| import OAI.Probability.FreeIsing.Energy.DiscrepancyRowEnergy |
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| namespace OAI |
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| noncomputable section |
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| open MeasureTheory Filter |
| open scoped Topology |
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| namespace Problem367.RowEnergyL1 |
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| private lemma abs_sq_sub_sq_le_young (x y : ℝ) {δ : ℝ} (hδ : 0 < δ) : |
| |x ^ 2 - y ^ 2| ≤ (1 + 1 / δ) * (x - y) ^ 2 + δ * y ^ 2 := by |
| have ha : |x ^ 2 - y ^ 2| ≤ (x - y) ^ 2 + 2 * |x - y| * |y| := by |
| calc |
| _ = |(x - y) ^ 2 + 2 * (x - y) * y| := by congr 1; ring |
| _ ≤ |(x - y) ^ 2| + |2 * (x - y) * y| := abs_add_le _ _ |
| _ = _ := by rw [abs_of_nonneg (sq_nonneg _), abs_mul, abs_mul]; norm_num |
| have hsq := sq_nonneg (|x - y| - δ * |y|) |
| rw [sub_sq, mul_pow, sq_abs, sq_abs] at hsq |
| have hdiv : δ * (1 / δ) = 1 := mul_one_div_cancel hδ.ne' |
| have hYoung : 2 * |x - y| * |y| ≤ (1 / δ) * (x - y) ^ 2 + δ * y ^ 2 := by |
| apply (mul_le_mul_iff_right₀ hδ).mp |
| rw [mul_add, ← mul_assoc δ (1 / δ), hdiv, one_mul] |
| nlinarith |
| linarith |
|
|
| theorem abs_tsum_sq_sub_le {ι : Type*} {a b : ι → ℝ} |
| (ha : Summable (fun i => (a i) ^ 2)) |
| (hb : Summable (fun i => (b i) ^ 2)) {δ : ℝ} (hδ : 0 < δ) : |
| |(∑' i, (a i) ^ 2) - ∑' i, (b i) ^ 2| ≤ |
| (1 + 1 / δ) * (∑' i, (a i - b i) ^ 2) + δ * ∑' i, (b i) ^ 2 := by |
| have hd := DiscrepancyEnergy.summable_sq_sub ha hb |
| have habs : Summable (fun i => |(a i) ^ 2 - (b i) ^ 2|) := (ha.sub hb).abs |
| rw [← ha.tsum_sub hb] |
| calc |
| |(∑' i, ((a i) ^ 2 - (b i) ^ 2))| ≤ |
| ∑' i, |(a i) ^ 2 - (b i) ^ 2| := by |
| simpa only [Real.norm_eq_abs] using |
| (norm_tsum_le_tsum_norm (f := fun i => (a i) ^ 2 - (b i) ^ 2) |
| (by simpa only [Real.norm_eq_abs] using habs)) |
| _ ≤ ∑' i, ((1 + 1 / δ) * (a i - b i) ^ 2 + δ * (b i) ^ 2) := |
| habs.tsum_le_tsum (fun i => abs_sq_sub_sq_le_young (a i) (b i) hδ) |
| ((hd.mul_left _).add (hb.mul_left _)) |
| _ = _ := by rw [Summable.tsum_add (hd.mul_left _) (hb.mul_left _), |
| tsum_mul_left, tsum_mul_left] |
|
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| variable {Ω ι : Type*} [MeasurableSpace Ω] [Countable ι] {μ : Measure Ω} |
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|
| theorem tendsto_integral_abs_tsum_sq_sub |
| {N : Type*} {l : Filter N} {a : N → ι → Ω → ℝ} {b : ι → Ω → ℝ} |
| (ha : ∀ n i, MemLp (a n i) 2 μ) (hb : ∀ i, MemLp (b i) 2 μ) |
| (haSum : ∀ n, Summable (fun i => ∫ ω, (a n i ω) ^ 2 ∂μ)) |
| (hbSum : Summable (fun i => ∫ ω, (b i ω) ^ 2 ∂μ)) |
| (hlim : Tendsto (fun n => ∑' i, ∫ ω, (a n i ω - b i ω) ^ 2 ∂μ) |
| l (𝓝 0)) : |
| Tendsto (fun n => ∫ ω, |(∑' i, (a n i ω) ^ 2) - ∑' i, (b i ω) ^ 2| ∂μ) |
| l (𝓝 0) := by |
| have hA n := DiscrepancyEnergy.sq_row_integrable_of_summable_integrals |
| (fun i => (ha n i).integrable_sq) (haSum n) |
| have hB := DiscrepancyEnergy.sq_row_integrable_of_summable_integrals |
| (fun i => (hb i).integrable_sq) hbSum |
| have hD n := DiscrepancyEnergy.integrable_sq_difference_row |
| (fun i => (ha n i).aestronglyMeasurable) (fun i => (hb i).aestronglyMeasurable) |
| (hA n).2 hB.2 ((hA n).1.and hB.1) |
| have hDsum n := DiscrepancyEnergy.hasSum_integral_sq_row |
| (fun i => ((ha n i).sub (hb i)).aestronglyMeasurable) |
| (((hA n).1.and hB.1).mono (fun ω hω => |
| DiscrepancyEnergy.summable_sq_sub hω.1 hω.2)) (hD n) |
| have hDlim : Tendsto (fun n => ∫ ω, ∑' i, (a n i ω - b i ω) ^ 2 ∂μ) l (𝓝 0) := by |
| have heq (n : N) : (∑' i, ∫ ω, (a n i ω - b i ω) ^ 2 ∂μ) = |
| ∫ ω, ∑' i, (a n i ω - b i ω) ^ 2 ∂μ := by |
| simpa only [Pi.sub_apply] using (hDsum n).tsum_eq |
| simpa only [heq] using hlim |
| let E : ℝ := ∫ ω, ∑' i, (b i ω) ^ 2 ∂μ |
| have hE : 0 ≤ E := integral_nonneg (fun ω => tsum_nonneg (fun i => sq_nonneg _)) |
| apply Metric.tendsto_nhds.mpr |
| intro ε hε |
| let δ : ℝ := ε / (2 * (E + 1)) |
| have hδ : 0 < δ := div_pos hε (by positivity) |
| have hcoef : 0 < 1 + 1 / δ := by positivity |
| have hevent := hDlim.eventually (gt_mem_nhds |
| (show 0 < ε / (2 * (1 + 1 / δ)) by positivity)) |
| filter_upwards [hevent] with n hn |
| have hbound : (∫ ω, |(∑' i, (a n i ω) ^ 2) - ∑' i, (b i ω) ^ 2| ∂μ) ≤ |
| (1 + 1 / δ) * (∫ ω, ∑' i, (a n i ω - b i ω) ^ 2 ∂μ) + δ * E := by |
| calc |
| _ ≤ ∫ ω, (1 + 1 / δ) * (∑' i, (a n i ω - b i ω) ^ 2) + |
| δ * (∑' i, (b i ω) ^ 2) ∂μ := by |
| apply integral_mono_ae ((hA n).2.sub hB.2).abs |
| (((hD n).const_mul _).add (hB.2.const_mul _)) |
| filter_upwards [(hA n).1, hB.1] with ω hω hω' |
| exact abs_tsum_sq_sub_le hω hω' hδ |
| _ = _ := by rw [integral_add ((hD n).const_mul _) (hB.2.const_mul _), |
| integral_const_mul, integral_const_mul] |
| have hfirst : (1 + 1 / δ) * (∫ ω, ∑' i, (a n i ω - b i ω) ^ 2 ∂μ) < ε / 2 := by |
| have hh := (lt_div_iff₀ (show 0 < 2 * (1 + 1 / δ) by positivity)).mp hn |
| nlinarith |
| have hsecond : δ * E < ε / 2 := by |
| dsimp [δ] |
| rw [div_mul_eq_mul_div] |
| apply (div_lt_iff₀ (show 0 < 2 * (E + 1) by positivity)).mpr |
| nlinarith |
| rw [Real.dist_eq, sub_zero, abs_of_nonneg (integral_nonneg (fun _ => abs_nonneg _))] |
| linarith |
|
|
| theorem tendsto_integral_abs_mul_sub |
| {E : ℕ → Ω → ℝ} {F : Ω → ℝ} {m : ℕ → Ω → ℝ} {q : Ω → ℝ} {C : ℝ} |
| (hE : ∀ n, Integrable (E n) μ) (hF : Integrable F μ) |
| (hm : ∀ n, AEStronglyMeasurable (m n) μ) (hq : AEStronglyMeasurable q μ) |
| (_hC : 0 ≤ C) (hmBound : ∀ n, ∀ᵐ ω ∂μ, |m n ω| ≤ C) |
| (hqBound : ∀ᵐ ω ∂μ, |q ω| ≤ C) |
| (hEconv : Tendsto (fun n => ∫ ω, |E n ω - F ω| ∂μ) atTop (𝓝 0)) |
| (hmconv : ∀ᵐ ω ∂μ, Tendsto (fun n => m n ω) atTop (𝓝 (q ω))) : |
| Tendsto (fun n => ∫ ω, |m n ω * E n ω - q ω * F ω| ∂μ) atTop (𝓝 0) := by |
| have hDiffBound (n : ℕ) : ∀ᵐ ω ∂μ, ‖m n ω - q ω‖ ≤ 2 * C := by |
| filter_upwards [hmBound n, hqBound] with ω hω hω' |
| rw [Real.norm_eq_abs] |
| exact (abs_sub _ _).trans (by linarith) |
| have hDint (n : ℕ) : Integrable (fun ω => |(m n ω - q ω) * F ω|) μ := |
| (hF.bdd_mul ((hm n).sub hq) (hDiffBound n)).abs |
| have hDlim : Tendsto (fun n => ∫ ω, |(m n ω - q ω) * F ω| ∂μ) atTop (𝓝 0) := by |
| have h := tendsto_integral_of_dominated_convergence (fun ω => (2 * C) * |F ω|) |
| (fun n => (hDint n).aestronglyMeasurable) (hF.abs.const_mul (2 * C)) |
| (fun n => ?_) (f := fun _ => (0 : ℝ)) ?_ |
| · simpa using h |
| · filter_upwards [hDiffBound n] with ω hω |
| simpa only [Real.norm_eq_abs, abs_abs, abs_mul] using |
| mul_le_mul_of_nonneg_right hω (abs_nonneg (F ω)) |
| · filter_upwards [hmconv] with ω hω |
| simpa using ((hω.sub_const (q ω)).mul_const (F ω)).abs |
| have hlim := (hEconv.const_mul C).add hDlim |
| simp only [mul_zero, add_zero] at hlim |
| apply tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds hlim |
| (fun n => integral_nonneg (fun _ => abs_nonneg _)) |
| intro n |
| have hmBoundNorm : ∀ᵐ ω ∂μ, ‖m n ω‖ ≤ C := by |
| simpa only [Real.norm_eq_abs] using hmBound n |
| have hqBoundNorm : ∀ᵐ ω ∂μ, ‖q ω‖ ≤ C := by |
| simpa only [Real.norm_eq_abs] using hqBound |
| have hfirstInt : Integrable (fun ω => C * |E n ω - F ω|) μ := by |
| simpa only [Pi.sub_apply] using ((hE n).sub hF).abs.const_mul C |
| calc |
| _ ≤ ∫ ω, C * |E n ω - F ω| + |(m n ω - q ω) * F ω| ∂μ := by |
| apply integral_mono_ae |
| (((hE n).bdd_mul (hm n) hmBoundNorm).sub (hF.bdd_mul hq hqBoundNorm)).abs |
| (hfirstInt.add (hDint n)) |
| filter_upwards [hmBound n] with ω hω |
| change |m n ω * E n ω - q ω * F ω| ≤ _ |
| calc |
| _ = |m n ω * (E n ω - F ω) + (m n ω - q ω) * F ω| := by |
| congr 1 |
| ring |
| _ ≤ |m n ω * (E n ω - F ω)| + |(m n ω - q ω) * F ω| := abs_add_le _ _ |
| _ ≤ _ := by |
| rw [abs_mul] |
| exact add_le_add (mul_le_mul_of_nonneg_right hω (abs_nonneg _)) le_rfl |
| _ = _ := by rw [integral_add hfirstInt (hDint n), |
| integral_const_mul] |
|
|
| theorem tendsto_integral_abs_multiplier_tsum_sq_sub |
| {a : ℕ → ι → Ω → ℝ} {b : ι → Ω → ℝ} |
| {m : ℕ → Ω → ℝ} {q : Ω → ℝ} {C : ℝ} |
| (ha : ∀ n i, MemLp (a n i) 2 μ) (hb : ∀ i, MemLp (b i) 2 μ) |
| (haSum : ∀ n, Summable (fun i => ∫ ω, (a n i ω) ^ 2 ∂μ)) |
| (hbSum : Summable (fun i => ∫ ω, (b i ω) ^ 2 ∂μ)) |
| (hlim : Tendsto (fun n => ∑' i, ∫ ω, (a n i ω - b i ω) ^ 2 ∂μ) |
| atTop (𝓝 0)) |
| (hm : ∀ n, AEStronglyMeasurable (m n) μ) (hq : AEStronglyMeasurable q μ) |
| (hC : 0 ≤ C) (hmBound : ∀ n, ∀ᵐ ω ∂μ, |m n ω| ≤ C) |
| (hqBound : ∀ᵐ ω ∂μ, |q ω| ≤ C) |
| (hmconv : ∀ᵐ ω ∂μ, Tendsto (fun n => m n ω) atTop (𝓝 (q ω))) : |
| Tendsto (fun n => ∫ ω, |
| |m n ω * (∑' i, (a n i ω) ^ 2) - q ω * (∑' i, (b i ω) ^ 2)| ∂μ) |
| atTop (𝓝 0) := by |
| apply tendsto_integral_abs_mul_sub |
| (fun n => (DiscrepancyEnergy.sq_row_integrable_of_summable_integrals |
| (fun i => (ha n i).integrable_sq) (haSum n)).2) |
| (DiscrepancyEnergy.sq_row_integrable_of_summable_integrals |
| (fun i => (hb i).integrable_sq) hbSum).2 |
| hm hq hC hmBound hqBound |
| · exact tendsto_integral_abs_tsum_sq_sub ha hb haSum hbSum hlim |
| · exact hmconv |
|
|
| theorem integral_abs_timePrimitive_le [SFinite μ] |
| {E : ℝ → Ω → ℝ} {T t : ℝ} (ht : t ∈ Set.Icc 0 T) |
| (hE : Integrable (fun p : ℝ × Ω => E p.1 p.2) |
| ((volume.restrict (Set.Ioc 0 T)).prod μ)) : |
| (∫ ω, |∫ s in (0 : ℝ)..t, E s ω| ∂μ) ≤ |
| ∫ p : ℝ × Ω, |E p.1 p.2| ∂(volume.restrict (Set.Ioc 0 T)).prod μ := by |
| have hmeasure : (volume.restrict (Set.Ioc 0 t)).prod μ ≤ |
| (volume.restrict (Set.Ioc 0 T)).prod μ := |
| Measure.prod_mono (Measure.restrict_mono_set _ (Set.Ioc_subset_Ioc_right ht.2)) le_rfl |
| have hEt := hE.mono_measure hmeasure |
| simp only [intervalIntegral.integral_of_le ht.1] |
| calc |
| _ ≤ ∫ ω, (∫ s in Set.Ioc 0 t, |E s ω|) ∂μ := by |
| apply integral_mono hEt.integral_prod_right.abs hEt.abs.integral_prod_right |
| intro ω |
| simpa only [Real.norm_eq_abs] using |
| norm_integral_le_integral_norm (fun s => E s ω) |
| _ = ∫ p : ℝ × Ω, |E p.1 p.2| ∂(volume.restrict (Set.Ioc 0 t)).prod μ := |
| (integral_prod_symm _ hEt.abs).symm |
| _ ≤ _ := integral_mono_measure hmeasure |
| (Eventually.of_forall (fun p => abs_nonneg (E p.1 p.2))) hE.abs |
|
|
| theorem integral_abs_timePrimitive_sub_le [SFinite μ] |
| {E F : ℝ → Ω → ℝ} {T t : ℝ} (ht : t ∈ Set.Icc 0 T) |
| (hE : Integrable (fun p : ℝ × Ω => E p.1 p.2) |
| ((volume.restrict (Set.Ioc 0 T)).prod μ)) |
| (hF : Integrable (fun p : ℝ × Ω => F p.1 p.2) |
| ((volume.restrict (Set.Ioc 0 T)).prod μ)) : |
| (∫ ω, |(∫ s in (0 : ℝ)..t, E s ω) - ∫ s in (0 : ℝ)..t, F s ω| ∂μ) ≤ |
| ∫ p : ℝ × Ω, |E p.1 p.2 - F p.1 p.2| |
| ∂(volume.restrict (Set.Ioc 0 T)).prod μ := by |
| have hmeasure : (volume.restrict (Set.Ioc 0 t)).prod μ ≤ |
| (volume.restrict (Set.Ioc 0 T)).prod μ := |
| Measure.prod_mono (Measure.restrict_mono_set _ (Set.Ioc_subset_Ioc_right ht.2)) le_rfl |
| have hEt := hE.mono_measure hmeasure |
| have hFt := hF.mono_measure hmeasure |
| calc |
| _ = ∫ ω, |∫ s in (0 : ℝ)..t, E s ω - F s ω| ∂μ := by |
| apply integral_congr_ae |
| filter_upwards [hEt.prod_left_ae, hFt.prod_left_ae] with ω hEω hFω |
| simp only [intervalIntegral.integral_of_le ht.1] |
| rw [integral_sub hEω hFω] |
| _ ≤ _ := integral_abs_timePrimitive_le ht (hE.sub hF) |
|
|
| theorem timePrimitive_sub_tendsto_uniform_L1 [SFinite μ] |
| {N : Type*} {l : Filter N} {E : N → ℝ → Ω → ℝ} {F : ℝ → Ω → ℝ} {T : ℝ} |
| (hE : ∀ n, Integrable (fun p : ℝ × Ω => E n p.1 p.2) |
| ((volume.restrict (Set.Ioc 0 T)).prod μ)) |
| (hF : Integrable (fun p : ℝ × Ω => F p.1 p.2) |
| ((volume.restrict (Set.Ioc 0 T)).prod μ)) |
| (hlim : Tendsto (fun n => ∫ p : ℝ × Ω, |E n p.1 p.2 - F p.1 p.2| |
| ∂(volume.restrict (Set.Ioc 0 T)).prod μ) l (𝓝 0)) : |
| ∀ ε > (0 : ℝ), ∀ᶠ n in l, ∀ t ∈ Set.Icc 0 T, |
| (∫ ω, |(∫ s in (0 : ℝ)..t, E n s ω) - ∫ s in (0 : ℝ)..t, F s ω| ∂μ) < ε := by |
| intro ε hε |
| filter_upwards [hlim.eventually (gt_mem_nhds hε)] with n hn |
| intro t ht |
| exact (integral_abs_timePrimitive_sub_le ht (hE n) hF).trans_lt hn |
|
|
| theorem tendsto_integral_abs_mul_sub_of_tendstoInMeasure |
| {E : ℕ → Ω → ℝ} {F : Ω → ℝ} {m : ℕ → Ω → ℝ} {q : Ω → ℝ} {C : ℝ} |
| (hE : ∀ n, Integrable (E n) μ) (hF : Integrable F μ) |
| (hm : ∀ n, AEStronglyMeasurable (m n) μ) (hq : AEStronglyMeasurable q μ) |
| (hC : 0 ≤ C) (hmBound : ∀ n, ∀ᵐ ω ∂μ, |m n ω| ≤ C) |
| (hqBound : ∀ᵐ ω ∂μ, |q ω| ≤ C) |
| (hEconv : Tendsto (fun n => ∫ ω, |E n ω - F ω| ∂μ) atTop (𝓝 0)) |
| (hmconv : TendstoInMeasure μ m atTop q) : |
| Tendsto (fun n => ∫ ω, |m n ω * E n ω - q ω * F ω| ∂μ) atTop (𝓝 0) := by |
| apply tendsto_of_subseq_tendsto |
| intro ns hns |
| obtain ⟨ms, hms, hmae⟩ := (hmconv.comp hns).exists_seq_tendsto_ae |
| refine ⟨ms, ?_⟩ |
| exact tendsto_integral_abs_mul_sub |
| (fun n => hE (ns (ms n))) hF (fun n => hm (ns (ms n))) hq hC |
| (fun n => hmBound (ns (ms n))) hqBound |
| (hEconv.comp (hns.comp hms.tendsto_atTop)) hmae |
|
|
| theorem tendstoInMeasure_of_integral_sq_sub |
| {m : ℕ → Ω → ℝ} {q : Ω → ℝ} |
| (hm : ∀ n, MemLp (m n) 2 μ) (hq : MemLp q 2 μ) |
| (hlim : Tendsto (fun n => ∫ ω, (m n ω - q ω) ^ 2 ∂μ) atTop (𝓝 0)) : |
| TendstoInMeasure μ m atTop q := by |
| apply tendstoInMeasure_of_tendsto_eLpNorm (p := 2) (by norm_num) |
| have heq n : eLpNorm (m n - q) 2 μ = |
| ENNReal.ofReal (Real.sqrt (∫ ω, (m n ω - q ω) ^ 2 ∂μ)) := by |
| rw [MemLp.eLpNorm_eq_integral_rpow_norm (by norm_num) (by norm_num) |
| ((hm n).sub hq)] |
| simp only [ENNReal.toReal_ofNat, Pi.sub_apply, Real.rpow_two, Real.norm_eq_abs, |
| sq_abs, Real.sqrt_eq_rpow, one_div] |
| simp only [heq] |
| simpa only [Function.comp_def, Real.sqrt_zero, ENNReal.ofReal_zero] using |
| ENNReal.continuous_ofReal.continuousAt.tendsto.comp hlim.sqrt |
|
|
| theorem tendsto_integral_abs_mul_sub_of_meanSquare |
| {E : ℕ → Ω → ℝ} {F : Ω → ℝ} {m : ℕ → Ω → ℝ} {q : Ω → ℝ} {C : ℝ} |
| (hE : ∀ n, Integrable (E n) μ) (hF : Integrable F μ) |
| (hm : ∀ n, MemLp (m n) 2 μ) (hq : MemLp q 2 μ) |
| (hC : 0 ≤ C) (hmBound : ∀ n, ∀ᵐ ω ∂μ, |m n ω| ≤ C) |
| (hqBound : ∀ᵐ ω ∂μ, |q ω| ≤ C) |
| (hEconv : Tendsto (fun n => ∫ ω, |E n ω - F ω| ∂μ) atTop (𝓝 0)) |
| (hmconv : Tendsto (fun n => ∫ ω, (m n ω - q ω) ^ 2 ∂μ) atTop (𝓝 0)) : |
| Tendsto (fun n => ∫ ω, |m n ω * E n ω - q ω * F ω| ∂μ) atTop (𝓝 0) := |
| tendsto_integral_abs_mul_sub_of_tendstoInMeasure hE hF |
| (fun n => (hm n).aestronglyMeasurable) hq.aestronglyMeasurable |
| hC hmBound hqBound hEconv (tendstoInMeasure_of_integral_sq_sub hm hq hmconv) |
|
|
| end Problem367.RowEnergyL1 |
|
|
| end |
|
|
| end OAI |
|
|
