''s''<sub>''x''</sub> ∈ ''H''<sub>''x''</sub> for all ''x'' ∈ ''X''. A cross-section is measurable if and only if its restriction to each partition element ''X''<sub>''n''</sub> 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

:<math> \int^\oplus_X H_x \, \mathrm{d} \mu(x) </math>

consists of equivalence classes (with respect to almost everywhere equality) of measurable square integrable cross-sections of {''H''<sub>''x''</sub>}<sub>''x''∈ ''X''</sub>. This is a Hilbert space under the inner product

:<math> \langle s | t \rangle = \int_X \langle s(x) | t(x) \rangle  \, \mathrm{d} \mu(x) </math>

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'' ''H''<sub>''x''</sub> 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

:<math> s \mapsto \left(\frac{\mathrm{d} \mu}{\mathrm{d} \nu}\right)^{1/2} s </math>

is a unitary operator

:<math> \int^\oplus_X H_x \, \mathrm{d} \mu(x) \rightarrow \int^\oplus_X H_x \, \mathrm{d} \nu(x). </math>

=== Example ===

The simplest example occurs when ''X'' is a [[countable set]] and μ is a [[discrete measure]]. Thus, when ''X'' = '''N''' and μ is counting measure on '''N''', then any sequence {''H''<sub>''k''</sub>} of separable Hilbert spaces can be considered as a measurable family. Moreover,

:<math> \int^\oplus_X H_x \, \mathrm{d} \mu(x) \cong \bigoplus_{k \in \mathbb{N}} H_k </math>