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b317718 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 | import OAI.Probability.FreeIsing.Conditioning.CutLevelConditionalMean
import OAI.Probability.FreeIsing.Estimates.BoundedAverageConcentration
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory
open scoped ENNReal
namespace Problem367
theorem integral_cond_fiber_le_of_condExp_le
{Ω S : Type*} [MeasurableSpace Ω] [MeasurableSpace S] [MeasurableSingletonClass S]
{μ : Measure Ω} [IsFiniteMeasure μ] {s : Ω → S} (hs : Measurable s)
{f : Ω → ℝ} (hf : Integrable f μ) {C : ℝ}
(hbound : ∀ᵐ ω ∂μ, (μ[f | MeasurableSpace.comap s inferInstance]) ω ≤ C)
(a : S) (ha : μ (s ⁻¹' {a}) ≠ 0) :
(∫ ω, f ω ∂μ[|s ⁻¹' {a}]) ≤ C := by
have : (ae (μ.restrict (s ⁻¹' {a}))).NeBot :=
ae_neBot.mpr (mt Measure.restrict_eq_zero.mp ha)
have hCE := condExp_comap_eq_cond_on_fiber hs a hf
have hB := ae_restrict_le (s := s ⁻¹' {a}) hbound
obtain ⟨ω, hωCE, hωB⟩ := (hCE.and hB).exists
exact hωCE.symm.le.trans hωB
theorem measureReal_le_of_cond_fibers
{Ω S : Type*} [MeasurableSpace Ω] [MeasurableSpace S]
[MeasurableSingletonClass S] [Countable S]
{μ : Measure Ω} [IsProbabilityMeasure μ] {s : Ω → S} (hs : Measurable s)
{E : Set Ω} (hE : MeasurableSet E) {C : ℝ}
(hbound : ∀ a : S, μ (s ⁻¹' {a}) ≠ 0 → μ[|s ⁻¹' {a}].real E ≤ C) :
μ.real E ≤ C := by
have hCE : ∀ᵐ ω ∂μ,
(μ[E.indicator (fun _ => (1 : ℝ)) | MeasurableSpace.comap s inferInstance]) ω ≤ C := by
apply ae_of_ae_restrict_fibers hs
intro a
by_cases ha : μ (s ⁻¹' {a}) = 0
· rw [Measure.restrict_eq_zero.mpr ha]
simp
filter_upwards [condExp_indicator_comap_eq_cond_on_fiber hs a hE] with ω hω
exact hω.le.trans (hbound a ha)
have hint := integral_mono_ae integrable_condExp (integrable_const C) hCE
rw [integral_condExp hs.comap_le] at hint
simpa only [integral_indicator_const (1 : ℝ) hE, smul_eq_mul, mul_one,
integral_const, probReal_univ, one_smul, one_mul] using hint
local instance : MeasurableSingletonClass IsingSpin := ⟨fun _ => trivial⟩
theorem HasFreeIsingLaw.measure_cutLevel_average_ge_cond_le
{d : ℕ} {theta : ℝ} {A : Type*} [MeasurableSpace A] [StandardBorelSpace A]
{μ : Measure (TreeVertex d → IsingSpin)} [IsProbabilityMeasure μ]
(hμ : HasFreeIsingLaw theta μ) (h0 : -1 ≤ theta) (h1 : theta ≤ 1) (n : ℕ)
(ν : TreeVertex d → Measure A) [∀ v, IsProbabilityMeasure (ν v)]
(F : ∀ c : TreeShell d n, (RootedBroadcast.DescendantVertex c.1 → IsingSpin) ×
(RootedBroadcast.DescendantVertex c.1 → A) → ℝ)
(hF : ∀ c, Measurable (F c)) (hbound : ∀ c p, 0 ≤ F c p ∧ F c p ≤ 1)
(I : Finset (TreeShell d n)) (hI : I.Nonempty) {τ : ℝ} (hτ : 0 ≤ τ)
(hmean : ∀ c ∈ I, (∫ z : (TreeVertex d → IsingSpin) × (TreeVertex d → A),
F c ((fun v => z.1 v.1), fun v => z.2 v.1) ∂μ.prod (Measure.infinitePi ν)) ≤ τ / 4)
(s : TreeShell d n → IsingSpin)
(hs : (μ.prod (Measure.infinitePi ν))
{z | (fun c : TreeShell d n => z.1 c.1) = s} ≠ 0) :
(μ.prod (Measure.infinitePi ν))[|{z | (fun c : TreeShell d n => z.1 c.1) = s}]
{z | τ ≤ (∑ c ∈ I, F c ((fun v => z.1 v.1), fun v => z.2 v.1)) / (I.card : ℝ)} ≤
ENNReal.ofReal (Real.exp (-(I.card : ℝ) * τ ^ 2 / 2)) := by
let P := μ.prod (Measure.infinitePi ν)
let S : (TreeVertex d → IsingSpin) × (TreeVertex d → A) → TreeShell d n → IsingSpin :=
fun z c => z.1 c.1
let X := fun (c : TreeShell d n) (z : (TreeVertex d → IsingSpin) × (TreeVertex d → A)) =>
F c ((fun v => z.1 v.1), fun v => z.2 v.1)
have hS : Measurable S := by fun_prop
have hX : ∀ c, Measurable (X c) := fun c => (hF c).comp (by fun_prop)
let : IsProbabilityMeasure ((μ.prod (Measure.infinitePi ν))[|
{z | (fun c : TreeShell d n => z.1 c.1) = s}]) := cond_isProbabilityMeasure hs
apply BoundedAverageConcentration.measure_average_ge_le
(hμ.iIndepFun_cutLevel_observables_cond h0 h1 n ν F hF s hs) I hI hτ
· intro c _
exact (hX c).aemeasurable
· intro c _
exact Filter.Eventually.of_forall (fun z => hbound c _)
· intro c hc
have hi : Integrable (X c) P := by
apply Integrable.of_bound (hX c).aestronglyMeasurable 1
exact Filter.Eventually.of_forall (fun z => by
rw [Real.norm_eq_abs, abs_of_nonneg (hbound c _).1]
exact (hbound c _).2)
have hb := integral_cond_fiber_le_of_condExp_le hS hi
(hμ.condExp_cutLevel_observable_le_two_integral h0 h1 n ν c (F c) (hF c)
(hbound c)) s hs
exact hb.trans (by linarith [hmean c hc])
theorem HasFreeIsingLaw.measure_cutLevel_average_ge_le
{d : ℕ} {theta : ℝ} {A : Type*} [MeasurableSpace A] [StandardBorelSpace A]
{μ : Measure (TreeVertex d → IsingSpin)} [IsProbabilityMeasure μ]
(hμ : HasFreeIsingLaw theta μ) (h0 : -1 ≤ theta) (h1 : theta ≤ 1) (n : ℕ)
(ν : TreeVertex d → Measure A) [∀ v, IsProbabilityMeasure (ν v)]
(F : ∀ c : TreeShell d n, (RootedBroadcast.DescendantVertex c.1 → IsingSpin) ×
(RootedBroadcast.DescendantVertex c.1 → A) → ℝ)
(hF : ∀ c, Measurable (F c)) (hbound : ∀ c p, 0 ≤ F c p ∧ F c p ≤ 1)
(I : Finset (TreeShell d n)) (hI : I.Nonempty) {τ : ℝ} (hτ : 0 ≤ τ)
(hmean : ∀ c ∈ I, (∫ z : (TreeVertex d → IsingSpin) × (TreeVertex d → A),
F c ((fun v => z.1 v.1), fun v => z.2 v.1) ∂μ.prod (Measure.infinitePi ν)) ≤ τ / 4) :
(μ.prod (Measure.infinitePi ν))
{z | τ ≤ (∑ c ∈ I, F c ((fun v => z.1 v.1), fun v => z.2 v.1)) / (I.card : ℝ)} ≤
ENNReal.ofReal (Real.exp (-(I.card : ℝ) * τ ^ 2 / 2)) := by
let P := μ.prod (Measure.infinitePi ν)
let S : (TreeVertex d → IsingSpin) × (TreeVertex d → A) → TreeShell d n → IsingSpin :=
fun z c => z.1 c.1
have hS : Measurable S := by fun_prop
have hsum : Measurable (fun z : (TreeVertex d → IsingSpin) × (TreeVertex d → A) =>
∑ c ∈ I, F c ((fun v => z.1 v.1), fun v => z.2 v.1)) :=
Finset.measurable_fun_sum I (fun c _ => (hF c).comp (by fun_prop))
rw [← ofReal_measureReal]
apply ENNReal.ofReal_le_ofReal
apply measureReal_le_of_cond_fibers hS (measurableSet_le measurable_const (hsum.div_const _))
intro s hs
have h := hμ.measure_cutLevel_average_ge_cond_le h0 h1 n ν F hF hbound I hI hτ hmean s hs
exact (ENNReal.toReal_le_toReal (measure_ne_top _ _) ENNReal.ofReal_ne_top).mpr h |>.trans_eq
(ENNReal.toReal_ofReal (Real.exp_pos _).le)
end Problem367
end
end OAI
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