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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 | import OAI.Analysis.FiniteFactor.Matrix.Entries
namespace OAI
namespace FiniteFactor
open MeasureTheory Filter
open scoped ENNReal ComplexConjugate
noncomputable section
theorem source_fiber_restrict (A : Set (SourceCoordinate → ℝ)) (n : ℤ) :
(sourceMeasure SourceCoordinate).restrict (A ×ˢ {n}) =
(sourceBaseMeasure.restrict A).map (fun x => (x,n)) := by
rw [← Measure.prod_restrict]
simp only [Measure.restrict_singleton, Measure.count_singleton, one_smul]
exact Measure.prod_dirac _
theorem source_fiber_integral (f : SourceHilbert SourceCoordinate)
(A : Set (SourceCoordinate → ℝ)) (n : ℤ) :
∫ z in A ×ˢ {n}, f z ∂sourceMeasure SourceCoordinate =
∫ x in A, f (x,n) ∂sourceBaseMeasure := by
rw [source_fiber_restrict]
exact integral_map (measurable_id.prodMk measurable_const).aemeasurable
(Lp.stronglyMeasurable f).aestronglyMeasurable
theorem source_fiber_integrable (f : SourceHilbert SourceCoordinate) (n : ℤ) :
Integrable (fun x => f (x,n)) sourceBaseMeasure := by
have hi := integrableOn_Lp_of_measure_ne_top f (by norm_num : (1 : ℝ≥0∞) ≤ 2)
(sourceFiberMeasure_ne_top Set.univ n)
rw [IntegrableOn, source_fiber_restrict, Measure.restrict_univ] at hi
exact (integrable_map_measure (Lp.stronglyMeasurable f).aestronglyMeasurable
(measurable_id.prodMk measurable_const).aemeasurable).mp hi
theorem sourceMatrixEntry_measurable
(H : SourceHilbert SourceCoordinate →L[ℂ] SourceHilbert SourceCoordinate) (k n : ℤ) :
Measurable (sourceMatrixEntry H k n) :=
(Lp.stronglyMeasurable (H (sourceBasis n))).measurable.comp
(measurable_id.prodMk measurable_const)
theorem sourceMatrixEntry_integrable
(H : SourceHilbert SourceCoordinate →L[ℂ] SourceHilbert SourceCoordinate) (k n : ℤ) :
Integrable (sourceMatrixEntry H k n) sourceBaseMeasure :=
source_fiber_integrable _ _
theorem sourceMatrixEntry_setIntegral
(H : SourceHilbert SourceCoordinate →L[ℂ] SourceHilbert SourceCoordinate)
(A : Set (SourceCoordinate → ℝ)) (hA : MeasurableSet A) (k n : ℤ) :
∫ x in A, sourceMatrixEntry H k n x ∂sourceBaseMeasure =
inner ℂ (sourceFiberIndicator A hA k) (H (sourceBasis n)) := by
rw [sourceFiberIndicator, L2.inner_indicatorConstLp_one, source_fiber_integral]
rfl
/-- The displayed trace integral is the actual previously proved normal state. -/
theorem sourceVacuumFunctional_matrixIntegral
(H : SourceHilbert SourceCoordinate →L[ℂ] SourceHilbert SourceCoordinate) :
sourceVacuumFunctional H = ∫ x, sourceMatrixEntry H 0 0 x ∂sourceBaseMeasure := by
rw [← setIntegral_univ, sourceMatrixEntry_setIntegral H Set.univ MeasurableSet.univ]
rfl
/-- Adjoint matrix coefficients agree almost everywhere, on the actual source base. -/
theorem sourceMatrixEntry_star (θ : SourceCoordinate → ℝ)
{H : SourceHilbert SourceCoordinate →L[ℂ] SourceHilbert SourceCoordinate}
(hH : H ∈ sourceAlgebra θ) (k n : ℤ) :
sourceMatrixEntry (star H) k n =ᵐ[sourceBaseMeasure]
fun x => star (sourceMatrixEntry H n k x) := by
apply Integrable.ae_eq_of_forall_setIntegral_eq _ _
(sourceMatrixEntry_integrable _ _ _)
(Complex.conjCLE.integrable_comp_iff.mpr (sourceMatrixEntry_integrable H n k))
intro A hA _
change (∫ x in A, sourceMatrixEntry (star H) k n x ∂sourceBaseMeasure) =
∫ x in A, conj (sourceMatrixEntry H n k x) ∂sourceBaseMeasure
rw [sourceMatrixEntry_setIntegral _ A hA, integral_conj, sourceMatrixEntry_setIntegral _ A hA,
inner_conj_symm]
rw [ContinuousLinearMap.star_eq_adjoint, ContinuousLinearMap.adjoint_inner_right]
have he : H (sourceFiberIndicator A hA k) =
sourceScalar (sourceBaseIndicator A hA) (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 hH (sourceBaseIndicator A hA))).symm
rw [he]
have hs : star (sourceScalar (sourceBaseIndicator A hA)) =
sourceScalar (sourceBaseIndicator A hA) := by
rw [sourceScalar_star]
congr 1
unfold sourceBaseStar sourceBaseIndicator
congr 1
funext x
by_cases hx : x ∈ A <;> simp [hx]
rw [← ContinuousLinearMap.adjoint_inner_right, ← ContinuousLinearMap.star_eq_adjoint,
hs, sourceScalar_indicator_basis]
end
end FiniteFactor
end OAI
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