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9eeb9c7 | 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 | 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
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