| import Mathlib |
| import OAI.Analysis.Conductivity.Sources.AngularGraph |
|
|
| namespace OAI |
|
|
| section |
|
|
| noncomputable section |
| namespace ScalarConductivity |
| open Set MeasureTheory Filter Topology UnitAddTorus |
| open scoped NNReal ENNReal |
|
|
| def faceRayMeasure (w : ℝ) (j : Fin 4) : Measure UnitAddCircle := |
| Measure.map (fun b : ℝ => ((faceRayAngle w j b:ℝ):UnitAddCircle)) |
| ((volume.restrict (Ioc (-1:ℝ) 1)).withDensity (fun b => ENNReal.ofReal (faceRayDensity w j b))) |
| |
| lemma continuous_faceRayCircle (w : ℝ) (j : Fin 4) : |
| Continuous (fun b : ℝ => ((faceRayAngle w j b:ℝ):UnitAddCircle)) := |
| (AddCircle.continuous_mk' 1).comp |
| (continuous_iff_continuousAt.mpr fun b => (hasDerivAt_faceRayAngle w j b).continuousAt) |
| |
| instance finite_faceRayMeasure (w : ℝ) (j : Fin 4) : IsFiniteMeasure (faceRayMeasure w j) := by |
| have h : IntegrableOn (faceRayDensity w j) (Icc (-1:ℝ) 1) volume := |
| (continuous_faceRayDensity w j).continuousOn.integrableOn_compact |
| (isCompact_Icc (a:=(-1:ℝ)) (b:=1)) |
| have := isFiniteMeasure_withDensity_ofReal (h.mono_set Ioc_subset_Icc_self).hasFiniteIntegral |
| unfold faceRayMeasure |
| infer_instance |
|
|
| lemma integral_faceRayMeasure {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| {w : ℝ} (hw : 0<w) (j : Fin 4) {F : UnitAddCircle → E} |
| (hF : AEStronglyMeasurable F (faceRayMeasure w j)) : |
| (∫ θ,F θ ∂faceRayMeasure w j)= |
| ∫ b in Ioc (-1:ℝ) 1,faceRayDensity w j b • F ((faceRayAngle w j b:ℝ):UnitAddCircle) := by |
| rw [faceRayMeasure,integral_map (continuous_faceRayCircle w j).measurable.aemeasurable hF] |
| rw [integral_withDensity_eq_integral_toReal_smul |
| (continuous_faceRayDensity w j).measurable.ennreal_ofReal |
| (Filter.Eventually.of_forall fun b => ENNReal.ofReal_lt_top)] |
| congr 1 |
| funext b |
| rw [ENNReal.toReal_ofReal (faceRayDensity_pos' hw j b).le] |
| |
| theorem haar_circle_eq_sum_faceRayMeasure {w : ℝ} (hw : 0<w) : |
| AddCircle.haarAddCircle=∑ j : Fin 4,faceRayMeasure w j := by |
| apply Measure.ext_of_integral_eq_on_compactlySupported |
| intro F |
| rw [integral_finsetSum_measure] |
| · rw [integral_circle_faces (F := fun θ => F θ) hw F.continuous] |
| apply Finset.sum_congr rfl |
| intro j _ |
| rw [integral_faceRayMeasure (F := fun θ => F θ) hw j F.continuous.aestronglyMeasurable] |
| rw [intervalIntegral.integral_of_le (by norm_num : (-1:ℝ)≤1)] |
| apply integral_congr_ae |
| filter_upwards [] with b |
| simp only [smul_eq_mul,mul_comm] |
| · intro j _ |
| exact F.integrable |
| |
| theorem integral_circle_faces_integrable {E : Type*} [NormedAddCommGroup E] |
| [NormedSpace ℝ E] {w : ℝ} (hw : 0<w) {F : UnitAddCircle → E} |
| (hF : Integrable F AddCircle.haarAddCircle) : |
| (∫ θ,F θ ∂AddCircle.haarAddCircle)= |
| ∑ j : Fin 4, ∫ b in Ioc (-1:ℝ) 1, |
| faceRayDensity w j b • F ((faceRayAngle w j b:ℝ):UnitAddCircle) := by |
| have hsum : Integrable F (∑ j : Fin 4,faceRayMeasure w j) := by |
| rwa [←haar_circle_eq_sum_faceRayMeasure hw] |
| have hint (j : Fin 4) : Integrable F (faceRayMeasure w j) := |
| hsum.mono_measure (Finset.single_le_sum (fun _ _ => Measure.zero_le _) (Finset.mem_univ j)) |
| rw [haar_circle_eq_sum_faceRayMeasure hw,integral_finsetSum_measure (fun j _ => hint j)] |
| apply Finset.sum_congr rfl |
| intro j _ |
| exact integral_faceRayMeasure hw j (hint j).aestronglyMeasurable |
| |
| end ScalarConductivity |
| |
| end |
| end |
| |
| end OAI |
| |
