File size: 3,674 Bytes
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 | 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
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