| import Mathlib |
| import OAI.Probability.SKBarriers.Calculus.ParameterCalculus |
|
|
| namespace OAI |
|
|
| section |
| section |
| noncomputable section |
| open scoped BigOperators Topology |
| open MeasureTheory ProbabilityTheory Filter |
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped Topology Interval |
| noncomputable section |
| open MeasureTheory Set |
| open scoped Interval |
| namespace SK.Analytic |
| |
| instance parameterMeasurable : (n : ℕ) → MeasurableSpace (ParameterSpace n) |
| | 0 => inferInstanceAs (MeasurableSpace ℝ) |
| | n+1 => letI := parameterMeasurable n; inferInstanceAs (MeasurableSpace (ParameterSpace n × ℝ)) |
| |
| instance parameterBorel : (n : ℕ) → BorelSpace (ParameterSpace n) |
| | 0 => inferInstanceAs (BorelSpace ℝ) |
| | n+1 => letI := parameterBorel n; inferInstanceAs (BorelSpace (ParameterSpace n × ℝ)) |
| |
| instance parameterSecondCountable : (n : ℕ) → SecondCountableTopology (ParameterSpace n) |
| | 0 => inferInstanceAs (SecondCountableTopology ℝ) |
| | n+1 => letI := parameterSecondCountable n; inferInstanceAs (SecondCountableTopology (ParameterSpace n × ℝ)) |
| |
| def fiberMeasure : (n : ℕ) → ℝ → Measure (ParameterSpace n) |
| | 0, x => Measure.dirac x |
| | n+1, x => (fiberMeasure n x).prod volume |
| |
| instance fiberSigmaFinite : (n : ℕ) → (x : ℝ) → SigmaFinite (fiberMeasure n x) |
| | 0, x => inferInstanceAs (SigmaFinite (Measure.dirac x)) |
| | n+1, x => letI := fiberSigmaFinite n x; inferInstanceAs (SigmaFinite ((fiberMeasure n x).prod volume)) |
| |
| def cubeMeasure : (n : ℕ) → ℝ → ℝ → ℝ → Measure (ParameterSpace n) |
| | 0, x, _, _ => Measure.dirac x |
| | n+1, x, a, b => (cubeMeasure n x a b).prod (volume.restrict (Ioc a b)) |
| |
| instance cubeFiniteMeasure : (n : ℕ) → (x a b : ℝ) → IsFiniteMeasure (cubeMeasure n x a b) |
| | 0, x, _, _ => inferInstanceAs (IsFiniteMeasure (Measure.dirac x)) |
| | n+1, x, a, b => letI := cubeFiniteMeasure n x a b |
| inferInstanceAs (IsFiniteMeasure ((cubeMeasure n x a b).prod (volume.restrict (Ioc a b)))) |
| |
| def cubeSupport : (n : ℕ) → ℝ → ℝ → ℝ → Set (ParameterSpace n) |
| | 0, x, _, _ => {x} |
| | n+1, x, a, b => (cubeSupport n x a b) ×ˢ Icc a b |
| |
| theorem cubeSupport_compact (n : ℕ) (x a b : ℝ) : IsCompact (cubeSupport n x a b) := by |
| induction n with |
| | zero => exact isCompact_singleton |
| | succ n ih => exact ih.prod isCompact_Icc |
| |
| theorem ae_mem_cubeSupport (n : ℕ) (x a b : ℝ) : |
| ∀ᵐ z ∂cubeMeasure n x a b, z ∈ cubeSupport n x a b := by |
| induction n with |
| | zero => simp [cubeMeasure, cubeSupport] |
| | succ n ih => |
| apply (Measure.ae_prod_mem_iff_ae_ae_mem (cubeSupport_compact (n+1) x a b).measurableSet).mpr |
| filter_upwards [ih] with z hz |
| filter_upwards [ae_restrict_mem measurableSet_Ioc] with y hy |
| exact ⟨hz, ⟨hy.1.le, hy.2⟩⟩ |
| |
| theorem integrable_cubeMeasure (n : ℕ) (f : ParameterSpace n → ℝ) |
| (hf : Continuous f) (x a b : ℝ) : Integrable f (cubeMeasure n x a b) := by |
| obtain ⟨C, hC⟩ := (cubeSupport_compact n x a b).exists_bound_of_continuousOn hf.continuousOn |
| apply (integrable_const C).mono' hf.aestronglyMeasurable |
| filter_upwards [ae_mem_cubeSupport n x a b] with z hz |
| exact hC z hz |
| |
| theorem cubeIntegral_eq_measureIntegral (n : ℕ) (f : ParameterSpace n → ℝ) |
| (hf : Continuous f) {a b : ℝ} (hab : a ≤ b) (x : ℝ) : |
| cubeIntegral n f a b x = ∫ z, f z ∂cubeMeasure n x a b := by |
| induction n with |
| | zero => exact (integral_dirac _ _).symm |
| | succ n ih => |
| change cubeIntegral n (fun z => ∫ y in a..b, f (z,y)) a b x = |
| ∫ z, f z ∂(cubeMeasure n x a b).prod (volume.restrict (Ioc a b)) |
| rw [integral_prod f (integrable_cubeMeasure (n+1) f hf x a b)] |
| rw [ih _ (contDiff_zero.mp (contDiff_intervalIntegral 0 f (contDiff_zero.mpr hf) a b))] |
| apply integral_congr_ae |
| exact ae_of_all _ (fun z => intervalIntegral.integral_of_le hab) |
| |
| def cubeSet : (n : ℕ) → ℝ → ℝ → Set (ParameterSpace n) |
| | 0, _, _ => univ |
| | n+1, a, b => cubeSet n a b ×ˢ Ioc a b |
| |
| theorem cubeSet_measurable (n : ℕ) (a b : ℝ) : MeasurableSet (cubeSet n a b) := by |
| induction n with |
| | zero => exact MeasurableSet.univ |
| | succ n ih => exact ih.prod measurableSet_Ioc |
| |
| theorem cubeMeasure_eq_restrict (n : ℕ) (x a b : ℝ) : |
| cubeMeasure n x a b = (fiberMeasure n x).restrict (cubeSet n a b) := by |
| induction n with |
| | zero => simp [cubeMeasure, fiberMeasure, cubeSet] |
| | succ n ih => |
| change (cubeMeasure n x a b).prod (volume.restrict (Ioc a b)) = _ |
| rw [ih, Measure.prod_restrict] |
| rfl |
| |
| end SK.Analytic |
| |
| end |
| end |
| end |
| end |
| end |
| |
| end OAI |
| |
