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f89a930 | 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 | import Mathlib.MeasureTheory.Integral.Bochner.Basic
import OAI.NumberTheory.Jacobsthal.Partitions.CompactTestGeometry
namespace OAI
namespace Erdos970
open scoped _root_.Erdos970
section
namespace NumberTheoryLean.CompactTestMeasurable
open _root_.Set _root_.Finset _root_.MeasureTheory ProbabilityTheory
open scoped ENNReal
open FinitePathGeometry FinitePathMeasures PrimeHistories PrimeKilledChain
open CompactTestGeometry RegeneratingInverseBands ActualCouplingUpdates ArrivalKernelGeometry
variable {w ell S : ℝ} {start : Node}
theorem primeTest_measurable (F : Side → ℝ×ℝ → ℝ) :
Measurable (primeTest (w:=w) (ell:=ell) (S:=S) (start:=start) F) := measurable_of_countable _
theorem continuousTest_measurable (v : ℝ) (F : Side → ℝ×ℝ → ℝ) (hF : ∀ i,Measurable (F i)) :
Measurable (continuousTest v F) := by
have hcoord : Measurable (fun z : CostState => (gapValue v z,stateRatio z.1)) :=
(gapValue_measurable v).prodMk (stateRatio_measurable.comp measurable_fst)
have hside : MeasurableSet {z : CostState | stateSide z.1 = .even} :=
measurable_fst (side_set_measurable .even)
have heq : (fun z : CostState => F (stateSide z.1) (gapValue v z,stateRatio z.1)) =
(fun z => if stateSide z.1 = .even then F .even (gapValue v z,stateRatio z.1)
else F .odd (gapValue v z,stateRatio z.1)) := by
funext z
rcases z with ⟨s,t⟩
cases s <;> rfl
have hb : Measurable (fun z : CostState => F (stateSide z.1) (gapValue v z,stateRatio z.1)) := by
rw [heq]
exact Measurable.ite hside ((hF .even).comp hcoord) ((hF .odd).comp hcoord)
exact hb.sumElim measurable_const
theorem primeTest_bound {F : Side → ℝ×ℝ → ℝ} {A : ℝ} (hA : 0 ≤ A)
(hF : ∀ i x,|F i x| ≤ A) (z : ChainState w ell S start) : |primeTest F z| ≤ A := by
cases z with
| none => simpa only [primeTest,abs_zero] using hA
| some h => exact hF _ _
theorem continuousTest_bound {F : Side → ℝ×ℝ → ℝ} {A : ℝ} (hA : 0 ≤ A)
(hF : ∀ i x,|F i x| ≤ A) (v : ℝ) (z : CemeteryKernel.Space CostState) :
|continuousTest v F z| ≤ A := by
cases z with
| inr u => simpa only [continuousTest,abs_zero] using hA
| inl y => exact hF _ _
theorem test_difference_bound {F : Side → ℝ×ℝ → ℝ} {A : ℝ} (hA : 0 ≤ A)
(hF : ∀ i x,|F i x| ≤ A) (v : ℝ) (p : ChainState w ell S start) (z : CemeteryKernel.Space CostState) :
|primeTest F p-continuousTest v F z| ≤ 2*A := by
exact (abs_sub _ _).trans (by linarith [primeTest_bound hA hF p,continuousTest_bound hA hF v z])
section Generic
variable {α : Type*} [MeasurableSpace α] {μ : Measure α}
theorem bounded_integrable [IsFiniteMeasure μ] {f : α → ℝ} (hf : Measurable f)
{A : ℝ} (hb : ∀ x,|f x| ≤ A) : Integrable f μ := by
refine (integrable_const A).mono' hf.aestronglyMeasurable ?_
exact Filter.Eventually.of_forall (fun x => by simpa only [Real.norm_eq_abs] using hb x)
theorem integral_difference_enorm {f g : α → ℝ} (hf : Integrable f μ) (hg : Integrable g μ) :
ENNReal.ofReal |(∫ x,f x ∂μ)-(∫ x,g x ∂μ)| ≤ ∫⁻ x,ENNReal.ofReal |f x-g x| ∂μ := by
rw [← integral_sub hf hg]
simpa only [Real.enorm_eq_ofReal_abs] using enorm_integral_le_lintegral_enorm (fun x => f x-g x)
theorem finite_integral_difference (μ : ℕ → Measure α) (f g : α → ℝ) (N : ℕ)
(hf : ∀ n ∈ range N,Integrable f (μ n)) (hg : ∀ n ∈ range N,Integrable g (μ n)) :
ENNReal.ofReal |(∑ n ∈ range N,∫ x,f x ∂μ n)-(∑ n ∈ range N,∫ x,g x ∂μ n)| ≤
∑ n ∈ range N,∫⁻ x,ENNReal.ofReal |f x-g x| ∂μ n := by
rw [← Finset.sum_sub_distrib]
refine (ENNReal.ofReal_le_ofReal (Finset.abs_sum_le_sum_abs _ _)).trans ?_
rw [ENNReal.ofReal_sum_of_nonneg (fun _ _ => abs_nonneg _)]
exact Finset.sum_le_sum (fun n hn => integral_difference_enorm (hf n hn) (hg n hn))
end Generic
end NumberTheoryLean.CompactTestMeasurable
end
end Erdos970
end OAI
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