| import OAI.Analysis.FiniteFactor.TailEstimate |
| import Mathlib.MeasureTheory.Measure.WithDensityFinite |
|
|
| namespace OAI |
|
|
| namespace FiniteFactor |
| open MeasureTheory Filter |
| open scoped ENNReal NNReal Topology |
| noncomputable section |
|
|
| /-- Cauchy--Schwarz bounds the mass of any Borel set under an actual L2 |
| density, in terms of the original reference measure of the set. -/ |
| theorem withDensity_L2_set_le {X : Type*} [MeasurableSpace X] |
| (μ : Measure X) [IsFiniteMeasure μ] (g : X → ℝ) |
| (hg : MemLp g 2 μ) (hg0 : ∀ x, 0 ≤ g x) |
| (A : Set X) (hA : MeasurableSet A) (C : ℝ) (hC : (∫ x, g x ^ 2 ∂μ) ≤ C) : |
| (μ.withDensity (fun x => ENNReal.ofReal (g x))) A ≤ |
| ENNReal.ofReal (Real.sqrt C * Real.sqrt ((μ A).toReal)) := by |
| have hi : MemLp (A.indicator (fun _ : X => (1 : ℝ))) 2 μ := |
| (memLp_const (1 : ℝ)).indicator hA |
| have hcs := integral_mul_le_sqrt_squares hg hi |
| (Eventually.of_forall hg0) |
| (Eventually.of_forall (fun x => Set.indicator_nonneg (fun _ _ => zero_le_one) x)) |
| have hint : (∫ x, g x * A.indicator (fun _ : X => (1 : ℝ)) x ∂μ) = ∫ x in A, g x ∂μ := by |
| rw [← integral_indicator hA] |
| apply integral_congr_ae |
| apply Eventually.of_forall |
| intro x |
| by_cases hx : x ∈ A <;> simp [hx] |
| have hsquare : (∫ x, (A.indicator (fun _ : X => (1 : ℝ)) x) ^ 2 ∂μ) = (μ A).toReal := by |
| have he : (fun x => (A.indicator (fun _ : X => (1 : ℝ)) x) ^ 2) = |
| A.indicator (fun _ : X => (1 : ℝ)) := by |
| funext x; by_cases hx : x ∈ A <;> simp [hx] |
| rw [he, integral_indicator hA] |
| simp [measureReal_def] |
| rw [hint, hsquare] at hcs |
| rw [withDensity_apply _ hA, ← ofReal_integral_eq_lintegral_ofReal |
| (hg.integrable (by norm_num)).restrict (Eventually.of_forall hg0)] |
| exact ENNReal.ofReal_le_ofReal (hcs.trans |
| (mul_le_mul_of_nonneg_right (Real.sqrt_le_sqrt hC) (Real.sqrt_nonneg _))) |
|
|
| /-- Uniform L2 density bounds imply uniform absolute continuity, with no |
| measurable-selection premise for the densities. -/ |
| theorem uniformAC_of_L2_densities {X : Type*} [MeasurableSpace X] |
| (μ : Measure X) [IsFiniteMeasure μ] (ν : ℕ → Measure X) (C : ℝ) |
| (hν : ∀ n, ∃ g : X → ℝ, MemLp g 2 μ ∧ (∀ x, 0 ≤ g x) ∧ |
| ν n = μ.withDensity (fun x => ENNReal.ofReal (g x)) ∧ (∫ x, g x^2 ∂μ) ≤ C) |
| (ε : ℝ≥0) (hε : 0 < ε) : |
| ∃ δ : ℝ≥0∞, 0 < δ ∧ ∀ A, MeasurableSet A → μ A < δ → ∀ n, ν n A ≤ ε := by |
| let r : ℝ := (ε : ℝ) / (Real.sqrt C + 1) |
| have hr : 0 < r := div_pos (by exact_mod_cast hε) (by positivity) |
| refine ⟨ENNReal.ofReal (r^2), ENNReal.ofReal_pos.mpr (sq_pos_of_pos hr), ?_⟩ |
| intro A hA hμ n |
| obtain ⟨g,hg,hg0,he,hgC⟩ := hν n |
| rw [he] |
| apply (withDensity_L2_set_le μ g hg hg0 A hA C hgC).trans |
| have hreal : (μ A).toReal ≤ r^2 := by |
| exact (ENNReal.toReal_le_of_le_ofReal (by positivity) hμ.le) |
| have hs : Real.sqrt ((μ A).toReal) ≤ r := by |
| exact (Real.sqrt_le_sqrt hreal).trans_eq (Real.sqrt_sq hr.le) |
| have hmul : Real.sqrt C * r ≤ (ε : ℝ) := by |
| have heq : r * (Real.sqrt C + 1) = (ε : ℝ) := div_mul_cancel₀ _ (by positivity) |
| nlinarith [Real.sqrt_nonneg C] |
| exact (ENNReal.ofReal_le_ofReal |
| ((mul_le_mul_of_nonneg_left hs (Real.sqrt_nonneg C)).trans hmul)).trans_eq (by simp) |
|
|
| end |
| end FiniteFactor |
|
|
| end OAI |
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