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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 58 59 60 | import OAI.Probability.FreeIsing.Model
namespace OAI
noncomputable section
open MeasureTheory
namespace Problem367
def vertexSampler {d : ℕ} {α β : Type*} (f : α → β)
(x : TreeVertex d → α) : TreeVertex d → β := fun v => f (x v)
@[simp] theorem vertexSampler_relabel {d : ℕ} {α β : Type*}
(f : α → β) (g : TreeAut d) (x : TreeVertex d → α) :
vertexSampler f (relabel g x) = relabel g (vertexSampler f x) := rfl
theorem measurable_vertexSampler {d : ℕ} {α β : Type*}
[MeasurableSpace α] [MeasurableSpace β] {f : α → β}
(hf : Measurable f) : Measurable (vertexSampler (d := d) f) := by
exact Measurable.of_eval fun vertex => hf.comp (measurable_pi_apply vertex)
theorem map_vertexSampler_labelMeasure {d : ℕ} {α : Type*}
[MeasurableSpace α] {f : ℝ → α} (hf : Measurable f) :
Measure.map (vertexSampler (d := d) f) (labelMeasure d) =
Measure.infinitePi (fun _ : TreeVertex d => Measure.map f uniform01) := by
exact Measure.infinitePi_map_pi (fun _ => uniform01) (fun _ => hf)
theorem isFactorOfIID_of_product_sampler {d : ℕ} {theta : ℝ} {α : Type*}
[MeasurableSpace α] (ν : Measure α) [IsProbabilityMeasure ν]
(f : ℝ → α) (hf : Measurable f) (hf_law : Measure.map f uniform01 = ν)
(H : (TreeVertex d → α) → (TreeVertex d → IsingSpin))
(hH : Measurable H)
(hLaw : HasFreeIsingLaw theta
(Measure.map H (Measure.infinitePi (fun _ : TreeVertex d => ν))))
(hEquiv : ∀ g : TreeAut d,
∀ᵐ W ∂Measure.infinitePi (fun _ : TreeVertex d => ν),
H (relabel g W) = relabel g (H W)) :
IsFactorOfIID d theta := by
have hs := measurable_vertexSampler (d := d) hf
have hslaw : Measure.map (vertexSampler (d := d) f) (labelMeasure d) =
Measure.infinitePi (fun _ : TreeVertex d => ν) := by
rw [map_vertexSampler_labelMeasure hf, hf_law]
refine ⟨H ∘ vertexSampler f, hH.comp hs, ?_, ?_⟩
· rw [← Measure.map_map hH hs, hslaw]
exact hLaw
· intro g
have h : ∀ᵐ W ∂Measure.map (vertexSampler (d := d) f) (labelMeasure d),
H (relabel g W) = relabel g (H W) := by
rw [hslaw]
exact hEquiv g
filter_upwards [ae_of_ae_map hs.aemeasurable h] with U hU
simpa only [Function.comp_apply, vertexSampler_relabel] using hU
end Problem367
end
end OAI
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