| import OAI.Analysis.Daugavet.WholeBallConcentration |
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| namespace OAI |
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| noncomputable section |
| open MeasureTheory ProbabilityTheory Set Filter |
| open scoped ENNReal NNReal Topology BigOperators |
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| universe uIota |
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| namespace ExactModuli.KW |
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| abbrev infiniteCube := ℕ → unitInterval |
| abbrev infiniteCubeMeasure : Measure infiniteCube := |
| Measure.infinitePi (fun _ : ℕ => (volume : Measure unitInterval)) |
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| lemma infinite_coordinate_mp (i : ℕ) : |
| MeasurePreserving (fun ω : infiniteCube => ω i) infiniteCubeMeasure volume := |
| ⟨measurable_pi_apply i, Measure.infinitePi_map_eval _ i⟩ |
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| lemma finite_coordinates_mp {ι : Type uIota} [Fintype ι] (c : ι → ℕ) (hc : Function.Injective c) : |
| MeasurePreserving (fun ω : infiniteCube => fun i => ω (c i)) infiniteCubeMeasure (cubeMeasure ι) := by |
| refine ⟨by fun_prop, ?_⟩ |
| rw [Measure.map_infinitePi_infinitePi_of_inj hc, Measure.infinitePi_eq_pi] |
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| / |
| This is Mathlib's L1 strong law, applied to the exact Pareto multiplier. -/ |
| lemma multiplier_averages {p h : ℝ} (hp : 1 < p) (hh : 0 < h) : |
| ∃ n : ℕ, 0 < n ∧ |
| (∫ v : cube (Fin n), |(n : ℝ)⁻¹ * (∑ j, multiplier p (v j)) - 1| |
| ∂cubeMeasure (Fin n)) < h := by |
| let X : ℕ → infiniteCube → ℝ := fun i ω => multiplier p (ω i) |
| have hXi (i : ℕ) : Integrable (X i) infiniteCubeMeasure := |
| ((infinite_coordinate_mp i).integrable_comp (integrable_multiplier hp).aestronglyMeasurable).mpr |
| (integrable_multiplier hp) |
| have hdist (i : ℕ) : IdentDistrib (X i) (multiplier p) infiniteCubeMeasure volume := |
| mp_ident (infinite_coordinate_mp i) (measurable_multiplier p) |
| have hind : iIndepFun X infiniteCubeMeasure := iIndepFun_infinitePi (fun _ => measurable_multiplier p) |
| have hmean : (∫ ω, X 0 ω ∂infiniteCubeMeasure) = 1 := |
| (hdist 0).integral_eq.trans (integral_multiplier hp) |
| have hLLN := strong_law_Lp (by norm_num : (1 : ℝ≥0∞) ≤ 1) (by norm_num : (1 : ℝ≥0∞) ≠ ∞) |
| X (memLp_one_iff_integrable.mpr (hXi 0)) (fun i j hij => hind.indepFun hij) (fun i => (hdist i).trans (hdist 0).symm) |
| have hreal := (ENNReal.tendsto_toReal (by norm_num : (0 : ℝ≥0∞) ≠ ∞)).comp hLLN |
| simp only [ENNReal.toReal_zero, hmean, smul_eq_mul] at hreal |
| have havg (n : ℕ) : Integrable |
| (fun ω => (n : ℝ)⁻¹ * (∑ i ∈ Finset.range n, X i ω) - 1) infiniteCubeMeasure := |
| ((integrable_finsetSum _ (fun i _ => hXi i)).const_mul _).sub (integrable_const 1) |
| have hevent : ∀ᶠ n : ℕ in atTop, |
| (∫ ω, |(n : ℝ)⁻¹ * (∑ i ∈ Finset.range n, X i ω) - 1| ∂infiniteCubeMeasure) < h := by |
| convert hreal.eventually (Iio_mem_nhds hh) using 1 |
| ext n |
| simp only [Function.comp_apply] |
| rw [eLpNorm_one_eq_lintegral_enorm (havg n).aestronglyMeasurable, |
| ← integral_norm_eq_lintegral_enorm (havg n).aestronglyMeasurable] |
| rfl |
| obtain ⟨n, hn, hn'⟩ := ((eventually_gt_atTop 0).and hevent).exists |
| refine ⟨n, hn, ?_⟩ |
| have hmp := finite_coordinates_mp (fun j : Fin n => (j : ℕ)) Fin.val_injective |
| have heq := mp_integral hmp |
| (((integrable_rowSum hp (fun _ : Fin n => 1)).const_mul (n : ℝ)⁻¹).sub (integrable_const 1)).abs.aestronglyMeasurable |
| simp only [rowSum, one_mul, Pi.sub_apply] at heq |
| rw [← heq] |
| have hefin (ω : infiniteCube) : (∑ j : Fin n, multiplier p (ω j)) = |
| ∑ i ∈ Finset.range n, X i ω := Fin.sum_univ_eq_sum_range (fun j => multiplier p (ω j)) n |
| simp only [hefin] |
| exact hn' |
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| end ExactModuli.KW |
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| end |
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| end OAI |
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