import Mathlib import OAI.Probability.Ballisticity.Geometry.WeightedConditionedTubeTransfer import OAI.Probability.Ballisticity.Estimates.RawConditionedWeight namespace OAI section section open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Pointwise Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Pointwise Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical BoundedContinuousFunction open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical BoundedContinuousFunction open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical namespace DirectionalTransience lemma finiteConditionalDensity_tube_transfer {d : ℕ} (ν : Measure (Row d)) [IsProbabilityMeasure ν] (e f : Direction d) (htrans : DirectionallyTransient ν (realPosition (step e))) (b : ℕ → ℝ) (H : ℕ → ℕ) (z : ℕ → ℝ) (hz : Tendsto z atTop atTop) (hsmall : ∀ᶠ i in atTop, (conditionedLaw ν (realPosition (step e))).real (MedianTubeFailure (realPosition (step e)) f b (H i) (z i/10)) ≤ 3/10) (N : ℕ) {ε : ℝ} (hε : 0 < ε) : ∀ᶠ i in atTop, (rawConditionedWeight ν (realPosition (step e)) (finiteConditionalDensity (realPosition (step e)) N)).real (MedianTubeFailure (realPosition (step e)) f b (H i) (z i)) ≤ (rawConditionedWeight ν (realPosition (step e)) (finiteConditionalDensity (realPosition (step e)) N)).real Set.univ * (conditionedLaw ν (realPosition (step e))).real (MedianTubeFailure (realPosition (step e)) f b (H i) (z i/10))+ε := by let ℓ := realPosition (step e) let p := annealedLaw ν (NoDrop ℓ 0) have hp : p ≠ 0 := ne_of_gt (noDrop_positive_of_directionallyTransient ν ℓ htrans) have hpfin : p ≠ ∞ := measure_ne_top _ _ let C : ℝ≥0∞ := (N+1:ℕ) have hC : C ≠ 0 := by simp [C] have hCfin : C ≠ ∞ := by simp [C] let g := fun ω => C⁻¹ * finiteConditionalDensity ℓ N ω have hg : @Measurable _ _ (rowSigma {x | dot (realPosition x) ℓ ≤ (N:ℝ)}) _ g := measurable_const.mul (finiteConditionalDensity_lower_rows e N) have hgle : ∀ ω, g ω ≤ 1 := by intro ω exact (mul_le_mul' (le_refl C⁻¹) (finiteConditionalDensity_le ℓ N ω)).trans (ENNReal.inv_mul_le_one C) let η := weightedConditioned ν ℓ g let ρ := rawConditionedWeight ν ℓ (finiteConditionalDensity ℓ N) have heq : ρ = (C*p) • η := by have he : finiteConditionalDensity ℓ N = fun ω => C*g ω := by funext ω dsimp only [g] rw [← mul_assoc,ENNReal.mul_inv_cancel hC hCfin,one_mul] dsimp only [ρ] rw [he,rawConditionedWeight_smul ν ℓ g (hg.mono (rowSigma_le _) le_rfl) C, rawConditionedWeight_eq_smul ν ℓ hp g,smul_smul] have hM : 0 < (C*p).toReal := ENNReal.toReal_pos (mul_ne_zero hC hp) (ENNReal.mul_ne_top hCfin hpfin) have hh := weightedConditioned_tube_transfer ν ℓ f htrans b H z hz hsmall (N:ℝ) (Nat.cast_nonneg N) g hg hgle (div_pos hε hM) filter_upwards [hh] with i hi have hm (A : Set (Path d)) : ρ.real A = (C*p).toReal*η.real A := by rw [heq,Measure.real,Measure.smul_apply,smul_eq_mul,ENNReal.toReal_mul] rfl change ρ.real (MedianTubeFailure ℓ f b (H i) (z i)) ≤ ρ.real Set.univ*(conditionedLaw ν ℓ).real (MedianTubeFailure ℓ f b (H i) (z i/10))+ε rw [hm,hm] have hi' := mul_le_mul_of_nonneg_left hi hM.le have hc : (C*p).toReal*(ε/(C*p).toReal) = ε := by field_simp rw [mul_add,hc] at hi' simpa only [mul_assoc,η] using hi' lemma measureReal_le_add_mass_defect {Ω : Type*} [MeasurableSpace Ω] (μ η : Measure Ω) [IsProbabilityMeasure μ] [IsFiniteMeasure η] (hη : η ≤ μ) (A : Set Ω) (hA : MeasurableSet A) : μ.real A ≤ η.real A+1-η.real Set.univ := by have hc : η.real Aᶜ ≤ μ.real Aᶜ := ENNReal.toReal_mono (measure_ne_top _ _) (hη Aᶜ) rw [measureReal_compl hA,measureReal_compl hA,show μ.real Set.univ = 1 from probReal_univ] at hc linarith theorem quenchedConditioned_tube_transfer_small {d : ℕ} (ν : Measure (Row d)) [IsProbabilityMeasure ν] (hue : UniformElliptic ν) (e f : Direction d) (htrans : DirectionallyTransient ν (realPosition (step e))) (b : ℕ → ℝ) (H : ℕ → ℕ) (z : ℕ → ℝ) (hz : Tendsto z atTop atTop) (hsmall : ∀ᶠ i in atTop, (conditionedLaw ν (realPosition (step e))).real (MedianTubeFailure (realPosition (step e)) f b (H i) (z i/10)) ≤ 3/10) {ε : ℝ} (hε : 0 < ε) : ∀ᶠ i in atTop, (quenchedConditionedAverage ν (realPosition (step e))).real (MedianTubeFailure (realPosition (step e)) f b (H i) (z i)) ≤ (conditionedLaw ν (realPosition (step e))).real (MedianTubeFailure (realPosition (step e)) f b (H i) (z i/10))+ε := by let ℓ := realPosition (step e) let μ := quenchedConditionedAverage ν ℓ let : IsProbabilityMeasure μ := quenchedConditionedAverage_probability ν hue ℓ (signed_direction_unit e) htrans let ρ := fun N => rawConditionedWeight ν ℓ (finiteConditionalDensity ℓ N) have hdom (N : ℕ) : ρ N ≤ μ := rawConditionedWeight_mono ν ℓ _ _ (finiteConditionalDensity_measurable ℓ N) (measurable_noDropQuenched ℓ 0).inv (finiteConditionalDensity_le_inverse ℓ N) let : ∀ N, IsFiniteMeasure (ρ N) := fun N => ⟨lt_of_le_of_lt (hdom N Set.univ) (measure_lt_top _ _)⟩ have ht : Tendsto (fun N => (ρ N).real Set.univ) atTop (𝓝 1) := by have hh := (ENNReal.continuousAt_toReal (by simp : (1:ℝ≥0∞) ≠ ∞)).tendsto.comp (finiteConditionalDensity_mass_tendsto ν hue ℓ (signed_direction_unit e) htrans) simpa only [Function.comp_def,ENNReal.toReal_one,Measure.real,ρ] using hh obtain ⟨N,hN⟩ := (ht.eventually (lt_mem_nhds (show 1-ε/2 < (1:ℝ) by linarith))).exists have hh := finiteConditionalDensity_tube_transfer ν e f htrans b H z hz hsmall N (show 0 < ε/2 by positivity) filter_upwards [hh] with i hi let A := MedianTubeFailure ℓ f b (H i) (z i) let B := MedianTubeFailure ℓ f b (H i) (z i/10) have ha := measureReal_le_add_mass_defect μ (ρ N) (hdom N) A (measurableSet_medianTubeFailure ℓ f b (H i) (z i)) have hmass : (ρ N).real Set.univ ≤ 1 := by have h := ENNReal.toReal_mono (measure_ne_top _ _) (hdom N Set.univ) simpa only [Measure.real,measure_univ,ENNReal.toReal_one] using h have hp := mul_le_mul_of_nonneg_right hmass (show 0 ≤ (conditionedLaw ν ℓ).real B from measureReal_nonneg) change (ρ N).real A ≤ (ρ N).real Set.univ*(conditionedLaw ν ℓ).real B+ε/2 at hi change μ.real A ≤ (conditionedLaw ν ℓ).real B+ε linarith end DirectionalTransience open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical end end end OAI