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f61e547 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 | import Mathlib
namespace OAI
noncomputable section
open MeasureTheory
namespace Problem367
section Probability
variable {Ω : Type*} {mΩ : MeasurableSpace Ω} {μ : Measure Ω}
variable [IsProbabilityMeasure μ]
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
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 _ _
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']
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
end Probability
end Problem367
end
end OAI
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