| import OAI.Analysis.IrrationalRotation.Model.FieldOperator |
| import Mathlib.Analysis.InnerProductSpace.Adjoint |
| import Mathlib.MeasureTheory.Function.LpSpace.Indicator |
| import Mathlib.Tactic |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open MeasureTheory Filter TopologicalSpace |
| open scoped ENNReal Topology |
|
|
| namespace Invariant233.DirectIntegral |
| variable {X E F G : Type*} [MeasurableSpace X] {μ : Measure X} |
| [NormedAddCommGroup E] [NormedSpace ℂ E] |
| [NormedAddCommGroup F] [NormedSpace ℂ F] |
| [NormedAddCommGroup G] [NormedSpace ℂ G] |
|
|
| variable (A : X → E →L[ℂ] F) (C : ℝ) |
| (hA : ∀ x, ‖A x‖ ≤ C) |
| (hm : ∀ v, AEStronglyMeasurable (fun x => A x v) μ) |
|
|
| theorem fieldOperator_ae (f : Lp E 2 μ) : |
| fieldOperator A C hA hm f =ᵐ[μ] fun x => A x (f x) := |
| fieldOperatorVec_ae A C hA hm f |
|
|
| theorem norm_fieldOperator_le (hC : 0 ≤ C) : ‖fieldOperator A C hA hm‖ ≤ C := |
| LinearMap.mkContinuous_norm_le _ hC _ |
|
|
| variable {A C hA hm} |
|
|
| theorem fieldOperator_congr |
| {B : X → E →L[ℂ] F} {D : ℝ} {hB : ∀ x, ‖B x‖ ≤ D} |
| {hn : ∀ v, AEStronglyMeasurable (fun x => B x v) μ} |
| (heq : A =ᵐ[μ] B) : fieldOperator A C hA hm = fieldOperator B D hB hn := by |
| ext f |
| filter_upwards [fieldOperator_ae A C hA hm f, |
| fieldOperator_ae B D hB hn f, heq] with x hx hy hxy |
| rw [hx, hy, hxy] |
|
|
| theorem fieldOperator_comp_of_ae_eq |
| {B : X → F →L[ℂ] G} {P : X → E →L[ℂ] G} {D R : ℝ} |
| {hB : ∀ x, ‖B x‖ ≤ D} {hP : ∀ x, ‖P x‖ ≤ R} |
| {hn : ∀ v, AEStronglyMeasurable (fun x => B x v) μ} |
| {hp : ∀ v, AEStronglyMeasurable (fun x => P x v) μ} |
| (heq : P =ᵐ[μ] fun x => (B x).comp (A x)) : |
| fieldOperator P R hP hp = |
| (fieldOperator B D hB hn).comp (fieldOperator A C hA hm) := by |
| ext f |
| filter_upwards [fieldOperator_ae P R hP hp f, |
| fieldOperator_ae B D hB hn (fieldOperator A C hA hm f), |
| fieldOperator_ae A C hA hm f, heq] with x hx hy hz he |
| change fieldOperator P R hP hp f x = |
| fieldOperator B D hB hn (fieldOperator A C hA hm f) x |
| rw [hx, hy, hz, he] |
| rfl |
|
|
| theorem fieldOperator_ae_eq_of_eq [IsFiniteMeasure μ] [SeparableSpace E] |
| {B : X → E →L[ℂ] F} {D : ℝ} {hB : ∀ x, ‖B x‖ ≤ D} |
| {hn : ∀ v, AEStronglyMeasurable (fun x => B x v) μ} |
| (heq : fieldOperator A C hA hm = fieldOperator B D hB hn) : A =ᵐ[μ] B := by |
| have hpoint (v : E) : ∀ᵐ x ∂μ, A x v = B x v := by |
| have heqv := congrArg (fun T : Lp E 2 μ →L[ℂ] Lp F 2 μ => T (Lp.const 2 μ v)) heq |
| have hae : fieldOperator A C hA hm (Lp.const 2 μ v) =ᵐ[μ] |
| fieldOperator B D hB hn (Lp.const 2 μ v) := by rw [heqv] |
| filter_upwards [fieldOperator_ae A C hA hm (Lp.const 2 μ v), |
| fieldOperator_ae B D hB hn (Lp.const 2 μ v), Lp.coeFn_const 2 μ v, hae] |
| with x hx hy hv he |
| simpa only [hx, hy, hv, Function.const_apply] using he |
| have hseq : ∀ᵐ x ∂μ, ∀ n : ℕ, A x (denseSeq E n) = B x (denseSeq E n) := |
| ae_all_iff.mpr fun n => hpoint (denseSeq E n) |
| filter_upwards [hseq] with x hx |
| have hfunctions : (A x : E → F) = (B x : E → F) := |
| (denseRange_denseSeq E).equalizer (A x).continuous (B x).continuous (funext hx) |
| exact DFunLike.ext' hfunctions |
| |
| end Invariant233.DirectIntegral |
| |
| namespace Invariant233.DirectIntegral |
| variable {X E F : Type*} [MeasurableSpace X] {μ : Measure X} |
| [NormedAddCommGroup E] [InnerProductSpace ℂ E] [CompleteSpace E] |
| [NormedAddCommGroup F] [InnerProductSpace ℂ F] [CompleteSpace F] |
| {A : X → E →L[ℂ] F} {C : ℝ} {hA : ∀ x, ‖A x‖ ≤ C} |
| {hm : ∀ v, AEStronglyMeasurable (fun x => A x v) μ} |
| |
| theorem fieldOperator_adjoint_of_ae_eq |
| {B : X → F →L[ℂ] E} {D : ℝ} {hB : ∀ x, ‖B x‖ ≤ D} |
| {hn : ∀ v, AEStronglyMeasurable (fun x => B x v) μ} |
| (heq : B =ᵐ[μ] fun x => ContinuousLinearMap.adjoint (A x)) : |
| fieldOperator B D hB hn = ContinuousLinearMap.adjoint (fieldOperator A C hA hm) := by |
| apply (ContinuousLinearMap.eq_adjoint_iff _ _).2 |
| intro f g |
| simp only [L2.inner_def] |
| apply integral_congr_ae |
| filter_upwards [fieldOperator_ae B D hB hn f, |
| fieldOperator_ae A C hA hm g, heq] with x hx hy he |
| rw [hx, hy, he] |
| exact (A x).adjoint_inner_left (g x) (f x) |
| |
| end Invariant233.DirectIntegral |
| |
| end |
| |
| end OAI |
| |
