| import OAI.Geometry.Kahler.BaseDensityMoments |
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| namespace OAI |
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| universe uKahler13116_1 |
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| open Complex |
| open scoped ContDiff Matrix Matrix.Norms.Elementwise |
| open scoped ContDiff Matrix Matrix.Norms.Elementwise ComplexOrder |
| open scoped ContDiff ComplexOrder |
| open Set Filter Topology |
| open scoped ContDiff ENNReal Pointwise |
| open scoped ContDiff |
| open Set Filter Topology MeasureTheory |
| open scoped ContDiff ENNReal |
| noncomputable section |
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| open Set Filter Topology MeasureTheory |
| open scoped ContDiff ENNReal |
| namespace PinchedHartogs.BaseConstruction |
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| def densityMeasure (W : Base → ℝ) : Measure Sphere := sigma.withDensity (fun ξ : Sphere => ENNReal.ofReal (W ξ)) |
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| lemma densityMeasure_integral {W : Base → ℝ} (hW : Continuous W) (hpos : ∀ ξ : Sphere, 0 ≤ W ξ) |
| {E : Type uKahler13116_1} [NormedAddCommGroup E] [NormedSpace ℝ E] (g : Sphere → E) : |
| (∫ ξ, g ξ ∂densityMeasure W) = ∫ ξ, W ξ • g ξ ∂sigma := by |
| change (∫ ξ, g ξ ∂sigma.withDensity (fun ξ : Sphere => ENNReal.ofReal (W ξ))) = _ |
| have hm : Measurable (fun ξ : Sphere => ENNReal.ofReal (W ξ)) := (hW.comp continuous_subtype_val).measurable.ennreal_ofReal |
| rw [integral_withDensity_eq_integral_toReal_smul hm (Eventually.of_forall (fun _ => ENNReal.ofReal_lt_top)) g] |
| simp_rw [ENNReal.toReal_ofReal (hpos _)] |
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| lemma densityMeasure_probability {W : Base → ℝ} (hW : Continuous W) (hpos : ∀ ξ : Sphere, 0 ≤ W ξ) |
| (hmass : (∫ ξ : Sphere, W ξ ∂sigma)=1) : IsProbabilityMeasure (densityMeasure W) := by |
| constructor |
| have hi : Integrable (fun ξ : Sphere => W ξ) sigma := continuous_integrable (hW.comp continuous_subtype_val) |
| rw [densityMeasure,withDensity_apply _ MeasurableSet.univ,Measure.restrict_univ, |
| ← ofReal_integral_eq_lintegral_ofReal hi (Eventually.of_forall hpos),hmass] |
| norm_num |
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| lemma densityMeasure_represents {W : Base → ℝ} (hW : Continuous W) (hpos : ∀ ξ : Sphere, 0 ≤ W ξ) |
| (hmoment : ∀ p : MvPolynomial (Fin 2) ℂ, |
| (∫ ξ : Sphere, MvPolynomial.eval (fun i => (ξ:Base) i) p*(W ξ:ℂ) ∂sigma)=MvPolynomial.eval (fun _ => 0) p) : |
| Represents (densityMeasure W) := by |
| intro p |
| rw [densityMeasure_integral hW hpos] |
| simpa only [Complex.real_smul,mul_comm] using hmoment p |
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| lemma density_orbit_mean {Q : ℕ} (hQ : 2 ≤ Q) {R E : ℝ} (hER : E < R) |
| {f b : ℝ → ℝ} (hf : ContDiff ℝ ∞ f) (hb : ContDiff ℝ ∞ b) |
| (htail : ∀ y, E ≤ y → f y=0 ∧ b y=0) (P : ℕ → Finset Sphere) (j : ℕ) : |
| OrbitMeanOne (density Q R f b P j) := by |
| classical |
| have hq : 0 < Q := by omega |
| induction j with |
| | zero => exact orbitMean_one |
| | succ j ih => |
| intro ξ |
| have hw := density_smooth hq hER hf hb htail P j |
| have hi : Integrable (fun z : Circle => density Q R f b P j ((z:ℂ) • (ξ:Base))) circleMeasure := by |
| apply compact_continuous_integrable |
| exact hw.continuous.comp ((continuous_subtype_val : Continuous (fun z : Circle => (z:ℂ))).smul continuous_const) |
| have hic : ∀ p ∈ P (Q^(j+1)), Integrable (fun z : Circle => densityCorrection (Q^(j+1)) R f b (density Q R f b P j) p ((z:ℂ) • (ξ:Base))) circleMeasure := by |
| intro p hp |
| apply compact_continuous_integrable |
| exact (densityCorrection_smooth (pow_pos hq _) hER hf hb hw htail p).continuous.comp ((continuous_subtype_val : Continuous (fun z : Circle => (z:ℂ))).smul continuous_const) |
| simp only [density] |
| rw [integral_add hi (integrable_finsetSum _ hic),integral_finsetSum _ hic,ih ξ] |
| have hz : (∑ p ∈ P (Q^(j+1)), ∫ z : Circle, densityCorrection (Q^(j+1)) R f b (density Q R f b P j) p ((z:ℂ) • (ξ:Base)) ∂circleMeasure)=0 := by |
| apply Finset.sum_eq_zero |
| intro p hp |
| exact densityCorrection_orbit_zero (hw.differentiable (by simp)) (density_bandwidth hq hER hf hb htail P j) (densityBandwidth_lt hQ j) R f b p ξ |
| rw [hz,add_zero] |
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| end PinchedHartogs.BaseConstruction |
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| end |
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| end OAI |
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