| import Mathlib.MeasureTheory.Function.L2Space |
| import Mathlib.MeasureTheory.Function.StronglyMeasurable.Lemmas |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open MeasureTheory Filter |
| open scoped ENNReal Topology |
|
|
| namespace Invariant233.DirectIntegral |
| variable {X E F : Type*} [MeasurableSpace X] {μ : Measure X} |
| [NormedAddCommGroup E] [NormedSpace ℂ E] |
| [NormedAddCommGroup F] [NormedSpace ℂ F] |
|
|
| theorem apply_simple_aestronglyMeasurable |
| (A : X → E →L[ℂ] F) |
| (hm : ∀ v, AEStronglyMeasurable (fun x => A x v) μ) (s : SimpleFunc X E) : |
| AEStronglyMeasurable (fun x => A x (s x)) μ := by |
| classical |
| induction s using SimpleFunc.induction' with |
| | const v => exact hm v |
| | @pcw s t u hu hs ht => |
| convert AEStronglyMeasurable.piecewise hu hs.restrict ht.restrict using 1 |
| funext x |
| by_cases hx : x ∈ u <;> simp [SimpleFunc.coe_piecewise, hx] |
|
|
| theorem apply_aestronglyMeasurable |
| (A : X → E →L[ℂ] F) |
| (hm : ∀ v, AEStronglyMeasurable (fun x => A x v) μ) |
| {f : X → E} (hf : AEStronglyMeasurable f μ) : |
| AEStronglyMeasurable (fun x => A x (f x)) μ := by |
| apply aestronglyMeasurable_of_tendsto_ae atTop |
| (f := fun n x => A x (hf.stronglyMeasurable_mk.approx n x)) |
| · intro n |
| exact apply_simple_aestronglyMeasurable A hm _ |
| · filter_upwards [hf.ae_eq_mk] with x hx |
| rw [hx] |
| exact ((A x).continuous.tendsto _).comp (hf.stronglyMeasurable_mk.tendsto_approx x) |
|
|
| def fieldOperatorVec (A : X → E →L[ℂ] F) (C : ℝ) |
| (hA : ∀ x, ‖A x‖ ≤ C) |
| (hm : ∀ v, AEStronglyMeasurable (fun x => A x v) μ) |
| (f : Lp E 2 μ) : Lp F 2 μ := |
| ((Lp.memLp f).of_le_mul (apply_aestronglyMeasurable A hm (Lp.aestronglyMeasurable f)) |
| (Filter.Eventually.of_forall fun x => (A x).le_of_opNorm_le (hA x) (f x))).toLp _ |
|
|
| theorem fieldOperatorVec_ae (A : X → E →L[ℂ] F) (C : ℝ) |
| (hA : ∀ x, ‖A x‖ ≤ C) |
| (hm : ∀ v, AEStronglyMeasurable (fun x => A x v) μ) |
| (f : Lp E 2 μ) : |
| fieldOperatorVec A C hA hm f =ᵐ[μ] fun x => A x (f x) := |
| MemLp.coeFn_toLp _ |
|
|
| def fieldOperatorLinear (A : X → E →L[ℂ] F) (C : ℝ) |
| (hA : ∀ x, ‖A x‖ ≤ C) |
| (hm : ∀ v, AEStronglyMeasurable (fun x => A x v) μ) : |
| Lp E 2 μ →ₗ[ℂ] Lp F 2 μ where |
| toFun := fieldOperatorVec A C hA hm |
| map_add' f g := by |
| apply Lp.ext |
| filter_upwards [fieldOperatorVec_ae A C hA hm (f + g), |
| fieldOperatorVec_ae A C hA hm f, fieldOperatorVec_ae A C hA hm g, |
| Lp.coeFn_add f g, |
| Lp.coeFn_add (fieldOperatorVec A C hA hm f) (fieldOperatorVec A C hA hm g)] |
| with x hfg hf hg hsum hout |
| simp only [Pi.add_apply] at hsum hout |
| rw [hfg, hout, hf, hg, hsum, map_add] |
| map_smul' c f := by |
| change fieldOperatorVec A C hA hm (c • f) = c • fieldOperatorVec A C hA hm f |
| apply Lp.ext |
| filter_upwards [fieldOperatorVec_ae A C hA hm (c • f), |
| fieldOperatorVec_ae A C hA hm f, Lp.coeFn_smul c f, |
| Lp.coeFn_smul c (fieldOperatorVec A C hA hm f)] with x hcf hf hsmul hout |
| simp only [Pi.smul_apply] at hsmul hout |
| rw [hcf, hout, hf, hsmul, map_smul] |
|
|
| def fieldOperator (A : X → E →L[ℂ] F) (C : ℝ) |
| (hA : ∀ x, ‖A x‖ ≤ C) |
| (hm : ∀ v, AEStronglyMeasurable (fun x => A x v) μ) : |
| Lp E 2 μ →L[ℂ] Lp F 2 μ := |
| (fieldOperatorLinear A C hA hm).mkContinuous C (by |
| intro f |
| apply Lp.norm_le_mul_norm_of_ae_le_mul |
| change ∀ᵐ x ∂μ, ‖fieldOperatorVec A C hA hm f x‖ ≤ C * ‖f x‖ |
| filter_upwards [fieldOperatorVec_ae A C hA hm f] with x hx |
| rw [hx] |
| exact (A x).le_of_opNorm_le (hA x) (f x)) |
|
|
| end Invariant233.DirectIntegral |
|
|
| end |
|
|
| end OAI |
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|
