5.2 Generalized Macroscopic Limits
109
Here C(E
) means “the G-measure E
satisfies the condition C ”, e.g. C(E
) :=
(dim(E
) = dim(E ◦ )), or C(E) := (E(F) = 0 ⇒ dim(F) = 0), etc. The classes
(5.2.13) could also be denoted by [E].
5.2.5 Lemma. The function dim : g
∗
→ R, F → dim(F) is lower semicontinuous.
Hence, the sets {F ∈ g
∗
: dim(F) ≥ n} are open and the sets {F ∈ g
∗
: dim(F) ≤ n}
are closed in g
∗ for any n ∈ Z + . Specifically, the set {F ∈ g
∗
: dim(F) = 0} is closed,
and the set {F ∈ g
∗
: dim(F) = n} is Borel.
Proof. It was assumed in 5.2.2 that the action ϕ G is a ‘maximal’ Poisson action,
i.e. the orbits of ϕ G coincide with the maximal integral manifolds of the Poisson
structure λ on g
∗ , [212]. The dimension dim(F) of ϕ G F is then given by the rank
of the skew-symmetric 2-tensor λ F (:= the value of λ in the point F ∈ g
∗ ), i.e. by
the rank of the mapping λ F : T F g
∗
→ T
∗
F g
∗
, v → λ F (v, ·), denoted by rank(λ F ).
Since λ depends smoothly on F, the function F → dim(F) = rank(λ F ) is lower
semicontinuous. The remaining assertions then follow.
5.2.6 Let S n := {F ∈ g
∗
: dim(F) ≤ n − 1}, 1 ≤ n ≤ dim G. For such a G
-measure E with p E = E(S n ) let r n E := ( p E − E(S n ))E, see (5.2.7). Clearly,
dim(E) = dim(r n E), if r n E = 0. E is a purely nontrivial G-measure, if 0 = p E
and r 1 E = E. If 0 = E = r n E and r n+1 E = 0, E is called a purely n-dimensional
G-measure. For n := dim(E) the measure r n E is purely n-dimensional. The Gmeasures E + E
, Es E
and q E (with a G-invariant projector q = qp E = 0) are
purely n-dimensional together with E and E
. Let the ordering (5.2.12) be given for
the set of classes [E] := {E
: p
E = p E and E
= r n E
, r n+1 E
= 0}. In any linearly
ordered subnet of such [E]’s there is a natural mapping
π E E : [E
] → [E], E
→ π E E (E
) := p E E
∈ [E] for p E ≥ p E .
(5.2.14)
The mappings π E E define a projective system [73, Definition 20.1] on the
linearly ordered subset J of classes [E] : π E E = π E E ◦ π E E for p E ≥ p E ≥ p E
and π E E = id [E] . If p ≤ p E , and E := pE
, then E
= Es E
. We want to show that
J has an upper bound in the set of classes [E] of purely n-dimensional G-measures
E. This would imply, by the Zorn’s lemma, the existence of the maximal element in
the set (uniqueness of the maximal element follows from the directedness of the set).
5.2.7 Lemma. Let L be a set of G-measures linearly ordered by E ≤ E
⇔ E
=
Es E
. Then L has an upper bound.
Proof. For any 0 ≤ f ∈ B(g
∗
) and E
≥ E it is E
( f ) ≥ E( f ). Denote
E L ( f ) := l.u.b.{E( f ) : E ∈ L} = s- lim{E( f ) : E ∈ L}.
(5.2.15)
The mapping E L can be extended by linearity to B(g
∗ ):
E L : B(g
∗
) → Z, f → E L ( f );
(5.2.16)
109
Here C(E
) means “the G-measure E
satisfies the condition C ”, e.g. C(E
) :=
(dim(E
) = dim(E ◦ )), or C(E) := (E(F) = 0 ⇒ dim(F) = 0), etc. The classes
(5.2.13) could also be denoted by [E].
5.2.5 Lemma. The function dim : g
∗
→ R, F → dim(F) is lower semicontinuous.
Hence, the sets {F ∈ g
∗
: dim(F) ≥ n} are open and the sets {F ∈ g
∗
: dim(F) ≤ n}
are closed in g
∗ for any n ∈ Z + . Specifically, the set {F ∈ g
∗
: dim(F) = 0} is closed,
and the set {F ∈ g
∗
: dim(F) = n} is Borel.
Proof. It was assumed in 5.2.2 that the action ϕ G is a ‘maximal’ Poisson action,
i.e. the orbits of ϕ G coincide with the maximal integral manifolds of the Poisson
structure λ on g
∗ , [212]. The dimension dim(F) of ϕ G F is then given by the rank
of the skew-symmetric 2-tensor λ F (:= the value of λ in the point F ∈ g
∗ ), i.e. by
the rank of the mapping λ F : T F g
∗
→ T
∗
F g
∗
, v → λ F (v, ·), denoted by rank(λ F ).
Since λ depends smoothly on F, the function F → dim(F) = rank(λ F ) is lower
semicontinuous. The remaining assertions then follow.
5.2.6 Let S n := {F ∈ g
∗
: dim(F) ≤ n − 1}, 1 ≤ n ≤ dim G. For such a G
-measure E with p E = E(S n ) let r n E := ( p E − E(S n ))E, see (5.2.7). Clearly,
dim(E) = dim(r n E), if r n E = 0. E is a purely nontrivial G-measure, if 0 = p E
and r 1 E = E. If 0 = E = r n E and r n+1 E = 0, E is called a purely n-dimensional
G-measure. For n := dim(E) the measure r n E is purely n-dimensional. The Gmeasures E + E
, Es E
and q E (with a G-invariant projector q = qp E = 0) are
purely n-dimensional together with E and E
. Let the ordering (5.2.12) be given for
the set of classes [E] := {E
: p
E = p E and E
= r n E
, r n+1 E
= 0}. In any linearly
ordered subnet of such [E]’s there is a natural mapping
π E E : [E
] → [E], E
→ π E E (E
) := p E E
∈ [E] for p E ≥ p E .
(5.2.14)
The mappings π E E define a projective system [73, Definition 20.1] on the
linearly ordered subset J of classes [E] : π E E = π E E ◦ π E E for p E ≥ p E ≥ p E
and π E E = id [E] . If p ≤ p E , and E := pE
, then E
= Es E
. We want to show that
J has an upper bound in the set of classes [E] of purely n-dimensional G-measures
E. This would imply, by the Zorn’s lemma, the existence of the maximal element in
the set (uniqueness of the maximal element follows from the directedness of the set).
5.2.7 Lemma. Let L be a set of G-measures linearly ordered by E ≤ E
⇔ E
=
Es E
. Then L has an upper bound.
Proof. For any 0 ≤ f ∈ B(g
∗
) and E
≥ E it is E
( f ) ≥ E( f ). Denote
E L ( f ) := l.u.b.{E( f ) : E ∈ L} = s- lim{E( f ) : E ∈ L}.
(5.2.15)
The mapping E L can be extended by linearity to B(g
∗ ):
E L : B(g
∗
) → Z, f → E L ( f );
(5.2.16)