| import Mathlib |
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| namespace OAI |
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| noncomputable section |
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| open MeasureTheory |
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| namespace Problem367 |
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| section Probability |
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| variable {Ω : Type*} {mΩ : MeasurableSpace Ω} {μ : Measure Ω} |
| variable [IsProbabilityMeasure μ] |
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| lemma integrable_of_measurable_unit_interval {f : Ω → ℝ} (hf : Measurable f) |
| (hbound : ∀ w, 0 ≤ f w ∧ f w ≤ 1) : Integrable f μ := by |
| apply (integrable_const (1 : ℝ)).mono' hf.aestronglyMeasurable |
| filter_upwards [] with w |
| rw [Real.norm_eq_abs, abs_of_nonneg (hbound w).1] |
| exact (hbound w).2 |
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| lemma integral_mul_le_of_condExp_le {m : MeasurableSpace Ω} (hm : m ≤ mΩ) |
| {f g : Ω → ℝ} {q : ℝ} |
| (hf : Integrable f μ) (hg : Measurable[m] g) |
| (hgBound : ∀ w, 0 ≤ g w ∧ g w ≤ 1) |
| (hcond : ∀ᵐ w ∂μ, (μ[f | m]) w ≤ q) : |
| (∫ w, f w * g w ∂μ) ≤ q * ∫ w, g w ∂μ := by |
| have hgb : @Measurable Ω ℝ mΩ _ g := hg.mono hm le_rfl |
| have hgInt := integrable_of_measurable_unit_interval (μ := μ) hgb hgBound |
| have hgNorm : ∀ᵐ w ∂μ, ‖g w‖ ≤ 1 := by |
| filter_upwards [] with w |
| rw [Real.norm_eq_abs, abs_of_nonneg (hgBound w).1] |
| exact (hgBound w).2 |
| have hprod := hf.mul_bdd hgb.aestronglyMeasurable hgNorm |
| have hpull := condExp_mul_of_stronglyMeasurable_right hg.stronglyMeasurable hprod hf |
| calc |
| (∫ w, f w * g w ∂μ) = ∫ w, (μ[fun v => f v * g v | m]) w ∂μ := |
| (integral_condExp hm).symm |
| _ = ∫ w, (μ[f | m]) w * g w ∂μ := integral_congr_ae hpull |
| _ ≤ ∫ w, q * g w ∂μ := by |
| apply integral_mono_ae |
| (integrable_condExp.mul_bdd hgb.aestronglyMeasurable hgNorm) |
| (hgInt.const_mul q) |
| filter_upwards [hcond] with w hw |
| exact mul_le_mul_of_nonneg_right hw (hgBound w).1 |
| _ = q * ∫ w, g w ∂μ := integral_const_mul _ _ |
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| lemma integral_prod_le_of_condExp_le {q : ℝ} (hq : 0 ≤ q) |
| (A : ℕ → Ω → ℝ) (m : ℕ → MeasurableSpace Ω) (s : Finset ℕ) |
| (hMeas : ∀ i ∈ s, Measurable (A i)) |
| (hBound : ∀ i ∈ s, ∀ w, 0 ≤ A i w ∧ A i w ≤ 1) |
| (hm : ∀ i ∈ s, m i ≤ mΩ) |
| (hCE : ∀ i ∈ s, ∀ᵐ w ∂μ, (μ[A i | m i]) w ≤ q) |
| (hLater : ∀ i ∈ s, ∀ j ∈ s, i < j → Measurable[m i] (A j)) : |
| (∫ w, ∏ i ∈ s, A i w ∂μ) ≤ q ^ s.card := by |
| classical |
| revert hMeas hBound hm hCE hLater |
| induction s using Finset.induction_on_min with |
| | empty => |
| intros |
| simp |
| | insert i s hmin ih => |
| intro hMeas hBound hm hCE hLater |
| have his : i ∉ s := by |
| intro hi |
| exact (lt_irrefl i) (hmin i hi) |
| have hiMem : i ∈ insert i s := Finset.mem_insert_self i s |
| have hsMem : ∀ j ∈ s, j ∈ insert i s := fun j hj => Finset.mem_insert_of_mem hj |
| have htail : (∫ w, ∏ j ∈ s, A j w ∂μ) ≤ q ^ s.card := |
| ih (fun j hj => hMeas j (hsMem j hj)) |
| (fun j hj => hBound j (hsMem j hj)) |
| (fun j hj => hm j (hsMem j hj)) |
| (fun j hj => hCE j (hsMem j hj)) |
| (fun j hj k hk => hLater j (hsMem j hj) k (hsMem k hk)) |
| have hprodMeas : Measurable[m i] (fun w => ∏ j ∈ s, A j w) := by |
| apply Finset.measurable_fun_prod |
| intro j hj |
| exact hLater i hiMem j (hsMem j hj) (hmin j hj) |
| have hprodBound : ∀ w, 0 ≤ (∏ j ∈ s, A j w) ∧ (∏ j ∈ s, A j w) ≤ 1 := by |
| intro w |
| exact ⟨Finset.prod_nonneg (fun j hj => (hBound j (hsMem j hj) w).1), |
| Finset.prod_le_one₀ (fun j hj => (hBound j (hsMem j hj) w).1) |
| (fun j hj => (hBound j (hsMem j hj) w).2)⟩ |
| calc |
| (∫ w, ∏ j ∈ insert i s, A j w ∂μ) = |
| ∫ w, A i w * (∏ j ∈ s, A j w) ∂μ := by simp [Finset.prod_insert, his] |
| _ ≤ q * ∫ w, ∏ j ∈ s, A j w ∂μ := |
| integral_mul_le_of_condExp_le (hm i hiMem) |
| (integrable_of_measurable_unit_interval (hMeas i hiMem) (hBound i hiMem)) |
| hprodMeas hprodBound (hCE i hiMem) |
| _ ≤ q * q ^ s.card := mul_le_mul_of_nonneg_left htail hq |
| _ = q ^ (insert i s).card := by rw [Finset.card_insert_of_notMem his, pow_succ'] |
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| lemma integral_prod_le_exp_of_condExp_le {delta : ℝ} (hδ : delta ≤ 1) |
| (A : ℕ → Ω → ℝ) (m : ℕ → MeasurableSpace Ω) (s : Finset ℕ) |
| (hMeas : ∀ i ∈ s, Measurable (A i)) |
| (hBound : ∀ i ∈ s, ∀ w, 0 ≤ A i w ∧ A i w ≤ 1) |
| (hm : ∀ i ∈ s, m i ≤ mΩ) |
| (hCE : ∀ i ∈ s, ∀ᵐ w ∂μ, (μ[A i | m i]) w ≤ 1 - delta) |
| (hLater : ∀ i ∈ s, ∀ j ∈ s, i < j → Measurable[m i] (A j)) : |
| (∫ w, ∏ i ∈ s, A i w ∂μ) ≤ Real.exp (-delta * (s.card : ℝ)) := by |
| calc |
| (∫ w, ∏ i ∈ s, A i w ∂μ) ≤ (1 - delta) ^ s.card := |
| integral_prod_le_of_condExp_le (sub_nonneg.mpr hδ) A m s hMeas hBound hm hCE hLater |
| _ ≤ (Real.exp (-delta)) ^ s.card := by |
| apply pow_le_pow_left₀ (sub_nonneg.mpr hδ) |
| have h := Real.add_one_le_exp (-delta) |
| linarith |
| _ = Real.exp (-delta * (s.card : ℝ)) := by |
| rw [← Real.exp_nat_mul] |
| congr 1 |
| ring |
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| end Probability |
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| end Problem367 |
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| end |
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| end OAI |
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