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59d1c80 | 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 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 | import Mathlib
import OAI.Geometry.IntegralFillings.Currents.Restriction
import OAI.Geometry.IntegralFillings.Charts.DomainRestriction
namespace OAI
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
end
end OAI
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