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f61e547 | 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 | import OAI.Probability.FreeIsing.Estimates.RelabelMeasure
import OAI.Probability.FreeIsing.Factors.Transport
namespace OAI
noncomputable section
open MeasureTheory
namespace Problem367
theorem isFactorOfIID_of_aemeasurable {d : ℕ} {theta : ℝ}
(Phi : (TreeVertex d → ℝ) → (TreeVertex d → IsingSpin))
(hPhi : AEMeasurable Phi (labelMeasure d))
(hLaw : HasFreeIsingLaw theta (Measure.map Phi (labelMeasure d)))
(hEquiv : ∀ g : TreeAut d,
∀ᵐ U ∂labelMeasure d, Phi (relabel g U) = relabel g (Phi U)) :
IsFactorOfIID d theta := by
refine ⟨hPhi.mk Phi, hPhi.measurable_mk, ?_, ?_⟩
· rw [← Measure.map_congr hPhi.ae_eq_mk]
exact hLaw
· intro g
exact FactorTransport.ae_equivariant_congr
(measurePreserving_relabel g).quasiMeasurePreserving
hPhi.ae_eq_mk (hEquiv g)
theorem isFactorOfIID_iff_aemeasurable (d : ℕ) (theta : ℝ) :
IsFactorOfIID d theta ↔
∃ Phi : (TreeVertex d → ℝ) → (TreeVertex d → IsingSpin),
AEMeasurable Phi (labelMeasure d) ∧
HasFreeIsingLaw theta (Measure.map Phi (labelMeasure d)) ∧
∀ g : TreeAut d,
∀ᵐ U ∂labelMeasure d, Phi (relabel g U) = relabel g (Phi U) := by
constructor
· rintro ⟨Phi, hPhi, hLaw, hEquiv⟩
exact ⟨Phi, hPhi.aemeasurable, hLaw, hEquiv⟩
· rintro ⟨Phi, hPhi, hLaw, hEquiv⟩
exact isFactorOfIID_of_aemeasurable Phi hPhi hLaw hEquiv
theorem isFactorOfIID_of_coordinatewise_ae {d : ℕ} {theta : ℝ}
(Phi : (TreeVertex d → ℝ) → (TreeVertex d → IsingSpin))
(hPhi : ∀ v, AEMeasurable (fun U => Phi U v) (labelMeasure d))
(hLaw : HasFreeIsingLaw theta (Measure.map Phi (labelMeasure d)))
(hEquiv : ∀ (g : TreeAut d) (v : TreeVertex d),
∀ᵐ U ∂labelMeasure d, Phi (relabel g U) v = relabel g (Phi U) v) :
IsFactorOfIID d theta := by
apply isFactorOfIID_of_aemeasurable Phi (aemeasurable_pi_iff.mpr hPhi) hLaw
intro g
filter_upwards [ae_all_iff.mpr (hEquiv g)] with U hU
exact funext hU
end Problem367
end
end OAI
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