| import OAI.Probability.FreeIsing.Estimates.UniformSampler | |
| import OAI.Probability.FreeIsing.Factors.Transfer | |
| namespace OAI | |
| noncomputable section | |
| open MeasureTheory | |
| namespace Problem367 | |
| theorem isFactorOfIID_of_standardBorelNoise {d : ℕ} {theta : ℝ} | |
| {A : Type*} [MeasurableSpace A] [Nonempty A] [StandardBorelSpace A] | |
| (ν : Measure A) [IsProbabilityMeasure ν] | |
| (H : (TreeVertex d → A) → (TreeVertex d → IsingSpin)) | |
| (hH : Measurable H) | |
| (hlaw : HasFreeIsingLaw theta | |
| (Measure.map H (Measure.infinitePi (fun _ : TreeVertex d => ν)))) | |
| (heq : ∀ g : TreeAut d, | |
| ∀ᵐ W ∂Measure.infinitePi (fun _ : TreeVertex d => ν), | |
| H (relabel g W) = relabel g (H W)) : | |
| IsFactorOfIID d theta := by | |
| obtain ⟨ξ, hξ, hν⟩ := exists_uniform01_sampler ν | |
| exact isFactorOfIID_of_product_sampler ν ξ hξ hν H hH hlaw heq | |
| end Problem367 | |
| end | |
| end OAI | |