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namespace OAI
noncomputable section
open MeasureTheory Filter
open scoped Topology
namespace Problem367.RowEnergyL1
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]
variable {Ω ι : Type*} [MeasurableSpace Ω] [Countable ι] {μ : Measure Ω}
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
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