File size: 2,641 Bytes
ddc84c5 | 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 | import OAI.Analysis.FiniteFactor.Coupling.Infinite
namespace OAI
namespace FiniteFactor
open MeasureTheory
open scoped ENNReal
noncomputable section
namespace CouplingParameters
variable (p : CouplingParameters)
def orderedPairs : Set (ℝ × ℝ) := {xy | xy.1 ≤ xy.2}
theorem orderedPairs_measurable : MeasurableSet orderedPairs :=
measurableSet_le measurable_fst measurable_snd
theorem componentPair_ordered (i : Fin 6) : p.pairComponentLaw i orderedPairs = 1 := by
refine Fin.cases ?_ (fun k => ?_) i
· change (unitIntervalMeasure.map (fun t => (t,t))) orderedPairs = 1
have hm : Measurable (fun t : ℝ => (t,t)) := measurable_id.prodMk measurable_id
rw [Measure.map_apply hm orderedPairs_measurable]
simp [orderedPairs]
· change ((p.endpointComponentLaw false k).prod (p.endpointComponentLaw true k)) orderedPairs = 1
have hU := p.endpointComponentLaw_of_contains false k (Set.Subset.refl _)
have hV := p.endpointComponentLaw_of_contains true k (Set.Subset.refl _)
have hsub : (p.endpointInterval false k ×ˢ p.endpointInterval true k) ⊆ orderedPairs := by
intro xy hxy
have h₁ : xy.1 < p.leftLength k := hxy.1.2
have h₂ : 1-p.rightLength k ≤ xy.2 := hxy.2.1
have hlen := p.lengths_sum_le k
change xy.1 ≤ xy.2
linarith
apply le_antisymm prob_le_one
have h := measure_mono (μ := (p.endpointComponentLaw false k).prod (p.endpointComponentLaw true k)) hsub
rw [Measure.prod_prod, hU, hV, mul_one] at h
exact h
theorem coupling_ordered : p.coupling orderedPairs = 1 := by
rw [p.coupling_eq_sum, Measure.finsetSum_apply]
simp only [Measure.smul_apply, smul_eq_mul, p.componentPair_ordered, mul_one]
rw [← ENNReal.ofReal_sum_of_nonneg (fun i _ => p.labelMass_nonneg i), p.labelMass_sum]
simp
theorem labelledCoupling_ordered : p.labelledCoupling {z | z.2.1 ≤ z.2.2} = 1 := by
have h := p.coupling_ordered
rw [coupling, Measure.map_apply measurable_snd orderedPairs_measurable] at h
exact h
/-- All infinitely many coordinates are ordered under the actual augmented law. -/
theorem productCoupling_ordered (p : ℕ → CouplingParameters) :
productCoupling p {ω | ∀ i, (ω i).2.1 ≤ (ω i).2.2} = 1 := by
have hset : {ω : CouplingSample | ∀ i, (ω i).2.1 ≤ (ω i).2.2} =
Set.pi Set.univ (fun _ : ℕ => {z : Fin 6 × (ℝ × ℝ) | z.2.1 ≤ z.2.2}) := by
ext ω
simp
rw [hset, productCoupling, Measure.infinitePi_pi_univ]
· simp [labelledCoupling_ordered]
· intro i
exact orderedPairs_measurable.preimage measurable_snd
end CouplingParameters
end
end FiniteFactor
end OAI
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