| 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 |
|
|
