openAI-math-dataset / lean /OAI /Analysis /IrrationalRotation /Couplings /ConditionalMaskedTail.lean
| import OAI.Analysis.IrrationalRotation.Analysis.MaskedCauchySchwarz | |
| import OAI.Analysis.IrrationalRotation.Analysis.MaskedTail | |
| namespace OAI | |
| open MeasureTheory | |
| open scoped ENNReal | |
| namespace InvariantProjection | |
| variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {μ : Measure Ω} | |
| theorem lintegral_tsum_masked_mul_le_of_condExp | |
| {ι : Type*} [mι : MeasurableSpace ι] [MeasurableSingletonClass ι] [Countable ι] | |
| [IsFiniteMeasure μ] | |
| {a b : Ω × ι → ℝ} (ha : Measurable a) (hb : Measurable b) | |
| (ha_nonneg : ∀ z, 0 ≤ a z) (hb_nonneg : ∀ z, 0 ≤ b z) | |
| {H : Set (Ω × ι)} (hH : MeasurableSet H) | |
| {m : MeasurableSpace Ω} (hm : m ≤ mΩ) {δ : ℝ} (hδ : 0 ≤ δ) | |
| (ha_endpoint : ∀ i, StronglyMeasurable[m] (fun ω => a (ω, i) ^ 2)) | |
| (hcond : ∀ i, μ[({ω | (ω, i) ∈ H}).indicator (fun _ => (1 : ℝ)) | m] | |
| ≤ᵐ[μ] fun _ => δ) | |
| (ha_energy : (∫⁻ ω, ∑' i, ENNReal.ofReal (a (ω, i) ^ 2) ∂μ) ≤ 1) | |
| (hb_energy : (∫⁻ ω, ∑' i, ENNReal.ofReal (b (ω, i) ^ 2) ∂μ) ≤ 1) : | |
| (∫⁻ ω, ∑' i, H.indicator | |
| (fun z => ENNReal.ofReal (a z) * ENNReal.ofReal (b z)) (ω, i) ∂μ) ≤ | |
| ENNReal.ofReal (Real.sqrt δ) := by | |
| have ha_sq_int : ∀ i, Integrable (fun ω => a (ω, i) ^ 2) μ := by | |
| intro i | |
| refine ⟨((ha.comp (@measurable_prodMk_right Ω ι mΩ mι i)).pow_const 2).aestronglyMeasurable, ?_⟩ | |
| apply (hasFiniteIntegral_iff_ofReal (Filter.Eventually.of_forall fun _ => sq_nonneg _)).2 | |
| calc | |
| (∫⁻ ω, ENNReal.ofReal (a (ω, i) ^ 2) ∂μ) ≤ | |
| ∫⁻ ω, ∑' j, ENNReal.ofReal (a (ω, j) ^ 2) ∂μ := | |
| lintegral_mono fun ω => ENNReal.le_tsum (f := fun j => ENNReal.ofReal (a (ω, j) ^ 2)) i | |
| _ ≤ 1 := ha_energy | |
| _ < ⊤ := by simp | |
| have hs : ∀ i, MeasurableSet[mΩ] {ω | (ω, i) ∈ H} := fun i => | |
| hH.preimage (@measurable_prodMk_right Ω ι mΩ mι i) | |
| have hsmall := lintegral_tsum_indicator_le_of_condExp_indicator_le_of_energy_le_one | |
| (mΩ := mΩ) hm hs hδ ha_sq_int ha_endpoint | |
| (fun _ => Filter.Eventually.of_forall fun _ => sq_nonneg _) hcond ha_energy | |
| have hmasked : (∫⁻ ω, ∑' i, H.indicator | |
| (fun z => ENNReal.ofReal (a z) ^ (2 : ℕ)) (ω, i) ∂μ) ≤ ENNReal.ofReal δ := by | |
| convert hsmall using 1 | |
| apply lintegral_congr | |
| intro ω | |
| apply tsum_congr | |
| intro i | |
| by_cases hi : (ω, i) ∈ H | |
| · simp [hi, ENNReal.ofReal_pow (ha_nonneg (ω, i))] | |
| · simp [hi] | |
| have hb2 : (∫⁻ ω, ∑' i, ENNReal.ofReal (b (ω, i)) ^ (2 : ℕ) ∂μ) ≤ 1 := by | |
| simpa only [← ENNReal.ofReal_pow (hb_nonneg _)] using hb_energy | |
| let : MeasurableSpace Ω := mΩ | |
| have hcs := Invariant233.MaskedTail.lintegral_tsum_indicator_mul_le_sqrt | |
| (μ := μ) hH ha.ennreal_ofReal.aemeasurable hb.ennreal_ofReal.aemeasurable hmasked hb2 | |
| simpa only [ENNReal.ofReal_rpow_of_nonneg hδ (by norm_num : (0 : ℝ) ≤ 1 / 2), | |
| ← Real.sqrt_eq_rpow] using hcs | |
| end InvariantProjection | |
| end OAI | |