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a44267b | 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 OAI.NumberTheory.DukePrimeDegree.Geometry.QuotientNeighborhoods
import OAI.NumberTheory.DukePrimeDegree.Geometry.QuotientSeparatingCuts
namespace OAI
/-! # Finite observations with uniformly small lifts on a compact set -/
noncomputable section
open MeasureTheory Set Filter
open scoped Topology
namespace DukePrimeDegree
open PrimeDegreePackets
theorem exists_finite_observation_small_lifts {n : ℕ}
(K : Set (X n)) (hK : IsCompact K) (W : Set (G n)) (hW : W ∈ 𝓝 1) :
∃ (N : ℕ) (Q : X n → Fin N), Measurable Q ∧
∀ x ∈ K, ∀ y, Q y=Q x → ∃ g ∈ W, rightAction x g=y := by
classical
choose U hU hxU hsmall using fun x : X n => exists_open_small_relative_lifts x W hW
obtain ⟨s,hs⟩ := hK.elim_finite_subcover U hU (fun x _ => mem_iUnion.mpr ⟨x,hxU x⟩)
let bits : X n → (s → Bool) := fun x i => decide (x ∈ U i.val)
have hbits : Measurable bits := by
apply measurable_pi_iff.mpr
intro i
apply measurable_to_bool
have he : (fun x => bits x i) ⁻¹' {true} = U i.val := by
ext x
simp [bits]
rw [he]
exact (hU i.val).measurableSet
let Q : X n → Fin (Fintype.card (s → Bool)) := fun x => Fintype.equivFin _ (bits x)
refine ⟨_,Q,(measurable_of_finite (Fintype.equivFin _)).comp hbits,?_⟩
intro x hx y heq
obtain ⟨i,hi,hxi⟩ := mem_iUnion₂.mp (hs hx)
have hxy : bits y=bits x := (Fintype.equivFin _).injective heq
have hbit := congrFun hxy ⟨i,hi⟩
have hyi : y ∈ U i := by
simpa only [bits,hxi,decide_true,decide_eq_true_eq] using hbit
exact hsmall i x hxi y hyi
end DukePrimeDegree
end
end OAI
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