| import Mathlib | |
| import OAI.Geometry.IntegralFillings.Currents.CanonicalMass | |
| import OAI.Geometry.IntegralFillings.Currents.WeightedAction | |
| namespace OAI | |
| universe u | |
| section | |
| open Set MeasureTheory Measure Filter Module | |
| open Set Filter MeasureTheory Measure ContinuousLinearMap | |
| open scoped Topology Convolution NNReal | |
| open Set Filter MeasureTheory Measure Metric | |
| open scoped Topology ContDiff | |
| open Set Filter Metric | |
| open Set MeasureTheory Filter | |
| open Set Filter MeasureTheory | |
| open scoped Topology ENNReal NNReal | |
| open Filter Set | |
| open scoped Topology NNReal | |
| open Set Filter MeasureTheory TopologicalSpace | |
| open scoped Topology ENNReal | |
| open MeasureTheory Filter Set Metric | |
| open scoped Topology Pointwise NNReal | |
| open Set MeasureTheory | |
| open scoped RealInnerProductSpace | |
| open Matrix | |
| open scoped RealInnerProductSpace MatrixOrder | |
| namespace SharpIntegralFillings | |
| attribute [local instance] Classical.propDecidable | |
| namespace BorelRestriction | |
| open BorelCoefficients MassMeasure | |
| variable {X : Type u} [MetricSpace X] [CompactSpace X] [MeasurableSpace X] [BorelSpace X] | |
| variable {k : ℕ} {T : Functional X k} | |
| noncomputable def restrictCurrent (hT : IsMetricCurrent T) (E : Set X) : Functional X k := | |
| fun b π => if Admissible b π then | |
| borelAction (currentMassMeasure hT) hT (E.indicator b) π else 0 | |
| lemma restrictCurrent_apply (hT : IsMetricCurrent T) (E : Set X) {b : X → ℝ} | |
| {π : Fin k → X → ℝ} (h : Admissible b π) : | |
| restrictCurrent hT E b π = borelAction (currentMassMeasure hT) hT (E.indicator b) π := | |
| ite_eq_left h | |
| lemma restrictCurrent_isMetricCurrent (hT : IsMetricCurrent T) {E : Set X} | |
| (hE : MeasurableSet E) : IsMetricCurrent (restrictCurrent hT E) := by | |
| have he : restrictCurrent hT E = | |
| weightedCurrent (currentMassMeasure hT) hT (E.indicator (fun _ => 1)) := by | |
| funext b π | |
| dsimp only [restrictCurrent, weightedCurrent] | |
| congr 2 | |
| funext x | |
| by_cases hx : x ∈ E <;> simp [hx] | |
| rw [he] | |
| apply weightedCurrent_isMetricCurrent _ hT (currentMassMeasure_controls hT) | |
| ((integrable_const (1 : ℝ)).indicator hE) (M := 1) | |
| exact Eventually.of_forall fun x => by | |
| by_cases hx : x ∈ E <;> simp [hx] | |
| lemma restrictCurrent_controls (hT : IsMetricCurrent T) {E : Set X} | |
| (hE : MeasurableSet E) : Controls (restrictCurrent hT E) ((currentMassMeasure hT).restrict E) := by | |
| intro b π hb hπ | |
| rw [restrictCurrent_apply hT E ⟨hb, fun i => ⟨1, hπ i⟩⟩] | |
| have h := borelAction_bound (currentMassMeasure hT) hT (currentMassMeasure_controls hT) | |
| ((integrable_boundedLip _ hb).indicator hE) π (fun _ => 1) hπ | |
| simp only [NNReal.coe_one, Finset.prod_const_one, one_mul] at h | |
| apply h.trans_eq | |
| rw [← integral_indicator hE] | |
| apply integral_congr_ae | |
| exact Eventually.of_forall fun x => by | |
| by_cases hx : x ∈ E <;> simp [hx] | |
| lemma restrictCurrent_complement_sum (hT : IsMetricCurrent T) {E : Set X} | |
| (hE : MeasurableSet E) (b : X → ℝ) (π : Fin k → X → ℝ) : | |
| restrictCurrent hT E b π + restrictCurrent hT Eᶜ b π = T b π := by | |
| by_cases h : Admissible b π | |
| · rw [restrictCurrent_apply hT E h, restrictCurrent_apply hT Eᶜ h, | |
| ← borelAction_add (currentMassMeasure hT) hT | |
| ((integrable_boundedLip _ h.1).indicator hE) | |
| ((integrable_boundedLip _ h.1).indicator hE.compl) π h.2] | |
| have he : E.indicator b + Eᶜ.indicator b = b := by | |
| funext x | |
| by_cases hx : x ∈ E <;> simp [hx] | |
| rw [he] | |
| exact borelAction_eq _ hT (currentMassMeasure_controls hT) h | |
| · simp only [restrictCurrent, ite_eq_right h, hT.offDomain b π h, add_zero] | |
| lemma restrictMass_le (hT : IsMetricCurrent T) {E : Set X} (hE : MeasurableSet E) : | |
| currentMassMeasure (restrictCurrent_isMetricCurrent hT hE) ≤ | |
| (currentMassMeasure hT).restrict E := | |
| currentMassMeasure_le _ (restrictCurrent_controls hT hE) | |
| lemma controls_restriction_sum (hT : IsMetricCurrent T) {E : Set X} | |
| (hE : MeasurableSet E) : | |
| Controls T (currentMassMeasure (restrictCurrent_isMetricCurrent hT hE) + | |
| currentMassMeasure (restrictCurrent_isMetricCurrent hT hE.compl)) := by | |
| intro b π hb hπ | |
| rw [← restrictCurrent_complement_sum hT hE b π] | |
| apply (abs_add_le _ _).trans | |
| rw [integral_add_measure (integrable_boundedLip _ hb).abs (integrable_boundedLip _ hb).abs] | |
| exact add_le_add (currentMassMeasure_controls (restrictCurrent_isMetricCurrent hT hE) b π hb hπ) | |
| (currentMassMeasure_controls (restrictCurrent_isMetricCurrent hT hE.compl) b π hb hπ) | |
| lemma restriction_massMeasure (hT : IsMetricCurrent T) {E : Set X} | |
| (hE : MeasurableSet E) : | |
| currentMassMeasure (restrictCurrent_isMetricCurrent hT hE) = | |
| (currentMassMeasure hT).restrict E := by | |
| apply le_antisymm (restrictMass_le hT hE) | |
| rw [Measure.le_iff] | |
| intro B hB | |
| let μ := currentMassMeasure hT | |
| let ν := currentMassMeasure (restrictCurrent_isMetricCurrent hT hE) | |
| let ξ := currentMassMeasure (restrictCurrent_isMetricCurrent hT hE.compl) | |
| have hle : μ ≤ ν + ξ := currentMassMeasure_le hT (controls_restriction_sum hT hE) | |
| have hξ : ξ (B ∩ E) = 0 := by | |
| apply le_antisymm _ bot_le | |
| apply (restrictMass_le hT hE.compl (B ∩ E)).trans | |
| rw [Measure.restrict_apply (hB.inter hE)] | |
| have he : B ∩ E ∩ Eᶜ = ∅ := by aesop | |
| rw [he, measure_empty] | |
| exact le_refl _ | |
| rw [Measure.restrict_apply hB] | |
| calc | |
| μ (B ∩ E) ≤ (ν + ξ) (B ∩ E) := hle _ | |
| _ = ν (B ∩ E) := by rw [Measure.add_apply, hξ, add_zero] | |
| _ ≤ ν B := measure_mono inter_subset_left | |
| lemma restriction_mass (hT : IsMetricCurrent T) {E : Set X} | |
| (hE : MeasurableSet E) : | |
| mass (restrictCurrent hT E) = (currentMassMeasure hT).real E := by | |
| rw [← currentMassMeasure_total (restrictCurrent_isMetricCurrent hT hE), | |
| Measure.real, restriction_massMeasure hT hE, Measure.restrict_apply_univ] | |
| rfl | |
| end BorelRestriction | |
| end SharpIntegralFillings | |
| namespace SharpIntegralFillings | |
| attribute [local instance] Classical.propDecidable | |
| end SharpIntegralFillings | |
| end | |
| end OAI | |