| import OAI.Analysis.IrrationalRotation.Couplings.ConditionalMaskedTail |
| import Mathlib.Analysis.InnerProductSpace.l2Space |
|
|
| namespace OAI |
|
|
| open MeasureTheory |
| open scoped ENNReal lp |
|
|
| namespace InvariantProjection |
|
|
| variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {μ : Measure Ω} |
|
|
| theorem l2_tsum_ofReal_sq (x : ℓ²(ℤ, ℂ)) : |
| (∑' i : ℤ, ENNReal.ofReal (‖x i‖ ^ 2)) = ENNReal.ofReal (‖x‖ ^ 2) := by |
| have hs : HasSum (fun i : ℤ => ‖x i‖ ^ 2) (‖x‖ ^ 2) := by |
| simpa using lp.hasSum_norm (p := (2 : ℝ≥0∞)) (by norm_num) x |
| rw [← ENNReal.ofReal_tsum_of_nonneg (fun _ => sq_nonneg _) hs.summable, hs.tsum_eq] |
|
|
| theorem lintegral_l2_sq_le_one [IsProbabilityMeasure μ] |
| (ξ : Ω → ℓ²(ℤ, ℂ)) (hξ : ∀ᵐ ω ∂μ, ‖ξ ω‖ ≤ 1) : |
| (∫⁻ ω, ∑' i : ℤ, ENNReal.ofReal (‖ξ ω i‖ ^ 2) ∂μ) ≤ 1 := by |
| calc |
| _ ≤ ∫⁻ _ : Ω, (1 : ℝ≥0∞) ∂μ := by |
| apply lintegral_mono_ae |
| filter_upwards [hξ] with ω hω |
| rw [l2_tsum_ofReal_sq] |
| apply ENNReal.ofReal_le_one.mpr |
| nlinarith [norm_nonneg (ξ ω)] |
| _ = 1 := by simp |
|
|
| theorem lintegral_l2_masked_tail_le_of_condExp |
| [IsProbabilityMeasure μ] |
| (ξ γ : Ω → ℓ²(ℤ, ℂ)) (hγ : StronglyMeasurable γ) |
| (hξ_ball : ∀ᵐ ω ∂μ, ‖ξ ω‖ ≤ 1) (hγ_ball : ∀ᵐ ω ∂μ, ‖γ ω‖ ≤ 1) |
| (e : ℤ ≃ ℤ) {H : Set (Ω × ℤ)} (hH : MeasurableSet H) |
| {m : MeasurableSpace Ω} (hm : m ≤ mΩ) (hξ : StronglyMeasurable[m] ξ) |
| {δ : ℝ} (hδ : 0 ≤ δ) |
| (hcond : ∀ i : ℤ, μ[({ω | (ω, i) ∈ H}).indicator (fun _ => (1 : ℝ)) | m] |
| ≤ᵐ[μ] fun _ => δ) : |
| (∫⁻ ω, ∑' i : ℤ, H.indicator |
| (fun z => ENNReal.ofReal ‖ξ z.1 z.2‖ * ENNReal.ofReal ‖γ z.1 (e z.2)‖) |
| (ω, i) ∂μ) ≤ ENNReal.ofReal (Real.sqrt δ) := by |
| let : MeasurableSpace Ω := mΩ |
| have hxcoord (i : ℤ) : StronglyMeasurable[m] (fun ω => ξ ω i) := |
| (lp.evalCLM ℂ (fun _ : ℤ => ℂ) 2 i).continuous.comp_stronglyMeasurable hξ |
| have hycoord (i : ℤ) : StronglyMeasurable (fun ω => γ ω (e i)) := |
| (lp.evalCLM ℂ (fun _ : ℤ => ℂ) 2 (e i)).continuous.comp_stronglyMeasurable hγ |
| apply lintegral_tsum_masked_mul_le_of_condExp |
| (a := fun z : Ω × ℤ => ‖ξ z.1 z.2‖) |
| (b := fun z : Ω × ℤ => ‖γ z.1 (e z.2)‖) |
| (measurable_from_prod_countable_left fun i => ((hxcoord i).mono hm).norm.measurable) |
| (measurable_from_prod_countable_left fun i => (hycoord i).norm.measurable) |
| (fun _ => norm_nonneg _) (fun _ => norm_nonneg _) hH hm hδ |
| (fun i => (hxcoord i).norm.pow 2) hcond |
| · exact lintegral_l2_sq_le_one ξ hξ_ball |
| · have h := lintegral_l2_sq_le_one γ hγ_ball |
| calc |
| _ = ∫⁻ ω, ∑' i : ℤ, ENNReal.ofReal (‖γ ω i‖ ^ 2) ∂μ := by |
| apply lintegral_congr |
| intro ω |
| exact e.tsum_eq (fun i => ENNReal.ofReal (‖γ ω i‖ ^ 2)) |
| _ ≤ 1 := h |
|
|
| theorem integral_l2_masked_tail_le_of_condExp |
| [IsProbabilityMeasure μ] |
| (ξ γ : Ω → ℓ²(ℤ, ℂ)) (hγ : StronglyMeasurable γ) |
| (hξ_ball : ∀ᵐ ω ∂μ, ‖ξ ω‖ ≤ 1) (hγ_ball : ∀ᵐ ω ∂μ, ‖γ ω‖ ≤ 1) |
| (e : ℤ ≃ ℤ) {H : Set (Ω × ℤ)} (hH : MeasurableSet H) |
| {m : MeasurableSpace Ω} (hm : m ≤ mΩ) (hξ : StronglyMeasurable[m] ξ) |
| {δ : ℝ} (hδ : 0 ≤ δ) |
| (hcond : ∀ i : ℤ, μ[({ω | (ω, i) ∈ H}).indicator (fun _ => (1 : ℝ)) | m] |
| ≤ᵐ[μ] fun _ => δ) : |
| (∀ᵐ ω ∂μ, Summable (fun i : ℤ => H.indicator |
| (fun z => ‖ξ z.1 z.2‖ * ‖γ z.1 (e z.2)‖) (ω, i))) ∧ |
| Integrable (fun ω => ∑' i : ℤ, H.indicator |
| (fun z => ‖ξ z.1 z.2‖ * ‖γ z.1 (e z.2)‖) (ω, i)) μ ∧ |
| (∫ ω, ∑' i : ℤ, H.indicator |
| (fun z => ‖ξ z.1 z.2‖ * ‖γ z.1 (e z.2)‖) (ω, i) ∂μ) ≤ Real.sqrt δ := by |
| let : MeasurableSpace Ω := mΩ |
| have hbound := lintegral_l2_masked_tail_le_of_condExp |
| (mΩ := mΩ) ξ γ hγ hξ_ball hγ_ball e hH hm hξ hδ hcond |
| have hxcoord (i : ℤ) : StronglyMeasurable (fun ω => ξ ω i) := |
| ((lp.evalCLM ℂ (fun _ : ℤ => ℂ) 2 i).continuous.comp_stronglyMeasurable hξ).mono hm |
| have hycoord (i : ℤ) : StronglyMeasurable (fun ω => γ ω (e i)) := |
| (lp.evalCLM ℂ (fun _ : ℤ => ℂ) 2 (e i)).continuous.comp_stronglyMeasurable hγ |
| have hf (i : ℤ) : Measurable (fun ω => H.indicator |
| (fun z => ‖ξ z.1 z.2‖ * ‖γ z.1 (e z.2)‖) (ω, i)) := by |
| have hs := hH.preimage (@measurable_prodMk_right Ω ℤ mΩ inferInstance i) |
| convert (((hxcoord i).norm.measurable.mul (hycoord i).norm.measurable).indicator hs) using 1 |
| funext ω |
| by_cases hi : (ω, i) ∈ H <;> simp [hi] |
| apply Invariant233.MaskedTail.real_tsum_integrable_of_lintegral_bound hf |
| (fun i ω => Set.indicator_nonneg (fun z _ => mul_nonneg (norm_nonneg _) (norm_nonneg _)) (ω, i)) |
| (Real.sqrt_nonneg δ) |
| convert hbound using 1 |
| apply lintegral_congr |
| intro ω |
| apply tsum_congr |
| intro i |
| by_cases hi : (ω, i) ∈ H |
| · simp [hi, ENNReal.ofReal_mul (norm_nonneg _)] |
| · simp [hi] |
|
|
| theorem integral_l2_masked_tail_le_of_condExp_right |
| [IsProbabilityMeasure μ] |
| (ξ γ : Ω → ℓ²(ℤ, ℂ)) (hξ : StronglyMeasurable ξ) |
| (hξ_ball : ∀ᵐ ω ∂μ, ‖ξ ω‖ ≤ 1) (hγ_ball : ∀ᵐ ω ∂μ, ‖γ ω‖ ≤ 1) |
| (e : ℤ ≃ ℤ) {H : Set (Ω × ℤ)} (hH : MeasurableSet H) |
| {m : MeasurableSpace Ω} (hm : m ≤ mΩ) (hγ : StronglyMeasurable[m] γ) |
| {δ : ℝ} (hδ : 0 ≤ δ) |
| (hcond : ∀ i : ℤ, μ[({ω | (ω, i) ∈ H}).indicator (fun _ => (1 : ℝ)) | m] |
| ≤ᵐ[μ] fun _ => δ) : |
| Integrable (fun ω => ∑' i : ℤ, H.indicator |
| (fun z => ‖ξ z.1 z.2‖ * ‖γ z.1 (e z.2)‖) (ω, i)) μ ∧ |
| (∫ ω, ∑' i : ℤ, H.indicator |
| (fun z => ‖ξ z.1 z.2‖ * ‖γ z.1 (e z.2)‖) (ω, i) ∂μ) ≤ Real.sqrt δ := by |
| let : MeasurableSpace Ω := mΩ |
| let H' : Set (Ω × ℤ) := {z | (z.1, e.symm z.2) ∈ H} |
| have hH' : MeasurableSet H' := |
| hH.preimage (measurable_fst.prodMk ((measurable_of_countable e.symm).comp measurable_snd)) |
| have hc' : ∀ j : ℤ, μ[({ω | (ω, j) ∈ H'}).indicator (fun _ => (1 : ℝ)) | m] |
| ≤ᵐ[μ] fun _ => δ := fun j => hcond (e.symm j) |
| have h := integral_l2_masked_tail_le_of_condExp (mΩ := mΩ) |
| γ ξ hξ hγ_ball hξ_ball e.symm hH' hm hγ hδ hc' |
| have heq : (fun ω => ∑' i : ℤ, H.indicator |
| (fun z => ‖ξ z.1 z.2‖ * ‖γ z.1 (e z.2)‖) (ω, i)) = |
| (fun ω => ∑' j : ℤ, H'.indicator |
| (fun z => ‖γ z.1 z.2‖ * ‖ξ z.1 (e.symm z.2)‖) (ω, j)) := by |
| funext ω |
| rw [← e.tsum_eq (fun j => H'.indicator |
| (fun z => ‖γ z.1 z.2‖ * ‖ξ z.1 (e.symm z.2)‖) (ω, j))] |
| apply tsum_congr |
| intro i |
| by_cases hi : (ω, i) ∈ H <;> simp [H', hi, mul_comm] |
| rw [heq] |
| exact h.2 |
|
|
| end InvariantProjection |
|
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| end OAI |
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