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a458723 | 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 | 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
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