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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 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 | import OAI.Analysis.FiniteFactor.Matrix.Integrals
import Mathlib.MeasureTheory.Integral.Bochner.SumMeasure
namespace OAI
namespace FiniteFactor
open MeasureTheory Filter
open scoped ENNReal ComplexConjugate
noncomputable section
def sourceFiberInner (f g : SourceHilbert SourceCoordinate) (x : SourceCoordinate → ℝ) : ℂ :=
∫ n : ℤ, inner ℂ (f (x,n)) (g (x,n)) ∂Measure.count
theorem sourceFiberInner_integrable (f g : SourceHilbert SourceCoordinate) :
Integrable (sourceFiberInner f g) sourceBaseMeasure :=
(L2.integrable_inner (𝕜 := ℂ) f g).integral_prod_left
theorem source_inner_fiber (f g : SourceHilbert SourceCoordinate) :
inner ℂ f g = ∫ x, sourceFiberInner f g x ∂sourceBaseMeasure :=
integral_prod _ (L2.integrable_inner (𝕜 := ℂ) f g)
theorem sourceScalar_indicator_inner (A : Set (SourceCoordinate → ℝ)) (hA : MeasurableSet A)
(f g : SourceHilbert SourceCoordinate) :
inner ℂ (sourceScalar (sourceBaseIndicator A hA) f) g =
∫ x in A, sourceFiberInner f g x ∂sourceBaseMeasure := by
rw [source_inner_fiber, ← integral_indicator hA]
apply integral_congr_ae
filter_upwards [source_ae_all_fibers (sourceScalar_apply_ae (sourceBaseIndicator A hA) f)] with x hx
unfold sourceFiberInner
by_cases ha : x ∈ A
· simp only [Set.indicator_of_mem ha]
apply integral_congr_ae
exact Eventually.of_forall fun n => by dsimp only; rw [hx]; simp [sourceBaseIndicator, ha]
· simp only [Set.indicator_of_notMem ha]
have he : (fun n : ℤ => inner ℂ ((sourceScalar (sourceBaseIndicator A hA) f) (x,n)) (g (x,n))) =
fun _ => (0 : ℂ) := by
funext n
rw [hx]
simp [sourceBaseIndicator, ha]
rw [he, integral_zero]
/-- The scalar distribution identity determining every output coordinate of H. -/
theorem source_coordinate_action_inner (θ : SourceCoordinate → ℝ)
{H : SourceHilbert SourceCoordinate →L[ℂ] SourceHilbert SourceCoordinate}
(hH : H ∈ sourceAlgebra θ) (f : SourceHilbert SourceCoordinate) (k : ℤ) :
(fun x => (H f) (x,k)) =ᵐ[sourceBaseMeasure]
sourceFiberInner ((star H) (sourceBasis k)) f := by
apply Integrable.ae_eq_of_forall_setIntegral_eq _ _ (source_fiber_integrable _ _)
(sourceFiberInner_integrable _ _)
intro A hA _
rw [← source_fiber_integral, ← L2.inner_indicatorConstLp_one]
change inner ℂ (sourceFiberIndicator A hA k) (H f) = _
rw [← ContinuousLinearMap.adjoint_inner_left, ← ContinuousLinearMap.star_eq_adjoint]
have he : (star H) (sourceFiberIndicator A hA k) =
sourceScalar (sourceBaseIndicator A hA) ((star H) (sourceBasis k)) := by
rw [← sourceScalar_indicator_basis]
exact (congrArg (fun T : SourceHilbert SourceCoordinate →L[ℂ] SourceHilbert SourceCoordinate =>
T (sourceBasis k)) (sourceAlgebra_commutes_scalar θ (star_mem hH)
(sourceBaseIndicator A hA))).symm
rw [he, sourceScalar_indicator_inner]
/-- Actual action on arbitrary L2 input, expressed through the constructed
matrix entries. The counting integral is an absolutely convergent series a.e. -/
theorem sourceMatrix_action (θ : SourceCoordinate → ℝ)
{H : SourceHilbert SourceCoordinate →L[ℂ] SourceHilbert SourceCoordinate}
(hH : H ∈ sourceAlgebra θ) (f : SourceHilbert SourceCoordinate) (k : ℤ) :
(fun x => (H f) (x,k)) =ᵐ[sourceBaseMeasure]
fun x => ∫ n : ℤ, sourceMatrixEntry H k n x * f (x,n) ∂Measure.count := by
have hs : ∀ᵐ x ∂sourceBaseMeasure, ∀ n : ℤ,
sourceMatrixEntry (star H) n k x = star (sourceMatrixEntry H k n x) := by
rw [ae_all_iff]
intro n
exact sourceMatrixEntry_star θ hH n k
filter_upwards [source_coordinate_action_inner θ hH f k, hs] with x hx hs
rw [hx]
unfold sourceFiberInner
apply integral_congr_ae
exact Eventually.of_forall fun n => by
change inner ℂ (sourceMatrixEntry (star H) n k x) (f (x,n)) = _
rw [hs]
simp [RCLike.inner_apply, mul_comm]
/-- The matrix action is absolutely integrable over the counting measure. -/
theorem sourceMatrix_action_integrable (θ : SourceCoordinate → ℝ)
{H : SourceHilbert SourceCoordinate →L[ℂ] SourceHilbert SourceCoordinate}
(hH : H ∈ sourceAlgebra θ) (f : SourceHilbert SourceCoordinate) (k : ℤ) :
∀ᵐ x ∂sourceBaseMeasure,
Integrable (fun n : ℤ => sourceMatrixEntry H k n x * f (x,n)) Measure.count := by
have hs : ∀ᵐ x ∂sourceBaseMeasure, ∀ n : ℤ,
sourceMatrixEntry (star H) n k x = star (sourceMatrixEntry H k n x) := by
rw [ae_all_iff]
intro n
exact sourceMatrixEntry_star θ hH n k
filter_upwards [(L2.integrable_inner (𝕜 := ℂ) ((star H) (sourceBasis k)) f).prod_right_ae,
hs] with x hi hs
apply hi.congr
exact Eventually.of_forall fun n => by
change inner ℂ (sourceMatrixEntry (star H) n k x) (f (x,n)) = _
rw [hs]
simp [RCLike.inner_apply, mul_comm]
theorem sourceMatrix_action_tsum (θ : SourceCoordinate → ℝ)
{H : SourceHilbert SourceCoordinate →L[ℂ] SourceHilbert SourceCoordinate}
(hH : H ∈ sourceAlgebra θ) (f : SourceHilbert SourceCoordinate) (k : ℤ) :
∀ᵐ x ∂sourceBaseMeasure,
(H f) (x,k) = ∑' n : ℤ, sourceMatrixEntry H k n x * f (x,n) := by
filter_upwards [sourceMatrix_action θ hH f k, sourceMatrix_action_integrable θ hH f k]
with x hx hi
rw [hx, integral_countable hi]
simp [measureReal_def]
/-- Products are given by the matrix product on a single conull base set. -/
theorem sourceMatrix_mul (θ : SourceCoordinate → ℝ)
{H B : SourceHilbert SourceCoordinate →L[ℂ] SourceHilbert SourceCoordinate}
(hH : H ∈ sourceAlgebra θ) :
∀ᵐ x ∂sourceBaseMeasure, ∀ k n : ℤ,
sourceMatrixEntry (H * B) k n x =
∫ r : ℤ, sourceMatrixEntry H k r x * sourceMatrixEntry B r n x ∂Measure.count := by
rw [ae_all_iff]
intro k
rw [ae_all_iff]
intro n
exact sourceMatrix_action θ hH (B (sourceBasis n)) k
end
end FiniteFactor
end OAI
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