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a89d282 | 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 | import OAI.Probability.InvariantIsing.Pressure.RandomSpectralBound
import Mathlib.MeasureTheory.Order.Lattice
namespace OAI
/-! Measurability of the finite-dimensional spectral norm bound. -/
noncomputable section
open MeasureTheory
namespace InvariantIsing
lemma spectralRadius_eq_sup_abs {n : ℕ} (eig : Fin (n+1) → ℝ) :
spectralRadius eig=Finset.univ.sup' Finset.univ_nonempty (fun i => |eig i|) := by
apply le_antisymm
· apply (spectralRadius_le_iff _ _).mpr
intro i
exact Finset.le_sup' (fun i => |eig i|) (Finset.mem_univ i)
· exact Finset.sup'_le Finset.univ_nonempty _ fun i _ => abs_le_spectralRadius eig i
lemma measurable_spectralRadius {Ω : Type*} [MeasurableSpace Ω] {n : ℕ}
(eig : Ω → Fin (n+1) → ℝ) (heig : Measurable eig) :
Measurable (fun ω => spectralRadius (eig ω)) := by
have hh := Finset.measurable_sup' (s := (Finset.univ : Finset (Fin (n+1))))
Finset.univ_nonempty (fun i _ => ((measurable_pi_apply i).comp heig).abs)
convert hh using 1
funext ω
rw [spectralRadius_eq_sup_abs,Finset.sup'_apply]
rfl
end InvariantIsing
end
end OAI
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