| import OAI.Analysis.CircleDomains.Sobolev.LocalComplexWitness |
| import Mathlib.MeasureTheory.Integral.DominatedConvergence |
|
|
| namespace OAI |
|
|
| /-! Compact test pairings converge for the locally bounded pointwise |
| limit of the actual continuous height functions. -/ |
| open Set Function Filter MeasureTheory |
| open scoped Topology |
| noncomputable section |
| namespace CircleDomainRigidity |
|
|
| theorem height_test_integral_tendsto |
| (f : ℕ → EuclideanPlane → ℝ) (hf : ∀ n, Continuous (f n)) |
| {v : EuclideanPlane → ℝ} |
| (hp : ∀ p, Tendsto (fun n => f n p) atTop (𝓝 (v p))) |
| (hb : ∀ K : Set EuclideanPlane, IsCompact K → |
| ∃ C : ℝ, ∀ n p, p ∈ K → ‖f n p‖ ≤ C) |
| {ψ : EuclideanPlane → ℝ} (hψ : Continuous ψ) (hs : HasCompactSupport ψ) : |
| Tendsto (fun n => ∫ p, f n p*ψ p) atTop (𝓝 (∫ p, v p*ψ p)) := by |
| obtain ⟨C, hC⟩ := hb (tsupport ψ) hs.isCompact |
| apply tendsto_integral_of_dominated_convergence (bound := fun p => C*‖ψ p‖) |
| · intro n |
| exact (hf n).aestronglyMeasurable.mul hψ.aestronglyMeasurable |
| · exact (hψ.integrable_of_hasCompactSupport hs).norm.const_mul C |
| · intro n |
| apply Eventually.of_forall |
| intro p |
| by_cases hp : p ∈ tsupport ψ |
| · rw [norm_mul] |
| exact mul_le_mul_of_nonneg_right (hC n p hp) (norm_nonneg _) |
| · simp only [image_eq_zero_of_notMem_tsupport hp, mul_zero, norm_zero, le_refl] |
| · exact Eventually.of_forall (fun p => (hp p).mul_const (ψ p)) |
|
|
| end CircleDomainRigidity |
|
|
| end |
|
|
| end OAI |
|
|
