| import OAI.Analysis.CircleDomains.Topology.SimpleArcToll |
| import OAI.Analysis.CircleDomains.Sobolev.ExtendedWeightedCoarea |
|
|
| namespace OAI |
|
|
| /-! Square-summable point weights and level-incidence bounds give finite |
| atomic payment on almost every prior level. -/ |
|
|
| open Set Function MeasureTheory |
| open scoped ENNReal Classical |
|
|
| noncomputable section |
| namespace CircleDomainRigidity |
|
|
| theorem ae_finite_atomic_level_payment |
| {X T : Type*} [MeasurableSpace X] [MeasurableSpace T] |
| (ν : Measure T) (H : T → Set X) |
| (hH : ∀t, MeasurableSet (H t)) |
| (hsection : ∀x, MeasurableSet {t | x ∈ H t}) |
| (m a : X → ℝ≥0∞) |
| (hm : (∑'x, m x ^ 2) ≠ ∞) (ha : (∑'x, a x ^ 2) ≠ ∞) |
| (hincidence : ∀x, ν {t | x ∈ H t} ≤ a x) : |
| ∀ᵐt ∂ν, atomicBoundaryMeasure m (H t) < ∞ := by |
| have hcount : (support m).Countable := by |
| simpa only [support, ne_eq, sq_eq_zero_iff] using |
| (Summable.countable_support_ennreal hm) |
| let : Countable (support m) := hcount.to_subtype |
| let L : T → ℝ≥0∞ := fun t => ∑'x : support m, |
| ({t | x.val ∈ H t}).indicator (fun _ => m x.val) t |
| have hL : Measurable L := Measurable.tsum fun x => |
| measurable_const.indicator (hsection x.val) |
| have hmass (t : T) : atomicBoundaryMeasure m (H t) = L t := by |
| rw [atomicBoundaryMeasure_apply m (hH t)] |
| have he : (∑'x : support m, (H t).indicator m x.val) = ∑'x, (H t).indicator m x := by |
| apply tsum_subtype_eq_of_support_subset |
| intro x hx |
| by_contra hmzero |
| have hz : m x = 0 := not_not.mp hmzero |
| simp only [mem_support] at hx |
| exact hx (by simp [hz]) |
| rw [← he] |
| apply tsum_congr |
| intro x |
| by_cases hx : x.val ∈ H t <;> simp [hx] |
| have hfinite : (∫⁻t, L t ∂ν) < ∞ := by |
| calc |
| _ = ∑'x : support m, m x.val * ν {t | x.val ∈ H t} := by |
| rw [show (∫⁻t, L t ∂ν) = |
| ∫⁻t, ∑'x : support m, ({t | x.val ∈ H t}).indicator (fun _ => m x.val) t ∂ν from rfl, |
| lintegral_tsum] |
| · exact tsum_congr fun x => lintegral_indicator_const (hsection x.val) _ |
| · intro x |
| exact (measurable_const.indicator (hsection x.val)).aemeasurable |
| _ ≤ ∑'x : support m, (m x.val ^ 2 + a x.val ^ 2) := by |
| apply ENNReal.tsum_le_tsum |
| intro x |
| exact (mul_le_mul_right (hincidence x.val) (m x.val)).trans |
| (ennreal_mul_le_sq_add_sq (m x.val) (a x.val)) |
| _ = (∑'x : support m, m x.val ^ 2) + ∑'x : support m, a x.val ^ 2 := |
| ENNReal.tsum_add |
| _ ≤ (∑'x, m x ^ 2) + ∑'x, a x ^ 2 := add_le_add |
| (ENNReal.tsum_comp_le_tsum_of_injective |
| (f := fun x : support m => (x.val : X)) Subtype.val_injective (fun x => m x ^ 2)) |
| (ENNReal.tsum_comp_le_tsum_of_injective |
| (f := fun x : support m => (x.val : X)) Subtype.val_injective (fun x => a x ^ 2)) |
| _ < ∞ := ENNReal.add_lt_top.mpr ⟨hm.lt_top, ha.lt_top⟩ |
| filter_upwards [ae_lt_top hL hfinite.ne] with t ht |
| rwa [hmass] |
|
|
| end CircleDomainRigidity |
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|
| end |
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| end OAI |
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|
