File size: 4,704 Bytes
ddc84c5 | 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 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 | import Mathlib.MeasureTheory.Integral.Bochner.Set
import Mathlib.MeasureTheory.Integral.Pi
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.Algebra.Order.Chebyshev
import Mathlib.Analysis.SpecialFunctions.Exp
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.Positivity
import Mathlib.Tactic.Ring
namespace OAI
namespace FiniteFactor
open MeasureTheory
open scoped ENNReal
noncomputable section
/-- Normalized restriction density, relative to the probability base measure. -/
def intervalDensity {X : Type*} (U : Set X) (length : ℝ) : X → ℝ :=
U.indicator (fun _ => length⁻¹)
variable {X : Type*} [MeasurableSpace X] {μ : Measure X} [IsProbabilityMeasure μ]
theorem intervalDensity_integrable {U : Set X} (hU : MeasurableSet U) (l : ℝ) :
Integrable (intervalDensity U l) μ := (integrable_const _).indicator hU
omit [MeasurableSpace X] in
theorem intervalDensity_nonneg [MeasurableSpace X] {U : Set X} {l : ℝ} (hl : 0 ≤ l) (x : X) :
0 ≤ intervalDensity U l x := by
classical
by_cases hx : x ∈ U <;> simp [intervalDensity, hx, inv_nonneg.mpr hl]
omit [MeasurableSpace X] in
theorem intervalDensity_le [MeasurableSpace X] {U : Set X} {l : ℝ} (hl : 0 ≤ l) (x : X) :
intervalDensity U l x ≤ l⁻¹ := by
classical
by_cases hx : x ∈ U <;> simp [intervalDensity, hx, inv_nonneg.mpr hl]
omit [IsProbabilityMeasure μ] in
theorem intervalDensity_integral [IsProbabilityMeasure μ] {U : Set X} (hU : MeasurableSet U)
{l : ℝ} (hl : 0 < l) (hm : μ.real U = l) :
∫ x, intervalDensity U l x ∂μ = 1 := by
rw [intervalDensity, integral_indicator hU, setIntegral_const, smul_eq_mul, hm]
exact mul_inv_cancel₀ hl.ne'
theorem intervalDensity_sq_integrable {U : Set X} (hU : MeasurableSet U) (l : ℝ) :
Integrable (fun x => intervalDensity U l x ^ 2) μ := by
have heq : (fun x => intervalDensity U l x ^ 2) = U.indicator (fun _ => (l⁻¹)^2) := by
funext x
by_cases hx : x ∈ U <;> simp [intervalDensity, hx]
rw [heq]
exact (integrable_const _).indicator hU
omit [IsProbabilityMeasure μ] in
theorem intervalDensity_sq_integral [IsProbabilityMeasure μ] {U : Set X} (hU : MeasurableSet U)
{l : ℝ} (hl : 0 < l) (hm : μ.real U = l) :
∫ x, intervalDensity U l x ^ 2 ∂μ = l⁻¹ := by
have heq : (fun x => intervalDensity U l x ^ 2) = U.indicator (fun _ => (l⁻¹)^2) := by
funext x
by_cases hx : x ∈ U <;> simp [intervalDensity, hx]
rw [heq, integral_indicator hU, setIntegral_const, smul_eq_mul, hm]
field_simp
/-- The mass-one identity cancels the linear term exactly. -/
theorem density_energy_identity {h : X → ℝ} (hh : Integrable h μ)
(hh2 : Integrable (fun x => h x ^ 2) μ) (hm : ∫ x, h x ∂μ = 1) :
(∫ x, (h x - 1)^2 ∂μ) = (∫ x, h x ^ 2 ∂μ) - 1 := by
have heq : (fun x => (h x - 1)^2) = fun x => h x ^ 2 - 2*h x + 1 := by
funext x; ring
have h₁ := integral_add (hh2.sub (hh.const_mul 2)) (integrable_const (1 : ℝ))
have h₂ := integral_sub hh2 (hh.const_mul 2)
dsimp only [Pi.sub_apply, Pi.add_apply] at h₁ h₂
rw [heq, h₁, h₂, integral_const_mul, hm]
simp
ring
/-- Each normalized interval perturbation has energy exactly 1/length - 1. -/
theorem intervalDensity_energy {U : Set X} (hU : MeasurableSet U)
{l : ℝ} (hl : 0 < l) (hm : μ.real U = l) :
(∫ x, (intervalDensity U l x - 1)^2 ∂μ) = l⁻¹ - 1 := by
rw [density_energy_identity (intervalDensity_integrable hU l)
(intervalDensity_sq_integrable hU l) (intervalDensity_integral hU hl hm),
intervalDensity_sq_integral hU hl hm]
/-- Finite independent endpoint selections: the square integral factors exactly. -/
theorem product_density_sq_integral {ι : Type*} [Fintype ι]
{E : ι → Type*} [∀ i, MeasurableSpace (E i)]
(ν : ∀ i, Measure (E i)) [∀ i, SigmaFinite (ν i)]
(h : ∀ i, E i → ℝ) :
(∫ x, (∏ i, h i (x i))^2 ∂Measure.pi ν) =
∏ i, ∫ x, h i x ^ 2 ∂ν i := by
simp_rw [← Finset.prod_pow]
exact integral_fintype_prod_eq_prod (fun i x => h i x ^ 2)
/-- Finite product estimate used in uniform-product-density; the bound has
no dependence on the choice of endpoint orientation. -/
theorem product_energy_le_exp {ι : Type*} [Fintype ι]
(A e : ι → ℝ) (hA0 : ∀ i, 0 ≤ A i) (hA : ∀ i, A i ≤ 1 + e i) :
(∏ i, A i) ≤ Real.exp (∑ i, e i) := by
calc
(∏ i, A i) ≤ ∏ i, Real.exp (e i) := by
apply Finset.prod_le_prod₀ (fun i _ => hA0 i)
intro i _
exact (hA i).trans (by linarith [Real.add_one_le_exp (e i)])
_ = _ := (Real.exp_sum _ _).symm
end
end FiniteFactor
end OAI
|