| import OAI.Analysis.FiniteFactor.CrossedProduct.FiberTotal |
| import Mathlib.MeasureTheory.Integral.Prod |
|
|
| namespace OAI |
|
|
| namespace FiniteFactor |
| open MeasureTheory Filter |
| open scoped ENNReal |
| noncomputable section |
|
|
| abbrev sourceBaseMeasure : Measure (SourceCoordinate → ℝ) := |
| Measure.infinitePi (fun _ => unitIntervalMeasure) |
|
|
| /-- A product almost-everywhere identity holds on every integer fiber. -/ |
| theorem source_ae_all_fibers {p : (SourceCoordinate → ℝ) × ℤ → Prop} |
| (hp : ∀ᵐ z ∂sourceMeasure SourceCoordinate, p z) : |
| ∀ᵐ x ∂sourceBaseMeasure, ∀ n : ℤ, p (x,n) := by |
| exact (Measure.ae_ae_of_ae_prod hp).mono fun x hx => Measure.ae_count_iff.mp hx |
|
|
| /-- Constant coordinate vector, realized in the original source L2 space. -/ |
| def sourceBasis (n : ℤ) : SourceHilbert SourceCoordinate := |
| sourceFiberIndicator Set.univ MeasurableSet.univ n |
|
|
| theorem sourceBasis_coe (n : ℤ) : sourceBasis n =ᵐ[sourceMeasure SourceCoordinate] |
| fun z => if z.2 = n then 1 else 0 := by |
| filter_upwards [sourceFiberIndicator_coe Set.univ MeasurableSet.univ n] with z hz |
| rw [show (sourceBasis n) z = _ from hz] |
| by_cases hn : z.2 = n <;> simp [hn] |
|
|
| theorem sourceBasis_zero : sourceBasis 0 = sourceVacuum := rfl |
|
|
| /-- Actual matrix coefficients obtained by applying H to constant e_n. -/ |
| def sourceMatrixEntry (H : SourceHilbert SourceCoordinate →L[ℂ] SourceHilbert SourceCoordinate) |
| (k n : ℤ) (x : SourceCoordinate → ℝ) : ℂ := (H (sourceBasis n)) (x,k) |
|
|
| theorem sourceScalar_apply_ae (g : SourceBaseFunction) (f : SourceHilbert SourceCoordinate) : |
| sourceScalar g f =ᵐ[sourceMeasure SourceCoordinate] fun z => g.toFun z.1 * f z := by |
| filter_upwards [sourcePi_apply_ae (fun _ => 0) g f, sourceHilbert_cube] with z hz hc |
| change (sourcePi (fun _ => 0) g f) z = _ |
| rw [hz] |
| have he : coordinateRotate (fun _ : SourceCoordinate => 0) z.2 z.1 = z.1 := by |
| funext i |
| simpa [coordinateRotate] using Int.fract_eq_self.mpr (hc i) |
| simp [sourcePiSymbol, he] |
|
|
| theorem sourceScalar_indicator_basis (A : Set (SourceCoordinate → ℝ)) (hA : MeasurableSet A) |
| (n : ℤ) : sourceScalar (sourceBaseIndicator A hA) (sourceBasis n) = |
| sourceFiberIndicator A hA n := by |
| apply Lp.ext |
| filter_upwards [sourceScalar_apply_ae (sourceBaseIndicator A hA) (sourceBasis n), |
| sourceBasis_coe n, sourceFiberIndicator_coe A hA n] with z hs hb hi |
| rw [hs, hb, hi] |
| by_cases ha : z.1 ∈ A <;> by_cases hn : z.2 = n <;> simp [sourceBaseIndicator, ha, hn] |
|
|
| /-- Localizing an input coordinate localizes every output coordinate by the |
| same base indicator. This uses the proved commutant of the actual algebra. -/ |
| theorem sourceMatrixEntry_localize (θ : SourceCoordinate → ℝ) |
| {H : SourceHilbert SourceCoordinate →L[ℂ] SourceHilbert SourceCoordinate} |
| (hH : H ∈ sourceAlgebra θ) (A : Set (SourceCoordinate → ℝ)) (hA : MeasurableSet A) |
| (n : ℤ) : ∀ᵐ x ∂sourceBaseMeasure, ∀ k : ℤ, |
| (H (sourceFiberIndicator A hA n)) (x,k) = |
| A.indicator (sourceMatrixEntry H k n) x := by |
| have he : H (sourceFiberIndicator A hA n) = |
| sourceScalar (sourceBaseIndicator A hA) (H (sourceBasis n)) := by |
| rw [← sourceScalar_indicator_basis] |
| exact (congrArg (fun T : SourceHilbert SourceCoordinate →L[ℂ] SourceHilbert SourceCoordinate => T (sourceBasis n)) (sourceAlgebra_commutes_scalar θ hH |
| (sourceBaseIndicator A hA))).symm |
| rw [he] |
| apply (source_ae_all_fibers (sourceScalar_apply_ae (sourceBaseIndicator A hA) |
| (H (sourceBasis n)))).mono |
| intro x hx k |
| rw [hx] |
| by_cases ha : x ∈ A <;> simp [sourceBaseIndicator, sourceMatrixEntry, ha] |
|
|
| end |
| end FiniteFactor |
|
|
| end OAI |
|
|
