| import Mathlib.MeasureTheory.Integral.Average |
|
|
| namespace OAI |
|
|
| /-! Elementary mean-oscillation bounds used for Sobolev square averages. -/ |
| open Set Function MeasureTheory |
| open scoped Topology |
| noncomputable section |
| namespace CircleDomainRigidity |
|
|
| lemma average_mono_real {X : Type*} [MeasurableSpace X] {μ : Measure X} |
| {f g : X → ℝ} (hf : Integrable f μ) (hg : Integrable g μ) (hfg : f ≤ᵐ[μ] g) : |
| (⨍ x, f x ∂μ) ≤ ⨍ x, g x ∂μ := by |
| rw [average_eq, average_eq] |
| exact mul_le_mul_of_nonneg_left (integral_mono_ae hf hg hfg) (by positivity) |
|
|
| lemma abs_average_le_average_abs {X : Type*} [MeasurableSpace X] |
| (μ : Measure X) (f : X → ℝ) : |⨍ x, f x ∂μ| ≤ ⨍ x, |f x| ∂μ := by |
| simpa only [average, Real.norm_eq_abs] using |
| (norm_integral_le_integral_norm (f := f) (μ := (μ univ)⁻¹ • μ)) |
|
|
| lemma abs_sub_average_le_average_abs_sub {X : Type*} [MeasurableSpace X] |
| {μ : Measure X} [IsFiniteMeasure μ] [NeZero μ] {f : X → ℝ} |
| (hf : Integrable f μ) (a : ℝ) : |
| |a-(⨍ x, f x ∂μ)| ≤ ⨍ x, |a-f x| ∂μ := by |
| have he : (⨍ x, a-f x ∂μ) = a-(⨍ x, f x ∂μ) := by |
| have hh := average_sub (integrable_const a : Integrable (fun _ : X => a) μ) hf |
| simpa only [Pi.sub_apply, average_const] using hh |
| rw [← he] |
| exact abs_average_le_average_abs μ _ |
|
|
| lemma abs_sub_average_le_of_ae {X : Type*} [MeasurableSpace X] |
| {μ : Measure X} [IsFiniteMeasure μ] [NeZero μ] {f : X → ℝ} |
| (hf : Integrable f μ) (a C : ℝ) (h : ∀ᵐ x ∂μ, |a-f x| ≤ C) : |
| |a-(⨍ x, f x ∂μ)| ≤ C := by |
| apply (abs_sub_average_le_average_abs_sub hf a).trans |
| simpa only [Pi.sub_apply, average_const] using |
| average_mono_real ((integrable_const a).sub hf).abs (integrable_const C) h |
|
|
| end CircleDomainRigidity |
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| end |
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| end OAI |
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