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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 35 36 37 38 | import OAI.Probability.InvariantIsing.Haar.HaarPolynomialValue
namespace OAI
/-! Compact domination for continuous derivatives along actual Haar rotations. -/
noncomputable section
open Matrix MeasureTheory
namespace InvariantIsing
lemma continuous_haar_integrable {N : ℕ} (μ : Measure (SpecialOrthogonal N)) [IsFiniteMeasure μ]
(f : SpecialOrthogonal N → ℝ) (hf : Continuous f) : Integrable f μ := by
obtain ⟨B,hB⟩ := isCompact_univ.exists_bound_of_continuousOn hf.continuousOn
exact Integrable.of_bound hf.measurable.aestronglyMeasurable B
(ae_of_all μ fun U => hB U (Set.mem_univ U))
lemma integral_haar_continuous_derivative {N : ℕ} (μ : Measure (SpecialOrthogonal N))
[IsFiniteMeasure μ] [μ.IsMulLeftInvariant] (i j : Fin N)
(f q : SpecialOrthogonal N → ℝ) (hf : Continuous f) (hq : Continuous q)
(hd : ∀ U : SpecialOrthogonal N, ∀ t : ℝ,
HasDerivAt (fun s => f (specialPlaneRotation i j s*U))
(q (specialPlaneRotation i j t*U)) t) :
(∫ U, q U ∂μ) = 0 := by
obtain ⟨B,hB⟩ := isCompact_univ.exists_bound_of_continuousOn hq.continuousOn
have he := integral_special_rotation_derivative_eq_zero μ
(specialPlaneRotation i j) (specialPlaneRotation_zero i j) f hf.measurable
(continuous_haar_integrable μ f hf) (fun t U => q (specialPlaneRotation i j t*U))
(by simpa only [specialPlaneRotation_zero,one_mul] using hq.measurable.aestronglyMeasurable)
(fun _ => B) (integrable_const B)
(ae_of_all μ fun U t _ => hB (specialPlaneRotation i j t*U) (Set.mem_univ _))
(ae_of_all μ fun U t _ => hd U t)
simpa only [specialPlaneRotation_zero,one_mul] using he
end InvariantIsing
end
end OAI
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