| import Mathlib |
| import OAI.Analysis.Conductivity.Model |
|
|
| namespace OAI |
|
|
| section |
|
|
| noncomputable section |
| namespace ScalarConductivity |
| open Set MeasureTheory Filter Topology |
| open scoped ENNReal |
|
|
| lemma defect_tendstoInMeasure_of_small_exceptions |
| {X V : Type*} [MeasurableSpace X] [NormedAddCommGroup V] [NormedSpace ℝ V] |
| (μ : Measure X) (E : ℕ → Lp V 2 μ) (r : ℕ → X → V) |
| (η : ℕ → ℝ) (hη : ∀ n, 0 ≤ η n) (hηt : Tendsto η atTop (𝓝 0)) |
| (B : ℕ → Set X) (hB : Tendsto (fun n => μ (B n)) atTop (𝓝 0)) |
| {C : ℝ} (hC : 0 ≤ C) (hEC : ∀ n, ‖E n‖ ≤ C) |
| (hdef : ∀ n, ∀ᵐ x ∂μ, x ∉ B n → ‖r n x‖ ≤ η n * ‖E n x‖) : |
| TendstoInMeasure μ r atTop (fun _ => 0) := by |
| let Z : ℕ → Lp V 2 μ := fun n => η n • E n |
| have hZ : Tendsto Z atTop (𝓝 0) := by |
| apply tendsto_zero_iff_norm_tendsto_zero.mpr |
| apply squeeze_zero (fun n => norm_nonneg _) _ (by simpa using hηt.mul_const C) |
| intro n |
| change ‖η n • E n‖ ≤ η n * C |
| rw [norm_smul, Real.norm_of_nonneg (hη n)] |
| simpa only [mul_comm] using mul_le_mul (hEC n) le_rfl (hη n) hC |
| have hZI : TendstoInMeasure μ (fun n x => Z n x) atTop (fun _ => 0) := by |
| have hh := tendstoInMeasure_of_tendsto_Lp hZ |
| apply hh.congr_right |
| filter_upwards [Lp.coeFn_zero (E := V) (p := 2) (μ := μ)] with x hx |
| exact hx |
| rw [tendstoInMeasure_iff_norm] at hZI ⊢ |
| intro ε hε |
| simp only [sub_zero] at hZI ⊢ |
| apply tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_const_nhds |
| (by simpa only [add_zero] using hB.add (hZI ε hε)) |
| · exact Eventually.of_forall (fun _ => bot_le) |
| · apply Eventually.of_forall |
| intro n |
| have hs : ∀ᵐ x ∂μ, x ∈ {x | ε ≤ ‖r n x‖} → x ∈ B n ∪ {x | ε ≤ ‖Z n x‖} := by |
| filter_upwards [hdef n, Lp.coeFn_smul (η n) (E n)] with x hx hz |
| intro hr |
| by_cases hb : x ∈ B n |
| · exact Or.inl hb |
| · right |
| have he : ‖Z n x‖ = η n * ‖E n x‖ := by |
| change ‖(η n • E n : Lp V 2 μ) x‖ = _ |
| rw [hz, Pi.smul_apply, norm_smul, Real.norm_of_nonneg (hη n)] |
| exact (hr.trans (hx hb)).trans_eq he.symm |
| exact (measure_mono_ae hs).trans (measure_union_le _ _) |
|
|
| end ScalarConductivity |
|
|
| end |
| end |
|
|
| end OAI |
|
|
