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a97875e | 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 | import OAI.Analysis.CircleDomains.Topology.MeanSigmaCompactPhysicalCount
import OAI.Analysis.CircleDomains.Modulus.SphericalChartCost
import OAI.Analysis.CircleDomains.Topology.SphereSliceChart
namespace OAI
/-! Finite generic intersections on a bounded physical chart follow
from the actual L2 occupation density and the coordinate's differential
energy. No simplicity of the sampled path is needed.
-/
open Set Function Filter MeasureTheory
open scoped Topology ENNReal NNReal
noncomputable section
namespace CircleDomainRigidity
attribute [local instance] sphereMetric sphereBorel
local instance : BorelSpace Sphere := ⟨rfl⟩
variable [MeasurableSpace C(PathTime, Sphere)] [BorelSpace C(PathTime, Sphere)]
theorem mean_real_chart_circle_count_finite
(hcoarea : PlanarLipschitzFiberCoarea)
{G U : Set Sphere} (e : G ≃ₜ U) (hU : IsOpen U) (y₀ : Sphere)
(π : Sphere → Sphere) (hπ : Continuous π) (he : ∀ y : G, (e y).val = π y.val)
{A D K : Set ℂ} (hA : IsSigmaCompact A) (hK : IsCompact K) (hAK : A ⊆ K)
(hAG : ∀ z ∈ A, (z : Sphere) ∈ G) (hAD : A ⊆ D) (hD : IsOpen D)
{f : ℂ → ℝ} (hf : ContDiffOn ℝ 1 f D)
(henergy : (∫⁻ z in D, ENNReal.ofReal ‖fderiv ℝ f z‖ ^ 2) < ∞)
(μ : ProbabilityMeasure C(PathTime, Sphere))
(hfinite : ∀ᵐ γ ∂(μ : Measure _), physicalDomainLength e y₀ γ < ∞)
(ρ : Sphere → ℝ≥0∞) (hρ : Measurable ρ)
(hocc : meanPhysicalOccupation e y₀ μ = sphericalArea.withDensity ρ)
(hρ2 : (∫⁻ y, ρ y ^ 2 ∂sphericalArea) < ∞) :
(∫⁻ t : UnitAddCircle, ∫⁻ γ,
pathImageCount ((fun z : ℂ => π (z : Sphere)) '' (A ∩ {z | (f z : UnitAddCircle) = t}))
γ ∂(μ : Measure _)) < ∞ := by
obtain ⟨B, hB⟩ := finiteCoord_lipschitz_on_compact_chart hK
have hAm : MeasurableSet A := by
obtain ⟨Q, hQ, hcov⟩ := hA
rw [← hcov]
exact MeasurableSet.iUnion fun n => (hQ n).measurableSet
have hcoeA : MeasurableSet (((↑) : ℂ → Sphere) '' A) :=
OnePoint.isOpenEmbedding_coe.measurableEmbedding.measurableSet_image.mpr hAm
let k : ℂ → ℝ≥0∞ := fun z => ENNReal.ofReal ‖fderiv ℝ f z‖
let g := localizedSphericalChartWeight B A k
have hk : Measurable k := (measurable_fderiv ℝ f).norm.ennreal_ofReal
have hg : Measurable g := measurable_localizedSphericalChartWeight B hAm hk
have hg2 : (∫⁻ y, g y ^ 2 ∂sphericalArea) < ∞ :=
localizedSphericalChartWeight_square_finite B hAm hk
((lintegral_mono_set hAD).trans_lt henergy)
have hcost : (∫⁻ y, g y ∂meanPhysicalOccupation e y₀ μ) < ∞ := by
rw [← lintegral_physicalOccupationCode e hU y₀ μ g hg]
exact (lintegral_physical_cost_le_square_norms e hU y₀ μ sphericalArea ρ g hρ hg hocc).trans_lt
(ENNReal.mul_lt_top
(ENNReal.rpow_lt_top_of_nonneg (by norm_num) hρ2.ne)
(ENNReal.rpow_lt_top_of_nonneg (by norm_num) hg2.ne))
have hbound := mean_sigmaCompact_physical_chart_circle_count_le hcoarea e hU y₀ π hπ he
((↑) : ℂ → Sphere) OnePoint.continuous_coe hA hAG hAD hD finiteCoord
measurable_finiteCoord (fun z _ => finiteCoord_coe z) B (hB.mono (image_mono hAK))
μ hfinite hf
have hi : (∫⁻ t in Ico (0 : ℝ) 1, ∫⁻ γ,
pathImageCount ((fun z : ℂ => π (z : Sphere)) ''
(A ∩ {z | (f z : UnitAddCircle) = (t : UnitAddCircle)})) γ ∂(μ : Measure _)) < ∞ := by
apply hbound.trans_lt
simpa only [g, localizedSphericalChartWeight, lintegral_const_mul' _ _ ENNReal.coe_ne_top,
lintegral_indicator hcoeA] using hcost
have hc := measurable_sigmaCompact_chart_circle_count
(fun z : ℂ => π (z : Sphere)) (hπ.comp OnePoint.continuous_coe) hA f
(hf.continuousOn.mono hAD)
have hm := (hc.comp measurable_swap).lintegral_prod_right' (ν := (μ : Measure _))
have heq := (AddCircle.measurePreserving_mk (1 : ℝ) 0).lintegral_comp hm
rw [restrict_Ico_eq_restrict_Ioc] at hi
simp only [zero_add] at heq
dsimp only [Function.comp_def, Prod.swap] at heq
exact heq ▸ hi
theorem ae_real_chart_circle_count_finite
(π : Sphere → Sphere) (hπ : Continuous π) {A : Set ℂ} (hA : IsSigmaCompact A)
(f : ℂ → ℝ) (hf : ContinuousOn f A)
(μ : ProbabilityMeasure C(PathTime, Sphere))
(hmean : (∫⁻ t : UnitAddCircle, ∫⁻ γ,
pathImageCount ((fun z : ℂ => π (z : Sphere)) '' (A ∩ {z | (f z : UnitAddCircle) = t}))
γ ∂(μ : Measure _)) < ∞) :
∀ᵐ γ ∂(μ : Measure _), ∀ᵐ t : UnitAddCircle ∂volume,
pathImageCount ((fun z : ℂ => π (z : Sphere)) '' (A ∩ {z | (f z : UnitAddCircle) = t})) γ < ∞ := by
have hc := measurable_sigmaCompact_chart_circle_count
(fun z : ℂ => π (z : Sphere)) (hπ.comp OnePoint.continuous_coe) hA f hf
have hm := (hc.comp measurable_swap).lintegral_prod_right' (ν := (μ : Measure _))
have hg : ∀ᵐ t : UnitAddCircle ∂volume, ∀ᵐ γ ∂(μ : Measure _),
pathImageCount ((fun z : ℂ => π (z : Sphere)) '' (A ∩ {z | (f z : UnitAddCircle) = t})) γ < ∞ := by
filter_upwards [ae_lt_top hm hmean.ne] with t ht
exact ae_lt_top (hc.comp (measurable_id.prodMk measurable_const)) ht.ne
have hmeas : MeasurableSet {p : C(PathTime, Sphere) × UnitAddCircle |
pathImageCount ((fun z : ℂ => π (z : Sphere)) ''
(A ∩ {z | (f z : UnitAddCircle) = p.2})) p.1 < ∞} :=
measurableSet_lt hc measurable_const
exact (Measure.ae_ae_comm hmeas).mpr hg
end CircleDomainRigidity
end
end OAI
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