| import OAI.Probability.InvariantIsing.Haar.HaarPolynomialValue |
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| namespace OAI |
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| /-! Compact domination for continuous derivatives along actual Haar rotations. -/ |
| noncomputable section |
| open Matrix MeasureTheory |
| namespace InvariantIsing |
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| 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)) |
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| 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 |
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| end InvariantIsing |
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| end |
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| end OAI |
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