| import OAI.Analysis.FiniteFactor.Convolution |
| import OAI.Analysis.FiniteFactor.TailEstimate |
|
|
| namespace OAI |
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|
| namespace FiniteFactor |
| open MeasureTheory Filter |
| open scoped ENNReal |
| noncomputable section |
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| / |
| theorem nonnegative_series_cauchy_schwarz {ι : Type*} (f g : ι → ℝ) |
| (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i) |
| (hfs : Summable (fun i => f i^2)) (hgs : Summable (fun i => g i^2)) : |
| Summable (fun i => f i*g i) ∧ |
| (∑' i, f i*g i) ≤ Real.sqrt (∑' i, f i^2)*Real.sqrt (∑' i, g i^2) := by |
| have hp : Real.HolderConjugate 2 2 := by norm_num [Real.holderConjugate_iff] |
| have hF : 0 ≤ ∑' i, f i^2 := tsum_nonneg (fun i => sq_nonneg _) |
| have hG : 0 ≤ ∑' i, g i^2 := tsum_nonneg (fun i => sq_nonneg _) |
| have hsf : HasSum (fun i => f i^(2 : ℝ)) ((Real.sqrt (∑' i, f i^2))^(2 : ℝ)) := by |
| simpa only [Real.rpow_two,Real.sq_sqrt hF] using hfs.hasSum |
| have hsg : HasSum (fun i => g i^(2 : ℝ)) ((Real.sqrt (∑' i, g i^2))^(2 : ℝ)) := by |
| simpa only [Real.rpow_two,Real.sq_sqrt hG] using hgs.hasSum |
| obtain ⟨C,_,hC,hs⟩ := Real.inner_le_Lp_mul_Lq_hasSum_of_nonneg hp |
| (Real.sqrt_nonneg _) (Real.sqrt_nonneg _) hf hg hsf hsg |
| exact ⟨hs.summable,hs.tsum_eq ▸ hC⟩ |
|
|
| / |
| Equivalent to the product-with-counting-measure argument; no independence or |
| finite truncation is assumed. -/ |
| theorem countable_error_integral_le {Ω ι : Type*} [MeasurableSpace Ω] [Countable ι] |
| {μ : Measure Ω} [IsProbabilityMeasure μ] (f g : ι → Ω → ℝ) (δ : ℝ) |
| (hf : ∀ i, Measurable (f i)) (hg : ∀ i, Measurable (g i)) |
| (hf0 : ∀ i ω, 0 ≤ f i ω) (hg0 : ∀ i ω, 0 ≤ g i ω) |
| (hfs : ∀ᵐ ω ∂μ, Summable (fun i => f i ω^2)) |
| (hgs : ∀ᵐ ω ∂μ, Summable (fun i => g i ω^2)) |
| (hfi : Integrable (fun ω => ∑' i, f i ω^2) μ) |
| (hfm : (∫ ω, (∑' i, f i ω^2) ∂μ) ≤ δ) |
| (hg1 : ∀ᵐ ω ∂μ, (∑' i, g i ω^2) ≤ 1) : |
| Integrable (fun ω => ∑' i, f i ω*g i ω) μ ∧ |
| (∫ ω, (∑' i, f i ω*g i ω) ∂μ) ≤ Real.sqrt δ := by |
| let F := fun ω => ∑' i, f i ω^2 |
| let S := fun ω => ∑' i, f i ω*g i ω |
| have hF0 (ω) : 0 ≤ F ω := tsum_nonneg (fun i => sq_nonneg _) |
| have hFm : Measurable F := Measurable.tsum (fun i => (hf i).pow_const 2) |
| have hSm : Measurable S := Measurable.tsum (fun i => (hf i).mul (hg i)) |
| have hS0 (ω) : 0 ≤ S ω := tsum_nonneg (fun i => mul_nonneg (hf0 i ω) (hg0 i ω)) |
| have hFroot : MemLp (fun ω => Real.sqrt (F ω)) 2 μ := by |
| apply (memLp_two_iff_integrable_sq (hFm.sqrt.aestronglyMeasurable)).mpr |
| simpa only [Real.sq_sqrt (hF0 _)] using hfi |
| have hS : ∀ᵐ ω ∂μ, S ω ≤ Real.sqrt (F ω) := by |
| filter_upwards [hfs,hgs,hg1] with ω hs ht h1 |
| have hh := (nonnegative_series_cauchy_schwarz (fun i => f i ω) (fun i => g i ω) |
| (fun i => hf0 i ω) (fun i => hg0 i ω) hs ht).2 |
| apply hh.trans |
| simpa only [Real.sqrt_one,mul_one] using |
| mul_le_mul_of_nonneg_left (Real.sqrt_le_sqrt h1) (Real.sqrt_nonneg (F ω)) |
| have hSi : Integrable S μ := (hFroot.integrable (by norm_num)).mono' |
| hSm.aestronglyMeasurable (hS.mono fun ω h => by rwa [Real.norm_of_nonneg (hS0 ω)]) |
| refine ⟨hSi,?_⟩ |
| have hh := integral_mul_le_sqrt_of_square_bounds hFroot (memLp_const (1 : ℝ)) |
| (Eventually.of_forall fun ω => Real.sqrt_nonneg (F ω)) (Eventually.of_forall fun _ => zero_le_one) |
| (by simpa only [Real.sq_sqrt (hF0 _)] using hfm) |
| (by simp : (∫ _ω : Ω, (1 : ℝ)^2 ∂μ) ≤ 1) |
| apply (integral_mono_ae hSi (hFroot.integrable (by norm_num)) hS).trans |
| simpa only [mul_one] using hh |
|
|
| end |
| end FiniteFactor |
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| end OAI |
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