| import Mathlib |
| import OAI.Geometry.IntegralFillings.Currents.Restriction |
| import OAI.Geometry.IntegralFillings.Charts.DomainRestriction |
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|
| namespace OAI |
|
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| 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 |
| open Set MeasureTheory Filter |
| open scoped Topology |
|
|
| namespace IntegerChart |
| variable {X : Type*} [MetricSpace X] [CompactSpace X] |
| [MeasurableSpace X] [BorelSpace X] [Nonempty X] {k : ℕ} (C : IntegerChart X k) |
| noncomputable def restrictImage (E : Set X) (hE : MeasurableSet E) : IntegerChart X k := |
| C.restrictDomain (C.borel.inter (C.measurable_paramExtended hE)) inter_subset_left |
|
|
| omit [CompactSpace X] in |
| lemma restrictImage_image (E : Set X) (hE : MeasurableSet E) : |
| (C.restrictImage E hE).image = C.image ∩ E := by |
| ext x |
| constructor |
| · rintro ⟨z,rfl⟩ |
| exact ⟨⟨⟨z,z.property.1⟩,rfl⟩,by |
| change C.param ⟨z,z.property.1⟩ ∈ E |
| simpa only [mem_preimage,paramExtended,dite_eq_left z.property.1] using z.property.2⟩ |
| · rintro ⟨⟨z,rfl⟩,hz⟩ |
| refine ⟨⟨z,⟨z.property,?_⟩⟩,rfl⟩ |
| simpa only [mem_preimage,paramExtended,dite_eq_left z.property] using hz |
|
|
| lemma restrictImage_action (E : Set X) (hE : MeasurableSet E) |
| (hC : IsMetricCurrent C.action) : |
| (C.restrictImage E hE).action = BorelRestriction.restrictCurrent hC E := by |
| classical |
| let t := C.domain ∩ C.paramExtended ⁻¹' E |
| have ht : MeasurableSet t := C.borel.inter (C.measurable_paramExtended hE) |
| have hts : t ⊆ C.domain := inter_subset_left |
| have him : MeasurableSet (C.paramExtended '' t) := C.measurableSet_paramExtended_image ht hts |
| change (C.restrictDomain ht hts).action = _ |
| rw [C.restrictDomain_action ht hts hC,←C.restrictMultiplicity_action hC him, |
| ←C.restrictMultiplicity_action hC hE] |
| funext b π |
| by_cases hab : Admissible b π |
| · simp only [action,ite_eq_left hab] |
| apply integral_congr_ae |
| filter_upwards [ae_restrict_mem C.borel] with z hz |
| have hi : C.paramExtended z ∈ C.paramExtended '' t ↔ C.paramExtended z ∈ E := by |
| rw [C.paramExtended_mem_image_iff hts hz] |
| exact and_iff_right hz |
| change (↑((C.paramExtended ⁻¹' (C.paramExtended '' t)).indicator C.multiplicity z) : ℝ) * |
| C.scalar b z * C.jacobian π z = |
| ↑((C.paramExtended ⁻¹' E).indicator C.multiplicity z) * C.scalar b z * C.jacobian π z |
| by_cases he : C.paramExtended z ∈ E |
| · rw [indicator_of_mem (show z ∈ C.paramExtended ⁻¹' (C.paramExtended '' t) from hi.mpr he), |
| indicator_of_mem (show z ∈ C.paramExtended ⁻¹' E from he)] |
| · rw [indicator_of_notMem (show z ∉ C.paramExtended ⁻¹' (C.paramExtended '' t) from mt hi.mp he), |
| indicator_of_notMem (show z ∉ C.paramExtended ⁻¹' E from he)] |
| · simp only [action,ite_eq_right hab] |
|
|
| lemma restrictImage_isMetricCurrent (E : Set X) (hE : MeasurableSet E) |
| (hC : IsMetricCurrent C.action) : IsMetricCurrent (C.restrictImage E hE).action := by |
| rw [C.restrictImage_action E hE hC] |
| exact BorelRestriction.restrictCurrent_isMetricCurrent hC hE |
|
|
| lemma restrictImage_mass_le (E : Set X) (hE : MeasurableSet E) |
| (hC : IsMetricCurrent C.action) : mass (C.restrictImage E hE).action ≤ mass C.action := by |
| rw [C.restrictImage_action E hE hC,BorelRestriction.restriction_mass hC hE, |
| ←MassMeasure.currentMassMeasure_total hC] |
| exact measureReal_mono (subset_univ E) |
|
|
| lemma restrictCurrent_image_eq (hC : IsMetricCurrent C.action) {E : Set X} |
| (hE : MeasurableSet E) (hCE : C.image ⊆ E) : |
| BorelRestriction.restrictCurrent hC E = C.action := by |
| classical |
| rw [←C.restrictMultiplicity_action hC hE] |
| funext b π |
| by_cases hab : Admissible b π |
| · simp only [action,ite_eq_left hab] |
| apply integral_congr_ae |
| filter_upwards [ae_restrict_mem C.borel] with z hz |
| change z ∈ C.domain at hz |
| have hi : C.paramExtended z ∈ E := hCE ⟨⟨z,hz⟩,by simp only [paramExtended,dite_eq_left hz]⟩ |
| change (↑((C.paramExtended ⁻¹' E).indicator C.multiplicity z) : ℝ) * |
| C.scalar b z * C.jacobian π z = _ |
| rw [indicator_of_mem (show z ∈ C.paramExtended ⁻¹' E from hi)] |
| · simp only [action,ite_eq_right hab] |
|
|
| end IntegerChart |
| end SharpIntegralFillings |
|
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| end |
|
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| end OAI |
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