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a2ee19a | 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 | import Mathlib
import OAI.Probability.Perceptron.Variational.CovarianceMatrix
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory Set Filter
open scoped Classical ENNReal NNReal BigOperators Topology
namespace SphericalPerceptronFreeEnergy
lemma array_law_eq_of_blocks {K : Type*} [MeasurableSpace K]
(μ ν : Measure (CompactArray K)) [IsFiniteMeasure μ] [IsFiniteMeasure ν]
(h : ∀ n, μ.map (compactBlock (n+1)) = ν.map (compactBlock (n+1))) : μ = ν := by
let e : CompactArray K ≃ᵐ (ℕ × ℕ → K) := (MeasurableEquiv.curry ℕ ℕ K).symm
apply e.map_measurableEquiv_injective
let P (I : Finset (ℕ × ℕ)) := (ν.map e).map I.restrict
have hp : IsProjectiveLimit (ν.map e) P := fun _ => rfl
apply IsProjectiveLimit.unique (P := P) ?_ hp
intro I
let n := I.sup (fun a => max a.1 a.2)
have hb (a : I) : a.val.1 < n+1 ∧ a.val.2 < n+1 := by
have he : max a.val.1 a.val.2 ≤ n := Finset.le_sup (f := fun a : ℕ × ℕ => max a.1 a.2) a.property
exact ⟨by omega,by omega⟩
let G : CompactBlock K (n+1) → (I → K) :=
fun B a => B ⟨a.val.1,(hb a).1⟩ ⟨a.val.2,(hb a).2⟩
have hG : Measurable G := by
apply Measurable.of_eval
intro a
exact (measurable_pi_apply _).comp (measurable_pi_apply _)
have hm : Measurable (compactBlock (K:=K) (n+1)) := by
unfold compactBlock
exact Measurable.of_eval fun _ => Measurable.of_eval fun _ => by fun_prop
change (μ.map e).map I.restrict = (ν.map e).map I.restrict
have he : I.restrict ∘ e = G ∘ compactBlock (n+1) := by
funext Q a
rfl
rw [Measure.map_map I.measurable_restrict e.measurable,
Measure.map_map I.measurable_restrict e.measurable,he,
← Measure.map_map hG hm,← Measure.map_map hG hm,h n]
end SphericalPerceptronFreeEnergy
end
end OAI
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