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159357f | 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 | import Mathlib
import OAI.Analysis.Conductivity.Sources.FaceMeasure
namespace OAI
section
noncomputable section
namespace ScalarConductivity
open Set MeasureTheory Filter Topology UnitAddTorus
open scoped ENNReal
local instance sourceFaceIntegralCircleMeasureSpace : MeasureSpace UnitAddCircle :=
⟨AddCircle.haarAddCircle⟩
local instance sourceFaceIntegralCircleIsProbabilityMeasure :
IsProbabilityMeasure (volume : Measure UnitAddCircle) :=
inferInstanceAs (IsProbabilityMeasure AddCircle.haarAddCircle)
lemma sourceFaceMeasure_le_cylinder {l r : ℝ} (hlr : l≤r) (i j : Fin 4) :
sourceFaceMeasure l r i j ≤
(volume.restrict (Ioc l r)).prod (volume : Measure (UnitAddTorus (Fin 2))) := by
rw [sourceCylinderMeasure_eq_faces hlr]
exact le_trans (Finset.single_le_sum (fun _ _ => Measure.zero_le _) (Finset.mem_univ j))
(Finset.single_le_sum (f := fun face => ∑ patch,sourceFaceMeasure l r face patch)
(fun _ _ => Measure.zero_le _) (Finset.mem_univ i))
lemma integrable_sourceFacePullback {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
{F : ℝ × UnitAddTorus (Fin 2) → E} {l r : ℝ} (hlr : l≤r)
(hF : Integrable F ((volume.restrict (Ioc l r)).prod volume)) (i j : Fin 4) :
IntegrableOn (fun x => sourceFaceWeight i j x • F (sourceFaceParameter i j x))
(sourceExtendedBox l r) volume := by
have hf := hF.mono_measure (sourceFaceMeasure_le_cylinder hlr i j)
have hg := (integrable_map_measure hf.aestronglyMeasurable
(continuous_sourceFaceParameter i j).aemeasurable).mp hf
have hh := (integrable_withDensity_iff_integrable_smul'
(continuous_sourceFaceWeight i j).measurable.ennreal_ofReal
(Eventually.of_forall fun _ => ENNReal.ofReal_lt_top)).mp hg
simpa only [IntegrableOn,Function.comp_apply,ENNReal.toReal_ofReal (sourceFaceWeight_pos i j _).le] using hh
theorem integral_sourceCylinder_faces {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
{F : ℝ × UnitAddTorus (Fin 2) → E} {l r : ℝ} (hlr : l≤r)
(hF : Integrable F ((volume.restrict (Ioc l r)).prod volume)) :
(∫ z,F z ∂(volume.restrict (Ioc l r)).prod volume)=
∑ i : Fin 4,∑ j : Fin 4,∫ x in sourceExtendedBox l r,
sourceFaceWeight i j x • F (sourceFaceParameter i j x) := by
have hf (i j : Fin 4) := hF.mono_measure (sourceFaceMeasure_le_cylinder hlr i j)
rw [sourceCylinderMeasure_eq_faces hlr,integral_finsetSum_measure (fun i _ =>
integrable_finsetSum_measure.mpr (fun j _ => hf i j))]
simp_rw [integral_finsetSum_measure (fun j _ => hf _ j)]
apply Finset.sum_congr rfl
intro i _
apply Finset.sum_congr rfl
intro j _
exact integral_sourceFaceMeasure l r i j (hf i j).aestronglyMeasurable
end ScalarConductivity
end
end
end OAI
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