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ddc84c5 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 | import OAI.Analysis.FiniteFactor.Fiber.StrongContinuity
import OAI.Analysis.FiniteFactor.CrossedProduct.FieldRepresentation
namespace OAI
namespace FiniteFactor
open MeasureTheory Filter
open scoped Topology
noncomputable section
/-- A bounded diagonal on the original l2(Z) fiber. -/
def fiberDiagonal (d : ℤ → ℂ) (hd : ∀ k, ‖d k‖ ≤ 1) :
SourceFiberHilbert →L[ℂ] SourceFiberHilbert :=
LinearMap.mkContinuous
{ toFun := fun v => ⟨fun k => d k * v k,
(lp.memℓp v).mono' (fun k => by
rw [norm_mul]
exact (mul_le_mul_of_nonneg_right (hd k) (norm_nonneg _)).trans_eq (one_mul _))⟩
map_add' := fun u v => by apply Subtype.ext; funext k; exact mul_add _ _ _
map_smul' := fun c v => by apply Subtype.ext; funext k; exact mul_left_comm _ _ _ }
1 (fun v => by
rw [one_mul]
apply lp.norm_mono (by norm_num)
intro k
change ‖d k * v k‖ ≤ ‖v k‖
rw [norm_mul]
exact (mul_le_mul_of_nonneg_right (hd k) (norm_nonneg _)).trans_eq (one_mul _))
@[simp] theorem fiberDiagonal_apply (d : ℤ → ℂ) (hd : ∀ k, ‖d k‖ ≤ 1)
(v : SourceFiberHilbert) (k : ℤ) : fiberDiagonal d hd v k = d k * v k := rfl
theorem fiberDiagonal_norm (d : ℤ → ℂ) (hd : ∀ k, ‖d k‖ ≤ 1) :
‖fiberDiagonal d hd‖ ≤ 1 :=
LinearMap.mkContinuous_norm_le _ (by norm_num) _
theorem fiberDiagonal_single (d : ℤ → ℂ) (hd : ∀ k, ‖d k‖ ≤ 1) (n : ℤ) :
fiberDiagonal d hd (lp.single 2 n (1 : ℂ)) = d n • lp.single 2 n (1 : ℂ) := by
classical
apply Subtype.ext
funext k
change d k * (lp.single 2 n (1 : ℂ) : SourceFiberHilbert) k =
d n * (lp.single 2 n (1 : ℂ) : SourceFiberHilbert) k
by_cases h : n = k
· subst k; rfl
· simp [lp.single_apply, h]
theorem fiberDiagonal_continuousOn {X : Type*} [TopologicalSpace X]
(d : X → ℤ → ℂ) (hd : ∀ x k, ‖d x k‖ ≤ 1) (Q : Set X)
(hc : ∀ k, ContinuousOn (fun x => d x k) Q) (v : SourceFiberHilbert) :
ContinuousOn (fun x => fiberDiagonal (d x) (hd x) v) Q := by
apply fiberField_continuousOn _ Q 1 (fun x _ => fiberDiagonal_norm _ _) _ v
intro k
simp_rw [fiberDiagonal_single]
exact (hc k).smul continuousOn_const
theorem fiberDiagonal_stronglyMeasurable {X : Type*} [MeasurableSpace X]
(d : X → ℤ → ℂ) (hd : ∀ x k, ‖d x k‖ ≤ 1)
(hm : ∀ k, Measurable (fun x => d x k)) (v : SourceFiberHilbert) :
StronglyMeasurable (fun x => fiberDiagonal (d x) (hd x) v) := by
apply fiberCoordinates_stronglyMeasurable
intro k
exact (hm k).mul_const (v k)
/-- Applying a measurable field to a measurable vector uses finite coordinate
approximations, without measurability in the operator norm topology. -/
theorem fiberField_stronglyMeasurable_apply {X : Type*} [MeasurableSpace X]
(A : X → SourceFiberHilbert →L[ℂ] SourceFiberHilbert)
(hA : ∀ v, StronglyMeasurable (fun x => A x v))
(v : X → SourceFiberHilbert) (hv : StronglyMeasurable v) :
StronglyMeasurable (fun x => A x (v x)) := by
classical
apply stronglyMeasurable_of_tendsto (atTop : Filter (Finset ℤ))
(f := fun s x => ∑ k ∈ s, (v x k) • A x (lp.single 2 k (1 : ℂ)))
· intro s
have h (k : ℤ) : StronglyMeasurable (fun x => (v x k) • A x (lp.single 2 k (1 : ℂ))) :=
(((lp.evalCLM ℂ (fun _ : ℤ => ℂ) 2 k).continuous.comp_stronglyMeasurable hv).smul
(hA (lp.single 2 k (1 : ℂ))))
convert Finset.stronglyMeasurable_sum s (fun k _ => h k) using 1
ext x
simp only [Finset.sum_apply]
· rw [tendsto_pi_nhds]
intro x
exact fiberOperator_hasSum (A x) (v x)
end
end FiniteFactor
end OAI
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