| import OAI.Probability.InvariantIsing.Spectral.MeasureInverseContinuity |
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| namespace OAI |
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| /-! Compact-support estimates for the general spectral transform. -/ |
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| noncomputable section |
| open MeasureTheory Filter Set |
| open scoped Topology Classical |
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| namespace InvariantIsing |
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| lemma integrable_spectralCoordinate (μ : Measure ℝ) [IsProbabilityMeasure μ] |
| {l e : ℝ} (hμ : ∀ᵐ y ∂μ, y ∈ Icc l e) : Integrable (fun y => y) μ := by |
| apply (integrable_const (|l| + |e|)).mono' measurable_id.aestronglyMeasurable |
| filter_upwards [hμ] with y hy |
| rw [Real.norm_eq_abs, abs_le] |
| simp only [id_eq] |
| constructor |
| · linarith [neg_abs_le l, abs_nonneg e, hy.1] |
| · linarith [le_abs_self e, abs_nonneg l, hy.2] |
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| theorem measureR_bounds_all (μ : Measure ℝ) [IsProbabilityMeasure μ] |
| {l e : ℝ} (hμ : ∀ᵐ y ∂μ, y ∈ Icc l e) (x : ℝ) : |
| l ≤ measureR μ e x ∧ measureR μ e x ≤ e := by |
| by_cases hx : 0 < x |
| · exact measureR_bounds μ hμ hx |
| · rw [measureR, ite_eq_right hx] |
| constructor |
| · simpa only [integral_const, probReal_univ, one_smul] using |
| integral_mono_ae (integrable_const l) (integrable_spectralCoordinate μ hμ) |
| (hμ.mono fun _ h => h.1) |
| · simpa only [integral_const, probReal_univ, one_smul] using |
| integral_mono_ae (integrable_spectralCoordinate μ hμ) (integrable_const e) |
| (hμ.mono fun _ h => h.2) |
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| theorem measureInverse_uncapped_small (μ : Measure ℝ) [IsProbabilityMeasure μ] |
| {K e x : ℝ} (hK : 0 < K) (he : e ≤ K) |
| (hμ : ∀ᵐ y ∂μ, y ∈ Icc (-K) e) (hx : 0 < x) (hsmall : 4 * K * x ≤ 1) : |
| e < measureInverse μ e x ∧ measureResolvent μ (measureInverse μ e x) = x := by |
| have htest : e < 1 / x - K := by |
| have h := (le_div_iff₀ hx).mpr hsmall |
| nlinarith |
| have hl := le_measureResolvent μ hμ htest |
| have hvalue : x ≤ measureResolvent μ (1 / x - K) := by |
| simpa only [sub_neg_eq_add, sub_add_cancel, one_div_one_div] using hl |
| have hex := exists_measureResolvent_solution μ (hμ.mono fun _ h => h.2) htest hx hvalue |
| obtain ⟨b, hb, hsol⟩ := hex |
| rw [measureInverse_eq_of_solution μ (hμ.mono fun _ h => h.2) hb hsol] |
| exact ⟨hb, hsol⟩ |
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| theorem measureR_sub_mean_identity (μ : Measure ℝ) [IsProbabilityMeasure μ] |
| {l e x : ℝ} (hμ : ∀ᵐ y ∂μ, y ∈ Icc l e) (hx : 0 < x) |
| (hs : e < measureInverse μ e x ∧ measureResolvent μ (measureInverse μ e x) = x) : |
| measureR μ e x - ∫ y, y ∂μ = |
| ∫ y, (measureR μ e x - y) ^ 2 / (measureInverse μ e x - y) ∂μ := by |
| let b := measureInverse μ e x |
| have hi := integrable_spectralCoordinate μ hμ |
| have hk : Integrable (fun y => 1 / (b - y)) μ := |
| integrable_resolventKernel μ (hμ.mono fun _ h => h.2) hs.1 |
| have ha : ∀ᵐ y ∂μ, (measureR μ e x - y) ^ 2 / (b - y) = |
| (b - y) - 2 / x + (1 / x ^ 2) * (1 / (b - y)) := by |
| filter_upwards [hμ] with y hy |
| have hby : b - y ≠ 0 := by dsimp only [b]; linarith [hs.1, hy.2] |
| simp only [measureR, ite_eq_left hx] |
| change (b - 1 / x - y) ^ 2 / (b - y) = _ |
| field_simp [hx.ne', hby] |
| ring |
| have hibase : Integrable (fun y => b - y) μ := (integrable_const b).sub hi |
| have hib : Integrable (fun y => b - y - 2 / x) μ := |
| hibase.sub (integrable_const (2 / x)) |
| have hir : Integrable (fun y => (1 / x ^ 2) * (1 / (b - y))) μ := hk.const_mul _ |
| have hsplit : (∫ y, b - y - 2 / x ∂μ) = b - (∫ y, y ∂μ) - 2 / x := by |
| calc |
| _ = (∫ y, b - y ∂μ) - (∫ _, 2 / x ∂μ) := |
| integral_sub hibase (integrable_const (2 / x)) |
| _ = _ := by |
| rw [show (∫ y, b - y ∂μ) = (∫ _, b ∂μ) - (∫ y, y ∂μ) from |
| integral_sub (integrable_const b) hi] |
| simp |
| rw [integral_congr_ae ha, integral_add hib hir, hsplit, integral_const_mul] |
| change measureR μ e x - (∫ y, y ∂μ) = |
| b - (∫ y, y ∂μ) - 2 / x + (1 / x ^ 2) * measureResolvent μ b |
| rw [hs.2] |
| simp only [measureR, ite_eq_left hx] |
| change b - 1 / x - (∫ y, y ∂μ) = _ |
| field_simp |
| ring |
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| theorem measureR_sub_mean_bound (μ : Measure ℝ) [IsProbabilityMeasure μ] |
| {K e x : ℝ} (hK : 0 < K) (he : e ≤ K) |
| (hμ : ∀ᵐ y ∂μ, y ∈ Icc (-K) e) (hx : 0 < x) (hsmall : 4 * K * x ≤ 1) : |
| 0 ≤ measureR μ e x - ∫ y, y ∂μ ∧ |
| measureR μ e x - ∫ y, y ∂μ ≤ 8 * K ^ 2 * x := by |
| let b := measureInverse μ e x |
| let r := measureR μ e x |
| have hs := measureInverse_uncapped_small μ hK he hμ hx hsmall |
| have hr := measureR_bounds μ hμ hx |
| have hbr : x * b = x * r + 1 := by |
| dsimp only [b, r] |
| simp only [measureR, ite_eq_left hx] |
| field_simp |
| ring |
| have hb : ∀ᵐ y ∂μ, 0 ≤ (r - y) ^ 2 / (b - y) ∧ |
| (r - y) ^ 2 / (b - y) ≤ 8 * K ^ 2 * x := by |
| filter_upwards [hμ] with y hy |
| have hden : 0 < b - y := by dsimp only [b]; linarith [hs.1, hy.2] |
| have hlo : -2 * K ≤ r - y := by dsimp only [r]; linarith [hr.1, hy.2] |
| have hhi : r - y ≤ 2 * K := by dsimp only [r]; linarith [hr.2, hy.1] |
| have hsq : (r - y) ^ 2 ≤ 4 * K ^ 2 := by |
| nlinarith [mul_nonneg (sub_nonneg.mpr hlo) (sub_nonneg.mpr hhi)] |
| have hinv : 1 / (b - y) ≤ 2 * x := by |
| apply (div_le_iff₀ hden).mpr |
| have := mul_le_mul_of_nonneg_left hlo hx.le |
| nlinarith |
| refine ⟨div_nonneg (sq_nonneg _) hden.le, ?_⟩ |
| calc |
| (r - y) ^ 2 / (b - y) = (r - y) ^ 2 * (1 / (b - y)) := by ring |
| _ ≤ (4 * K ^ 2) * (2 * x) := mul_le_mul hsq hinv (one_div_pos.mpr hden).le (by positivity) |
| _ = 8 * K ^ 2 * x := by ring |
| have hm : Measurable (fun y : ℝ => (r - y) ^ 2 / (b - y)) := |
| ((measurable_const.sub measurable_id).pow_const 2).div (measurable_const.sub measurable_id) |
| have hi : Integrable (fun y => (r - y) ^ 2 / (b - y)) μ := by |
| apply (integrable_const (8 * K ^ 2 * x)).mono' hm.aestronglyMeasurable |
| filter_upwards [hb] with y hy |
| simpa only [Real.norm_eq_abs, abs_of_nonneg hy.1] using hy.2 |
| rw [measureR_sub_mean_identity μ hμ hx hs] |
| constructor |
| · exact integral_nonneg_of_ae (hb.mono fun _ h => h.1) |
| · have hu := integral_mono_ae hi (integrable_const (8 * K ^ 2 * x)) (hb.mono fun _ h => h.2) |
| simpa only [integral_const, probReal_univ, one_smul] using hu |
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| theorem continuousAt_measureR_zero (μ : Measure ℝ) [IsProbabilityMeasure μ] |
| {K e : ℝ} (hK : 0 < K) (he : e ≤ K) (hμ : ∀ᵐ y ∂μ, y ∈ Icc (-K) e) : |
| ContinuousAt (measureR μ e) 0 := by |
| have herr : ∀ᶠ x : ℝ in 𝓝 0, |
| ‖measureR μ e x - ∫ y, y ∂μ‖ ≤ 8 * K ^ 2 * |x| := by |
| filter_upwards [Iio_mem_nhds (show (0 : ℝ) < 1 / (4 * K) by positivity)] with x hxsmall |
| by_cases hx : 0 < x |
| · have hsmall : 4 * K * x ≤ 1 := by |
| have := (lt_div_iff₀ (show (0 : ℝ) < 4 * K by positivity)).mp hxsmall |
| nlinarith |
| have hb := measureR_sub_mean_bound μ hK he hμ hx hsmall |
| simpa only [Real.norm_eq_abs, abs_of_nonneg hb.1, abs_of_pos hx] using hb.2 |
| · simp only [measureR, ite_eq_right hx, sub_self, norm_zero] |
| positivity |
| have ht : Tendsto (fun x : ℝ => 8 * K ^ 2 * |x|) (𝓝 0) (𝓝 0) := by |
| simpa using (continuous_abs.tendsto (0 : ℝ)).const_mul (8 * K ^ 2) |
| have hl := (tendsto_sub_nhds_zero_iff).mp (squeeze_zero_norm' herr ht) |
| simpa only [ContinuousAt, measureR, lt_self_iff_false, ite_false] using hl |
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| theorem continuous_measureR (μ : Measure ℝ) [IsProbabilityMeasure μ] |
| {K e : ℝ} (hK : 0 < K) (he : e ≤ K) (hμ : ∀ᵐ y ∂μ, y ∈ Icc (-K) e) : |
| Continuous (measureR μ e) := by |
| apply continuous_iff_continuousAt.mpr |
| intro x |
| rcases lt_trichotomy x 0 with hx | hx | hx |
| · have hc : ContinuousAt (fun _ : ℝ => ∫ y, y ∂μ) x := continuousAt_const |
| apply hc.congr_of_eventuallyEq |
| filter_upwards [Iio_mem_nhds hx] with y hy |
| simp only [measureR, ite_eq_right (not_lt.mpr (show y < 0 from hy).le)] |
| · subst x |
| exact continuousAt_measureR_zero μ hK he hμ |
| · exact continuousAt_measureR_pos μ (hμ.mono fun _ h => h.2) hx |
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| end InvariantIsing |
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| end |
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| end OAI |
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