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9655094 | 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 | 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
end
end OAI
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