| 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 |
| |
