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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 | import Mathlib
import OAI.Geometry.IntegralFillings.Charts.Differentiability
import OAI.Geometry.IntegralFillings.Charts.Jacobian
namespace OAI
section
open Set Filter MeasureTheory
open scoped Topology ENNReal NNReal
open MeasureTheory Filter Set Metric
open scoped Topology Pointwise NNReal
namespace SharpIntegralFillings
namespace IntegerChart
variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X] [Nonempty X]
{k : ℕ} (C : IntegerChart X k)
noncomputable def paramExtended : Euc k → X :=
by classical exact fun z => if hz : z ∈ C.domain then C.param ⟨z,hz⟩ else Classical.arbitrary X
lemma measurable_paramExtended : Measurable C.paramExtended := by
classical
obtain ⟨L,U,hL,_⟩ := C.bilipschitz
exact hL.continuous.measurable.dite measurable_const C.borel
omit [MeasurableSpace X] [BorelSpace X] in
lemma scalar_ae_paramExtended (b : X → ℝ) :
C.scalar b =ᵐ[volume.restrict C.domain] b ∘ C.paramExtended := by
filter_upwards [ae_restrict_mem C.borel] with z hz
simp only [scalar,paramExtended,Function.comp_apply,dite_eq_left hz]
variable [CompactSpace X]
omit [Nonempty X] in
lemma measurableEmbedding_param : MeasurableEmbedding C.param := by
let := C.borel.standardBorel
obtain ⟨L,U,hL,hU⟩ := C.bilipschitz
exact hL.continuous.measurable.measurableEmbedding hU.injective
omit [MeasurableSpace X] [BorelSpace X] [Nonempty X] [CompactSpace X] in
lemma integrable_weighted_jacobian {π : Fin k → X → ℝ}
(hπ : ∀ i, ∃ K : ℝ≥0, LipschitzWith K (π i)) :
Integrable (fun z => (C.multiplicity z : ℝ)*C.jacobian π z)
(volume.restrict C.domain) := by
obtain ⟨B,hB⟩ := C.exists_jacobian_bound hπ
exact C.integrable.mul_bdd (C.jacobian_aestronglyMeasurable hπ) hB
lemma weighted_jacobian_le_measure
(hC : IsMetricCurrent C.action) {ν : Measure X} [IsFiniteMeasure ν]
(hν : Controls C.action ν) (π : Fin k → X → ℝ) (K : Fin k → ℝ≥0)
(hK : ∀ i, LipschitzWith (K i) (π i)) :
densityPush (volume.restrict C.domain) C.paramExtended
(fun z => |(C.multiplicity z : ℝ)*C.jacobian π z|) ≤ (∏ i, K i) • ν := by
have he : MeasurableEmbedding (fun z : C.domain => C.paramExtended (z : Euc k)) := by
convert C.measurableEmbedding_param using 1
funext z
simp [paramExtended,z.property]
apply signed_chart_control_dominates_on (volume.restrict C.domain) ((∏ i,K i) • ν)
C.measurable_paramExtended (ae_restrict_mem C.borel) he
(C.integrable_weighted_jacobian (fun i => ⟨K i,hK i⟩))
intro b hb
have heq : (∫ z, (C.multiplicity z : ℝ)*C.jacobian π z*b (C.paramExtended z)
∂volume.restrict C.domain) = C.action b π := by
rw [action,ite_eq_left ⟨hb,fun i => ⟨K i,hK i⟩⟩]
apply integral_congr_ae
filter_upwards [C.scalar_ae_paramExtended b] with z hz
dsimp only [Function.comp_apply] at hz
rw [hz]
ring
rw [heq,integral_smul_nnreal_measure]
simpa only [NNReal.smul_def,NNReal.coe_prod,smul_eq_mul] using hC.mass_bound hν hb K hK
end IntegerChart
end SharpIntegralFillings
namespace SharpIntegralFillings
namespace IntegerChart
variable {X : Type*} [MetricSpace X] {k : ℕ} (C : IntegerChart X k)
lemma ae_fderivWithin_scalar_eq_linear {t : Set (Euc k)}
(ht : MeasurableSet t) (hts : t ⊆ C.domain)
{f : X → ℝ} {K : ℝ≥0} (hf : LipschitzWith K f) (ℓ : Euc k →L[ℝ] ℝ)
(heq : ∀ z ∈ t, C.scalar f z = ℓ z) :
∀ᵐ z ∂volume.restrict t, fderivWithin ℝ (C.scalar f) C.domain z = ℓ := by
have hd := (C.ae_differentiableWithinAt_scalar hf).filter_mono
(ae_mono (Measure.restrict_mono hts le_rfl))
filter_upwards [hd,ae_uniqueDiffWithinAt volume t,ae_restrict_mem ht] with z hz hu hzt
rw [←fderivWithin_subset hts hu hz,fderivWithin_congr' heq hzt,ℓ.fderivWithin hu]
lemma exists_inverse_coordinate_tests {t : Set (Euc k)}
(ht : MeasurableSet t) (hts : t ⊆ C.domain)
(ℓ : Fin k → Euc k →L[ℝ] ℝ) (K : ℝ≥0)
(hrow : ∀ i y (hy : y ∈ t) z (hz : z ∈ t),
|ℓ i y - ℓ i z| ≤ (K : ℝ)*dist (C.param ⟨y,hts hy⟩) (C.param ⟨z,hts hz⟩)) :
∃ π : Fin k → X → ℝ, (∀ i, LipschitzWith K (π i)) ∧
(∀ i, ∀ z ∈ t, C.scalar (π i) z = ℓ i z) ∧
∀ᵐ z ∂volume.restrict t,
C.jacobian π z = (Matrix.of fun i j => ℓ i (EuclideanSpace.single j 1)).det := by
classical
by_cases hne : t.Nonempty
· let := hne.to_subtype
let φ : t → X := fun z => C.param ⟨z,hts z.property⟩
have hinj : Function.Injective φ := by
obtain ⟨L,U,hL,hU⟩ := C.bilipschitz
intro a b hab
apply Subtype.ext
exact congrArg (fun z : C.domain => (z : Euc k)) (hU.injective hab)
let q : Fin k → X → ℝ := fun i x => ℓ i ((Function.invFun φ x : t) : Euc k)
have hq (i : Fin k) : LipschitzOnWith K (q i) (Set.range φ) := by
apply lipschitzOnWith_iff_dist_le_mul.mpr
rintro _ ⟨a,rfl⟩ _ ⟨b,rfl⟩
simpa only [q,Function.leftInverse_invFun hinj a,
Function.leftInverse_invFun hinj b,Real.dist_eq] using
hrow i a a.property b b.property
choose π hπ heq using fun i => (hq i).extend_real
have hagree (i : Fin k) (z : Euc k) (hz : z ∈ t) : C.scalar (π i) z = ℓ i z := by
rw [C.scalar_eq (hts hz)]
change π i (φ ⟨z,hz⟩) = _
rw [←heq i (Set.mem_range_self (⟨z,hz⟩ : t))]
dsimp only [q]
rw [Function.leftInverse_invFun hinj]
refine ⟨π,hπ,hagree,?_⟩
have hae : ∀ᵐ z ∂volume.restrict t, ∀ i,
fderivWithin ℝ (C.scalar (π i)) C.domain z = ℓ i :=
ae_all_iff.mpr fun i => C.ae_fderivWithin_scalar_eq_linear ht hts (hπ i) (ℓ i) (hagree i)
filter_upwards [hae] with z hz
simp only [jacobian,hz]
rfl
· have ht0 : t = ∅ := Set.not_nonempty_iff_eq_empty.mp hne
subst t
exact ⟨fun _ _ => 0,fun _ => LipschitzWith.const' (0 : ℝ),by simp,by simp⟩
end IntegerChart
end SharpIntegralFillings
end
end OAI
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