90
5 Macroscopic Limits
n ≥ n B it is B ⊂ B n , and B n ⊂ B n+1 for n ∈ Z + . If f : R → C is any Borel function
which is uniformly bounded on each bounded Borel subset B of R, then
E ξ (B) f (X ξ ) :=
B
f (λ)E ξ (dλ) = f (E ξ (B)X ξ )
(5.1.79)
is a well defined element of N G , [274, 1.11.3]. Since ω is normal, we can write for
such ‘locally finite’ functions f ∈ L
1
(R, μ
ω
ξ ) :
ω( f (X ξ )) = lim
n→∞
ω(E ξ (B n ) f (X ξ )),
(5.1.80)
ω(E ξ (B) f (X ξ )) =
M
m(E ξ (B) f (X ξ ))μ ω (dm),
(5.1.81)
m(E ξ (B) f (X ξ )) = m(E ξ (B))m(E ξ (B) f (X ξ ));
(5.1.82)
in (5.1.82) we have used the character-property of m ∈ M := ES(M G ).
For n ∈ Z + , the function χ ξn : m → m(E ξ (B n )) is continuous characteristic
function of a clopen set M ξn ⊂ M. From the monotonicity property of spectral
measures, we have M ξ(n+1) ⊃ M ξn . The union
n∈Z +
M ξn =: M ξ
(5.1.83)
is open, hence measurable together with all the M ξn . We see from (5.1.80), (5.1.81)
and (5.1.82) that μ ω is concentrated on M ξ :
μ ω (M ξ ) = μ ω (M) = 1, ∀ξ ∈ g;
(5.1.84)
it suffices to set for f a (nonzero) constant function. But
M g :=
ξ∈g
M ξ =
n
j=1
M ξ j ⊃ M ∗ \{m ◦ } = N ∗ ,
(5.1.85)
where {ξ j : j = 1, 2, . . . n} is a basis of g, 5.1.13, and μ ω is concentrated on states
in M g ,
μ ω (M g ) = μ ω (M) = 1, for any ω ∈ S ∗ (M G ).
(5.1.86)
Let
F ξ : M ξ → R, m → F ξ (m) := F m (ξ) := lim
n
m(E ξ (B n )X ξ ),
(5.1.87)
what is a bounded continuous function on each M ξn , and due to monotonicity it is
continuous on the whole M ξ . For f in (5.1.80) we have: