File size: 9,984 Bytes
9eeb9c7 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 | import OAI.Analysis.IrrationalRotation.Analysis.CrossingAvoidance
import OAI.Analysis.IrrationalRotation.Couplings.ConditionalKernelSupport
import Mathlib.Probability.Independence.Conditional
namespace OAI
open MeasureTheory ProbabilityTheory
open scoped BigOperators
namespace Invariant233.CrossingAvoidance
section KernelBounds
variable {ι Ω Y : Type*} [Countable ι] [MeasurableSpace Ω] [MeasurableSpace Y]
theorem kernel_iIndepSet_ae (κ : Kernel Y Ω) (ν : Measure Y)
(events : ι → Set Ω) (hmeas : ∀ i, MeasurableSet (events i))
(hind : Kernel.iIndepSet events κ ν) :
∀ᵐ y ∂ν, iIndepSet events (κ y) := by
have hall : ∀ᵐ y ∂ν, ∀ s : Finset ι,
κ y (⋂ i ∈ s, events i) = ∏ i ∈ s, κ y (events i) :=
ae_all_iff.mpr (fun s => hind.meas_biInter s)
filter_upwards [hall] with y hy
exact (iIndepSet_iff_meas_biInter hmeas).mpr hy
theorem kernel_avoid_le_exp (κ : Kernel Y Ω) (ν : Measure Y)
(events : ι → Set Ω) (hmeas : ∀ i, MeasurableSet (events i))
(hind : Kernel.iIndepSet events κ ν) (selected : Y → Finset ι) (p : Y → ℝ)
(hp : ∀ᵐ y ∂ν, ∀ i ∈ selected y, p y ≤ (κ y).real (events i)) :
∀ᵐ y ∂ν, (κ y).real (⋂ i ∈ selected y, (events i)ᶜ) ≤
Real.exp (-((selected y).card : ℝ) * p y) := by
filter_upwards [kernel_iIndepSet_ae κ ν events hmeas hind, hp] with y hy hp
let : IsProbabilityMeasure (κ y) := hy.isProbabilityMeasure
exact measureReal_avoid_le_exp (κ y) events hy hmeas (selected y) (p y) hp
theorem kernel_bad_le_exp_intensity (κ : Kernel Y Ω) (ν : Measure Y)
(events : ι → Set Ω) (hmeas : ∀ i, MeasurableSet (events i))
(hind : Kernel.iIndepSet events κ ν) (selected : Y → Finset ι)
(m ρ L : ℝ) (hρ : 0 ≤ ρ) (hL : L ≤ m * ρ)
(hcard : ∀ᵐ y ∂ν, m / 40 ≤ ((selected y).card : ℝ))
(hp : ∀ᵐ y ∂ν, ∀ i ∈ selected y, ρ / 10 ≤ (κ y).real (events i))
(bad : Y → Set Ω)
(hbad : ∀ᵐ y ∂ν, ∀ᵐ ω ∂κ y,
ω ∈ bad y → ∀ i ∈ selected y, ω ∉ events i) :
∀ᵐ y ∂ν, (κ y).real (bad y) ≤ Real.exp (-L / 400) := by
filter_upwards [kernel_iIndepSet_ae κ ν events hmeas hind, hcard, hp, hbad]
with y hy hcard hp hbad
let : IsProbabilityMeasure (κ y) := hy.isProbabilityMeasure
have hsubset : bad y ≤ᵐ[κ y] ⋂ i ∈ selected y, (events i)ᶜ := by
filter_upwards [hbad] with ω hω
intro hb
exact Set.mem_iInter.mpr (fun i => Set.mem_iInter.mpr (fun hi => hω hb i hi))
have hmono : (κ y).real (bad y) ≤ (κ y).real (⋂ i ∈ selected y, (events i)ᶜ) :=
ENNReal.toReal_mono (measure_ne_top _ _) (measure_mono_ae hsubset)
refine hmono.trans ?_
exact measureReal_le_exp_intensity (κ y) events hy hmeas (selected y) m ρ L
hρ hcard hL hp _ (Set.Subset.refl _)
theorem kernel_bad_le_exp_intensity_on (κ : Kernel Y Ω) (ν : Measure Y)
(events : ι → Set Ω) (hmeas : ∀ i, MeasurableSet (events i))
(hind : Kernel.iIndepSet events κ ν) (selected : Y → Finset ι)
(m ρ L : ℝ) (hρ : 0 ≤ ρ) (hL : L ≤ m * ρ) (active : Y → Prop)
(hcard : ∀ᵐ y ∂ν, active y → m / 40 ≤ ((selected y).card : ℝ))
(hp : ∀ᵐ y ∂ν, active y → ∀ i ∈ selected y, ρ / 10 ≤ (κ y).real (events i))
(bad : Y → Set Ω)
(hzero : ∀ᵐ y ∂ν, ¬ active y → κ y (bad y) = 0)
(hbad : ∀ᵐ y ∂ν, ∀ᵐ ω ∂κ y,
ω ∈ bad y → ∀ i ∈ selected y, ω ∉ events i) :
∀ᵐ y ∂ν, (κ y).real (bad y) ≤ Real.exp (-L / 400) := by
filter_upwards [kernel_iIndepSet_ae κ ν events hmeas hind, hcard, hp, hzero, hbad]
with y hy hcard hp hzero hbad
let : IsProbabilityMeasure (κ y) := hy.isProbabilityMeasure
by_cases ha : active y
· have hsubset : bad y ≤ᵐ[κ y] ⋂ i ∈ selected y, (events i)ᶜ := by
filter_upwards [hbad] with ω hω
intro hb
exact Set.mem_iInter.mpr (fun i => Set.mem_iInter.mpr (fun hi => hω hb i hi))
have hmono : (κ y).real (bad y) ≤ (κ y).real (⋂ i ∈ selected y, (events i)ᶜ) :=
ENNReal.toReal_mono (measure_ne_top _ _) (measure_mono_ae hsubset)
exact hmono.trans (measureReal_le_exp_intensity (κ y) events hy hmeas (selected y) m ρ L
hρ (hcard ha) hL (hp ha) _ (Set.Subset.refl _))
· change (κ y (bad y)).toReal ≤ _
rw [hzero ha, ENNReal.toReal_zero]
exact (Real.exp_pos _).le
end KernelBounds
section ConditionalExpectation
variable {ι Ω : Type*} [Countable ι]
theorem iCondIndepSet_ae_condExpKernel (m' : MeasurableSpace Ω) {mΩ : MeasurableSpace Ω}
[StandardBorelSpace Ω] (hm' : m' ≤ mΩ) (μ : Measure Ω) [IsFiniteMeasure μ]
(events : ι → Set Ω) (hmeas : ∀ i, MeasurableSet (events i))
(hind : iCondIndepSet m' hm' events μ) :
∀ᵐ x ∂μ, iIndepSet events (condExpKernel μ m' x) := by
exact ae_of_ae_trim hm'
(@kernel_iIndepSet_ae ι Ω Ω ‹Countable ι› mΩ m'
(condExpKernel μ m') (μ.trim hm') events hmeas hind)
theorem conditional_bad_le_exp_intensity (m' : MeasurableSpace Ω) {mΩ : MeasurableSpace Ω}
[StandardBorelSpace Ω] (hm' : m' ≤ mΩ) (μ : Measure Ω) [IsFiniteMeasure μ]
(events : ι → Set Ω) (hmeas : ∀ i, MeasurableSet (events i))
(hind : iCondIndepSet m' hm' events μ) (selected : Ω → Finset ι)
(m ρ L : ℝ) (hρ : 0 ≤ ρ) (hL : L ≤ m * ρ) (active : Ω → Prop)
(hcard : ∀ᵐ x ∂μ, active x → m / 40 ≤ ((selected x).card : ℝ))
(hp : ∀ᵐ x ∂μ, active x → ∀ i ∈ selected x,
ρ / 10 ≤ (condExpKernel μ m' x).real (events i))
(bad : Set Ω) (hbadmeas : MeasurableSet bad)
(hzero : ∀ᵐ x ∂μ, ¬ active x → condExpKernel μ m' x bad = 0)
(hbad : ∀ᵐ x ∂μ, ∀ᵐ ω ∂condExpKernel μ m' x,
ω ∈ bad → ∀ i ∈ selected x, ω ∉ events i) :
∀ᵐ x ∂μ, (μ[bad.indicator (fun _ => (1 : ℝ)) | m']) x ≤
Real.exp (-L / 400) := by
filter_upwards [iCondIndepSet_ae_condExpKernel m' hm' μ events hmeas hind,
hcard, hp, hzero, hbad, condExpKernel_ae_eq_condExp hm' hbadmeas]
with x hx hcard hp hzero hbad heq
rw [← heq]
let : IsProbabilityMeasure (condExpKernel μ m' x) := hx.isProbabilityMeasure
by_cases ha : active x
· have hsubset : bad ≤ᵐ[condExpKernel μ m' x] ⋂ i ∈ selected x, (events i)ᶜ := by
filter_upwards [hbad] with ω hω
intro hb
exact Set.mem_iInter.mpr (fun i => Set.mem_iInter.mpr (fun hi => hω hb i hi))
have hmono : (condExpKernel μ m' x).real bad ≤
(condExpKernel μ m' x).real (⋂ i ∈ selected x, (events i)ᶜ) :=
ENNReal.toReal_mono (measure_ne_top _ _) (measure_mono_ae hsubset)
exact hmono.trans (measureReal_le_exp_intensity (condExpKernel μ m' x) events hx hmeas
(selected x) m ρ L hρ (hcard ha) hL (hp ha) _ (Set.Subset.refl _))
· change (condExpKernel μ m' x bad).toReal ≤ _
rw [hzero ha, ENNReal.toReal_zero]
exact (Real.exp_pos _).le
theorem conditional_bad_le_exp_of_endpoint {Y : Type*}
(m' : MeasurableSpace Ω) {mΩ : MeasurableSpace Ω} [StandardBorelSpace Ω]
[MeasurableSpace Y] [MeasurableEq Y]
(hm' : m' ≤ mΩ) (μ : Measure Ω) [IsFiniteMeasure μ]
(events : ι → Set Ω) (hmeas : ∀ i, MeasurableSet (events i))
(hind : iCondIndepSet m' hm' events μ)
(endpoint : Ω → Y) (hendpoint : Measurable[m'] endpoint)
(selected : Y → Finset ι) (active : Y → Prop)
(m ρ L : ℝ) (hρ : 0 ≤ ρ) (hL : L ≤ m * ρ)
(hcard : ∀ y, active y → m / 40 ≤ ((selected y).card : ℝ))
(hp : ∀ᵐ x ∂μ, active (endpoint x) → ∀ i ∈ selected (endpoint x),
ρ / 10 ≤ (μ[(events i).indicator (fun _ => (1 : ℝ)) | m']) x)
(bad : Set Ω) (hbadmeas : MeasurableSet bad)
(hbad : ∀ᵐ ω ∂μ, ω ∈ bad → active (endpoint ω) ∧
∀ i ∈ selected (endpoint ω), ω ∉ events i) :
∀ᵐ x ∂μ, (μ[bad.indicator (fun _ => (1 : ℝ)) | m']) x ≤
Real.exp (-L / 400) := by
have heq : ∀ᵐ x ∂μ, ∀ i, (condExpKernel μ m' x).real (events i) =
(μ[(events i).indicator (fun _ => (1 : ℝ)) | m']) x :=
ae_all_iff.mpr (fun i => condExpKernel_ae_eq_condExp hm' (hmeas i))
have hpKernel : ∀ᵐ x ∂μ, active (endpoint x) → ∀ i ∈ selected (endpoint x),
ρ / 10 ≤ (condExpKernel μ m' x).real (events i) := by
filter_upwards [hp, heq] with x hx heq
intro ha i hi
rw [heq i]
exact hx ha i hi
have hfixed := condExpKernel_ae_eq_of_measurable m' hm' μ endpoint hendpoint
have hlift := ae_ae_condExpKernel_of_ae m' hm' μ hbad
have hsupport : ∀ᵐ x ∂μ, ∀ᵐ ω ∂condExpKernel μ m' x,
ω ∈ bad → active (endpoint x) ∧ ∀ i ∈ selected (endpoint x), ω ∉ events i := by
filter_upwards [hfixed, hlift] with x hfixed hlift
filter_upwards [hfixed, hlift] with ω hfixed hlift
intro hb
simpa only [hfixed] using hlift hb
have hzero : ∀ᵐ x ∂μ, ¬ active (endpoint x) → condExpKernel μ m' x bad = 0 := by
filter_upwards [hsupport] with x hx
intro ha
have hn : ∀ᵐ ω ∂condExpKernel μ m' x, ω ∉ bad := by
filter_upwards [hx] with ω hω
exact fun hb => ha (hω hb).1
simpa only [not_not, Set.ofPred_mem_eq] using (ae_iff.mp hn)
have havoid : ∀ᵐ x ∂μ, ∀ᵐ ω ∂condExpKernel μ m' x,
ω ∈ bad → ∀ i ∈ selected (endpoint x), ω ∉ events i := by
filter_upwards [hsupport] with x hx
filter_upwards [hx] with ω hω
exact fun hb => (hω hb).2
exact conditional_bad_le_exp_intensity m' hm' μ events hmeas hind
(fun x => selected (endpoint x)) m ρ L hρ hL (fun x => active (endpoint x))
(Filter.Eventually.of_forall (fun x => hcard (endpoint x))) hpKernel
bad hbadmeas hzero havoid
end ConditionalExpectation
end Invariant233.CrossingAvoidance
end OAI
|