| 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 |
|
|
