[page 633, §1]

[633.1.1] The set of BMO-states is
weak*-compact.
[633.1.2] Its open subsets are the elements of the weak*-topology
restricted to .
[633.1.3] They generate the -algebra of Borel sets
on .
[633.1.4] Let denote an invariant
state so that eq. (15)
holds for all .
[633.1.5] An invariant probability measure on
corresponding to the invariant can be
constructed with the help of a resolution of
the identity on .

[633.2.1] Let denote the cyclic representation canonically associated with an invariant state and the time evolution on . [633.2.2] It is uniquely determined by the two requirements

(26) |

for , and

(27) |

for . [633.2.3] Let denotes the scalar product in .

[633.3.1] A resolution of the identity[13, p.301] on the Borel -algebra is a mapping

(28) |

with the properties

,

Each is a self-adjoint projector.

If then

For every and the set function defined by

(29) is a complex regular Borel measure on .

[633.3.2] Because the projectors are self-adjoint the set function is a positive measure for every . [633.3.3] For the resulting measure

(30) |

is an invariant probability measure on the measurable
space associated with the
invariant BMO-state .
[633.3.4] The triple
is a probability space.
[633.3.5] The probability measure is invariant under
the adjoint time evolution on .