| import OAI.Probability.GaussianReplacement.Basic |
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| namespace OAI |
|
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| open MeasureTheory ProbabilityTheory Set Filter |
| open scoped ENNReal NNReal RealInnerProductSpace Topology |
| noncomputable section |
| namespace CurrentProjection |
|
|
| section InverseMoments |
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|
| variable {X : Type*} [MeasurableSpace X] |
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|
| omit [MeasurableSpace X] in |
| lemma inverse_pow_dyadic_majorant (f : X → ℝ) (m : ℕ) (hf : ∀ x, 0 ≤ f x) (x : X) : |
| ENNReal.ofReal ((f x)⁻¹ ^ m) ≤ |
| 1 + ∑' k : ℕ, ({x | f x ≤ (1 / 2 : ℝ) ^ k}).indicator |
| (fun _ => ENNReal.ofReal ((2 : ℝ) ^ (m * (k + 1)))) x := by |
| classical |
| by_cases hzero : f x = 0 |
| · simp only [hzero, inv_zero] |
| exact le_add_of_le_left (by cases m <;> simp) |
| have hxpos : 0 < f x := lt_of_le_of_ne (hf x) (Ne.symm hzero) |
| by_cases hxone : 1 ≤ f x |
| · apply le_add_of_le_left |
| have h : (f x)⁻¹ ^ m ≤ 1 := |
| pow_le_one₀ (inv_nonneg.mpr (hf x)) (inv_le_one_of_one_le₀ hxone) |
| simpa using ENNReal.ofReal_le_ofReal h |
| obtain ⟨k, hklo, hkhi⟩ := exists_nat_pow_near_of_lt_one hxpos |
| (le_of_not_ge hxone) (by norm_num : (0 : ℝ) < 1 / 2) (by norm_num : (1 / 2 : ℝ) < 1) |
| have hinv : (f x)⁻¹ ≤ (2 : ℝ) ^ (k + 1) := by |
| have h := (inv_le_inv₀ hxpos (pow_pos (by norm_num : (0 : ℝ) < 1 / 2) _)).2 hklo.le |
| simpa using h |
| have hpow : ENNReal.ofReal ((f x)⁻¹ ^ m) ≤ |
| ENNReal.ofReal ((2 : ℝ) ^ (m * (k + 1))) := by |
| apply ENNReal.ofReal_le_ofReal |
| simpa [← pow_mul, Nat.mul_comm] using pow_le_pow_left₀ (inv_nonneg.mpr (hf x)) hinv m |
| calc |
| ENNReal.ofReal ((f x)⁻¹ ^ m) ≤ ENNReal.ofReal ((2 : ℝ) ^ (m * (k + 1))) := hpow |
| _ = ({x | f x ≤ (1 / 2 : ℝ) ^ k}).indicator |
| (fun _ => ENNReal.ofReal ((2 : ℝ) ^ (m * (k + 1)))) x := by |
| rw [Set.indicator_of_mem (show x ∈ {x | f x ≤ (1 / 2 : ℝ) ^ k} from hkhi)] |
| _ ≤ ∑' j : ℕ, ({x | f x ≤ (1 / 2 : ℝ) ^ j}).indicator |
| (fun _ => ENNReal.ofReal ((2 : ℝ) ^ (m * (j + 1)))) x := |
| ENNReal.le_tsum (f := fun j => ({x | f x ≤ (1 / 2 : ℝ) ^ j}).indicator |
| (fun _ => ENNReal.ofReal ((2 : ℝ) ^ (m * (j + 1)))) x) k |
| _ ≤ 1 + ∑' j : ℕ, ({x | f x ≤ (1 / 2 : ℝ) ^ j}).indicator |
| (fun _ => ENNReal.ofReal ((2 : ℝ) ^ (m * (j + 1)))) x := le_add_left le_rfl |
|
|
| lemma dyadic_power_cancel (m k : ℕ) : |
| (2 : ℝ) ^ (m * (k + 1)) * ((1 / 2 : ℝ) ^ k) ^ (m + 1) = |
| (2 : ℝ) ^ m * (1 / 2 : ℝ) ^ k := by |
| rw [Nat.mul_add, mul_one, pow_add, pow_succ] |
| have h : (2 : ℝ) ^ (m * k) * ((1 / 2 : ℝ) ^ k) ^ m = 1 := by |
| rw [Nat.mul_comm m k, pow_mul, ← mul_pow, ← mul_pow] |
| norm_num |
| calc |
| 2 ^ (m * k) * 2 ^ m * (((1 / 2 : ℝ) ^ k) ^ m * (1 / 2 : ℝ) ^ k) = |
| (2 ^ (m * k) * ((1 / 2 : ℝ) ^ k) ^ m) * (2 ^ m * (1 / 2 : ℝ) ^ k) := by ring |
| _ = _ := by rw [h, one_mul] |
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|
| theorem inverse_moment_of_tube_bound (μ : Measure X) (f : X → ℝ) (m : ℕ) |
| (hf : Measurable f) (hnonneg : ∀ x, 0 ≤ f x) (C : ℝ≥0∞) |
| (htube : ∀ r : ℝ, 0 < r → r ≤ 1 → |
| μ {x | f x ≤ r} ≤ C * ENNReal.ofReal (r ^ (m + 1))) : |
| ∫⁻ x, ENNReal.ofReal ((f x)⁻¹ ^ m) ∂μ ≤ |
| μ Set.univ + C * (2 : ℝ≥0∞) ^ (m + 1) := by |
| classical |
| let g : ℕ → X → ℝ≥0∞ := fun k => |
| ({x | f x ≤ (1 / 2 : ℝ) ^ k}).indicator |
| (fun _ => ENNReal.ofReal ((2 : ℝ) ^ (m * (k + 1)))) |
| have hg : ∀ k, Measurable (g k) := fun k => |
| measurable_const.indicator (measurableSet_le hf measurable_const) |
| calc |
| ∫⁻ x, ENNReal.ofReal ((f x)⁻¹ ^ m) ∂μ ≤ ∫⁻ x, 1 + ∑' k, g k x ∂μ := |
| lintegral_mono (inverse_pow_dyadic_majorant f m hnonneg) |
| _ = μ Set.univ + ∑' k, ∫⁻ x, g k x ∂μ := by |
| rw [lintegral_add_left measurable_const, lintegral_const, one_mul, lintegral_tsum (fun k => (hg k).aemeasurable)] |
| _ ≤ μ Set.univ + ∑' k, C * ENNReal.ofReal ((2 : ℝ) ^ m * (1 / 2 : ℝ) ^ k) := by |
| apply add_le_add le_rfl |
| apply ENNReal.tsum_le_tsum |
| intro k |
| dsimp only [g] |
| rw [lintegral_indicator (measurableSet_le hf measurable_const), lintegral_const, |
| Measure.restrict_apply_univ] |
| calc |
| ENNReal.ofReal ((2 : ℝ) ^ (m * (k + 1))) * μ {x | f x ≤ (1 / 2 : ℝ) ^ k} ≤ |
| ENNReal.ofReal ((2 : ℝ) ^ (m * (k + 1))) * |
| (C * ENNReal.ofReal (((1 / 2 : ℝ) ^ k) ^ (m + 1))) := by |
| gcongr |
| exact htube _ (by positivity) (pow_le_one₀ (by norm_num) (by norm_num)) |
| _ = _ := by |
| rw [← mul_assoc, mul_comm _ C, mul_assoc, ← ENNReal.ofReal_mul (by positivity), |
| dyadic_power_cancel] |
| _ = μ Set.univ + C * (2 : ℝ≥0∞) ^ (m + 1) := by |
| simp_rw [ENNReal.ofReal_mul (by positivity : (0 : ℝ) ≤ 2 ^ m), |
| ENNReal.ofReal_pow (by norm_num : (0 : ℝ) ≤ 1 / 2), |
| ENNReal.ofReal_pow (by norm_num : (0 : ℝ) ≤ 2)] |
| rw [ENNReal.tsum_mul_left, ENNReal.tsum_mul_left, ENNReal.tsum_geometric] |
| norm_num [pow_succ, mul_assoc, ENNReal.ofReal_div_of_pos] |
|
|
| end InverseMoments |
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|
| section FrostmanGeometry |
|
|
| variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] |
|
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|
| def HasLocalMassBound (ν : Measure E) (a : ℝ) : Prop := |
| ∀ t ∈ ν.support, ∀ r : ℝ, 0 < r → |
| ν (Metric.closedBall t r) ≤ ENNReal.ofReal (r ^ a) |
|
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| omit [BorelSpace E] in |
|
|
| theorem HasLocalMassBound.any_center {ν : Measure E} {a : ℝ} |
| (hν : HasLocalMassBound ν a) (z : E) {r : ℝ} (hr : 0 < r) : |
| ν (Metric.closedBall z r) ≤ ENNReal.ofReal ((2 * r) ^ a) := by |
| by_cases hmass : ν (Metric.closedBall z r) = 0 |
| · simp [hmass] |
| obtain ⟨t, htball, htsupp⟩ := ν.nonempty_inter_support_of_pos (pos_iff_ne_zero.mpr hmass) |
| refine (measure_mono ?_).trans (hν t htsupp (2 * r) (by positivity)) |
| intro x hx |
| apply Metric.mem_closedBall.mpr |
| calc |
| dist x t ≤ dist x z + dist z t := dist_triangle _ _ _ |
| _ ≤ r + r := add_le_add (Metric.mem_closedBall.mp hx) |
| (by simpa [dist_comm] using Metric.mem_closedBall.mp htball) |
| _ = 2 * r := by ring |
|
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|
| theorem card_le_volume_ratio (s : Finset E) {r R : ℝ} (hr : 0 < r) (hR : 0 ≤ R) |
| (hs : ∀ x ∈ s, ‖x‖ ≤ R) |
| (hsep : ∀ x ∈ s, ∀ y ∈ s, x ≠ y → r ≤ dist x y) : |
| (s.card : ℝ) ≤ ((R + r / 2) / (r / 2)) ^ Module.finrank ℝ E := by |
| classical |
| let ν : Measure E := Measure.addHaar |
| let δ : ℝ := r / 2 |
| let ρ : ℝ := R + r / 2 |
| have hδ : 0 < δ := by dsimp [δ]; positivity |
| have hρ : 0 < ρ := by dsimp [ρ]; positivity |
| let A : Set E := ⋃ c ∈ s, Metric.ball c δ |
| have hd : Set.Pairwise (s : Set E) (fun c d => Disjoint (Metric.ball c δ) (Metric.ball d δ)) := by |
| intro c hc d hd hcd |
| apply Metric.ball_disjoint_ball |
| simpa [δ] using hsep c hc d hd hcd |
| have hsub : A ⊆ Metric.ball (0 : E) ρ := by |
| refine iUnion₂_subset fun x hx => ?_ |
| apply Metric.ball_subset_ball' |
| dsimp [δ, ρ] |
| rw [dist_zero_right] |
| linarith [hs x hx] |
| have I : |
| (s.card : ℝ≥0∞) * ENNReal.ofReal (δ ^ Module.finrank ℝ E) * ν (Metric.ball 0 1) ≤ |
| ENNReal.ofReal (ρ ^ Module.finrank ℝ E) * ν (Metric.ball 0 1) := by |
| calc |
| (s.card : ℝ≥0∞) * ENNReal.ofReal (δ ^ Module.finrank ℝ E) * ν (Metric.ball 0 1) = ν A := by |
| dsimp only [A] |
| rw [measure_biUnion_finset hd (fun _ _ => Metric.isOpen_ball.measurableSet)] |
| simp only [ν.addHaar_ball_of_pos _ hδ, Finset.sum_const, nsmul_eq_mul, mul_assoc] |
| _ ≤ ν (Metric.ball (0 : E) ρ) := measure_mono hsub |
| _ = _ := ν.addHaar_ball_of_pos 0 hρ |
| have J : (s.card : ℝ≥0∞) * ENNReal.ofReal (δ ^ Module.finrank ℝ E) ≤ |
| ENNReal.ofReal (ρ ^ Module.finrank ℝ E) := |
| (ENNReal.mul_le_mul_iff_left (Metric.measure_ball_pos ν 0 zero_lt_one).ne' |
| measure_ball_lt_top.ne).mp I |
| have K : (s.card : ℝ) * δ ^ Module.finrank ℝ E ≤ ρ ^ Module.finrank ℝ E := by |
| simpa only [ENNReal.toReal_mul, ENNReal.toReal_natCast, |
| ENNReal.toReal_ofReal (pow_nonneg hδ.le _)] using |
| ENNReal.toReal_le_of_le_ofReal (pow_nonneg hρ.le _) J |
| rw [div_pow] |
| exact (le_div_iff₀ (pow_pos hδ _)).mpr K |
|
|
| end FrostmanGeometry |
|
|
| section FiniteNets |
|
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| variable {E : Type*} [PseudoMetricSpace E] |
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| theorem exists_net_of_finite_packing_bound (A : Set E) {r : ℝ} (hr : 0 ≤ r) (N : ℕ) |
| (hcard : ∀ s : Finset E, (∀ x ∈ s, x ∈ A) → |
| (∀ x ∈ s, ∀ y ∈ s, x ≠ y → r < dist x y) → s.card ≤ N) : |
| ∃ s : Finset E, (∀ x ∈ s, x ∈ A) ∧ |
| (∀ x ∈ s, ∀ y ∈ s, x ≠ y → r < dist x y) ∧ |
| ∀ x ∈ A, ∃ y ∈ s, dist x y ≤ r := by |
| classical |
| let S : Set ℕ := {n | ∃ s : Finset E, (∀ x ∈ s, x ∈ A) ∧ |
| (∀ x ∈ s, ∀ y ∈ s, x ≠ y → r < dist x y) ∧ s.card = n} |
| have hS : S.Finite := (Set.finite_Iic N).subset (by |
| rintro n ⟨s, hs, hsep, rfl⟩ |
| exact hcard s hs hsep) |
| have hS0 : S.Nonempty := ⟨0, ∅, by simp⟩ |
| obtain ⟨n, hn, hmax⟩ := Set.exists_max_image S id hS hS0 |
| obtain ⟨s, hs, hsep, rfl⟩ := hn |
| refine ⟨s, hs, hsep, ?_⟩ |
| intro x hx |
| by_contra! hnot |
| have hxs : x ∉ s := by |
| intro h |
| have := hnot x h |
| simpa using lt_of_le_of_lt hr this |
| have hins : ∀ y ∈ insert x s, y ∈ A := by |
| intro y hy |
| rcases Finset.mem_insert.mp hy with rfl | hy |
| · exact hx |
| · exact hs y hy |
| have hsep' : ∀ y ∈ insert x s, ∀ z ∈ insert x s, y ≠ z → r < dist y z := by |
| intro y hy z hz hne |
| rcases Finset.mem_insert.mp hy with hxy | hys |
| · subst y |
| rcases Finset.mem_insert.mp hz with hxz | hzs |
| · exact (hne hxz.symm).elim |
| · exact hnot z hzs |
| · rcases Finset.mem_insert.mp hz with hxz | hzs |
| · subst z |
| simpa [dist_comm] using hnot y hys |
| · exact hsep y hys z hzs hne |
| have hinS : (insert x s).card ∈ S := ⟨insert x s, hins, hsep', rfl⟩ |
| have hh := hmax _ hinS |
| simp only [id_eq, Finset.card_insert_of_notMem hxs] at hh |
| omega |
|
|
| end FiniteNets |
|
|
| section EuclideanNets |
|
|
| variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] |
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| theorem exists_unit_ball_net {r : ℝ} (hr : 0 < r) (hr1 : r ≤ 1) : |
| ∃ s : Finset E, (∀ x ∈ s, ‖x‖ ≤ 1) ∧ |
| (s.card : ℝ) ≤ (3 / r) ^ Module.finrank ℝ E ∧ |
| ∀ x : E, ‖x‖ ≤ 1 → ∃ y ∈ s, dist x y ≤ r := by |
| classical |
| have hbound : ∀ s : Finset E, (∀ x ∈ s, ‖x‖ ≤ 1) → |
| (∀ x ∈ s, ∀ y ∈ s, x ≠ y → r < dist x y) → |
| (s.card : ℝ) ≤ (3 / r) ^ Module.finrank ℝ E := by |
| intro s hs hsep |
| refine (card_le_volume_ratio s hr zero_le_one hs |
| (fun x hx y hy hne => (hsep x hx y hy hne).le)).trans ?_ |
| apply pow_le_pow_left₀ (by positivity) |
| apply (div_le_div_iff₀ (by positivity : 0 < r / 2) hr).mpr |
| nlinarith |
| obtain ⟨s, hs, hsep, hnet⟩ := exists_net_of_finite_packing_bound |
| {x : E | ‖x‖ ≤ 1} hr.le ⌈(3 / r) ^ Module.finrank ℝ E⌉₊ |
| (by |
| intro s hs hsep |
| exact_mod_cast (hbound s hs hsep).trans (Nat.le_ceil _)) |
| exact ⟨s, hs, hbound s hs hsep, hnet⟩ |
|
|
| end EuclideanNets |
|
|
| section AffineTubes |
|
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| variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] |
| [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] |
|
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| open EuclideanGeometry |
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| theorem exists_affine_tube_cover (L : AffineSubspace ℝ E) [Nonempty L] |
| {r : ℝ} (hr : 0 < r) (hr1 : r ≤ 1) : |
| ∃ s : Finset L.direction, |
| (s.card : ℝ) ≤ (3 / r) ^ Module.finrank ℝ L.direction ∧ |
| ∀ t : E, ‖t‖ ≤ 1 → Metric.infDist t L ≤ r → |
| ∃ y ∈ s, dist t ((y : E) + (orthogonalProjection L 0 : E)) ≤ 2 * r := by |
| classical |
| obtain ⟨s, _, hcard, hnet⟩ := exists_unit_ball_net (E := L.direction) hr hr1 |
| refine ⟨s, hcard, ?_⟩ |
| intro t ht hdist |
| obtain ⟨y, hy, hty⟩ := hnet (L.direction.orthogonalProjectionOnto t) |
| ((L.direction.norm_orthogonalProjectionOnto_apply_le t).trans ht) |
| refine ⟨y, hy, ?_⟩ |
| have hproj : (orthogonalProjection L t : E) = |
| (L.direction.orthogonalProjectionOnto t : E) + (orthogonalProjection L 0 : E) := by |
| have h := (orthogonalProjection L).map_vadd (0 : E) t |
| simpa only [vadd_eq_add, add_zero, orthogonalProjection_contLinear, |
| AffineSubspace.coe_vadd] using congrArg (fun x : L => (x : E)) h |
| have hpdist : dist (orthogonalProjection L t : E) |
| ((y : E) + (orthogonalProjection L 0 : E)) ≤ r := by |
| rw [hproj, dist_add_right] |
| exact hty |
| have hd : dist t (orthogonalProjection L t : E) ≤ r := by |
| rwa [dist_orthogonalProjection_eq_infDist] |
| calc |
| dist t ((y : E) + (orthogonalProjection L 0 : E)) ≤ |
| dist t (orthogonalProjection L t : E) + |
| dist (orthogonalProjection L t : E) ((y : E) + (orthogonalProjection L 0 : E)) := |
| dist_triangle _ _ _ |
| _ ≤ r + r := add_le_add hd hpdist |
| _ = 2 * r := by ring |
|
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|
|
| theorem HasLocalMassBound.affine_tube {ν : Measure E} {a : ℝ} |
| (hν : HasLocalMassBound ν a) (hsupport : ν.support ⊆ Metric.closedBall 0 1) |
| (L : AffineSubspace ℝ E) [Nonempty L] {r : ℝ} (hr : 0 < r) (hr1 : r ≤ 1) : |
| ν {t | Metric.infDist t L ≤ r} ≤ |
| ENNReal.ofReal ((3 / r) ^ Module.finrank ℝ L.direction * (4 * r) ^ a) := by |
| classical |
| obtain ⟨s, hcard, hcover⟩ := exists_affine_tube_cover L hr hr1 |
| let c : L.direction → E := fun y => (y : E) + (orthogonalProjection L 0 : E) |
| have hsub : {t | Metric.infDist t L ≤ r} ≤ᵐ[ν] |
| ⋃ y ∈ s, Metric.closedBall (c y) (2 * r) := by |
| filter_upwards [ν.support_mem_ae] with t ht |
| intro hdist |
| have htunit : ‖t‖ ≤ 1 := by simpa using hsupport ht |
| obtain ⟨y, hy, hty⟩ := hcover t htunit hdist |
| exact mem_iUnion₂.mpr ⟨y, hy, hty⟩ |
| calc |
| ν {t | Metric.infDist t L ≤ r} ≤ ν (⋃ y ∈ s, Metric.closedBall (c y) (2 * r)) := |
| measure_mono_ae hsub |
| _ ≤ ∑ y ∈ s, ν (Metric.closedBall (c y) (2 * r)) := measure_biUnion_finset_le _ _ |
| _ ≤ ∑ _y ∈ s, ENNReal.ofReal ((4 * r) ^ a) := by |
| refine Finset.sum_le_sum fun y _ => ?_ |
| simpa only [show (2 : ℝ) * (2 * r) = 4 * r by ring] using |
| hν.any_center (c y) (r := 2 * r) (by positivity) |
| _ = (s.card : ℝ≥0∞) * ENNReal.ofReal ((4 * r) ^ a) := by simp |
| _ ≤ ENNReal.ofReal ((3 / r) ^ Module.finrank ℝ L.direction) * |
| ENNReal.ofReal ((4 * r) ^ a) := by |
| gcongr |
| exact_mod_cast ENNReal.ofReal_le_ofReal hcard |
| _ = _ := (ENNReal.ofReal_mul (by positivity)).symm |
|
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|
|
| lemma tube_radius_factors (ℓ : ℕ) (a : ℝ) {r : ℝ} (hr : 0 < r) : |
| (3 / r) ^ ℓ * (4 * r) ^ a = |
| (3 : ℝ) ^ ℓ * (4 : ℝ) ^ a * r ^ (a - ℓ) := by |
| rw [div_pow, Real.mul_rpow (by norm_num) hr.le, |
| Real.rpow_sub_natCast hr.ne'] |
| ring |
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|
|
| theorem HasLocalMassBound.affine_tube_pow {ν : Measure E} {a : ℝ} |
| (hν : HasLocalMassBound ν a) (hsupport : ν.support ⊆ Metric.closedBall 0 1) |
| (L : AffineSubspace ℝ E) [Nonempty L] (k : ℕ) |
| (hmargin : (Module.finrank ℝ L.direction : ℝ) + k ≤ a) |
| {r : ℝ} (hr : 0 < r) (hr1 : r ≤ 1) : |
| ν {t | Metric.infDist t L ≤ r} ≤ |
| ENNReal.ofReal ((3 : ℝ) ^ Module.finrank ℝ L.direction * (4 : ℝ) ^ a) * |
| ENNReal.ofReal (r ^ k) := by |
| refine (hν.affine_tube hsupport L hr hr1).trans ?_ |
| rw [tube_radius_factors _ _ hr, ← ENNReal.ofReal_mul (by positivity)] |
| apply ENNReal.ofReal_le_ofReal |
| apply mul_le_mul_of_nonneg_left _ (by positivity) |
| rw [← Real.rpow_natCast] |
| exact Real.rpow_le_rpow_of_exponent_ge hr hr1 (by linarith) |
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|
| theorem HasLocalMassBound.affine_inverse_moment {ν : Measure E} {a : ℝ} |
| (hν : HasLocalMassBound ν a) (hsupport : ν.support ⊆ Metric.closedBall 0 1) |
| (L : AffineSubspace ℝ E) [Nonempty L] (m : ℕ) |
| (hmargin : (Module.finrank ℝ L.direction : ℝ) + (m + 1 : ℕ) ≤ a) : |
| ∫⁻ t, ENNReal.ofReal ((Metric.infDist t L)⁻¹ ^ m) ∂ν ≤ |
| ν Set.univ + |
| ENNReal.ofReal ((3 : ℝ) ^ Module.finrank ℝ L.direction * (4 : ℝ) ^ a) * |
| (2 : ℝ≥0∞) ^ (m + 1) := by |
| apply inverse_moment_of_tube_bound ν _ m (Metric.continuous_infDist_pt _).measurable |
| (fun _ => Metric.infDist_nonneg) |
| intro r hr hr1 |
| exact hν.affine_tube_pow hsupport L (m + 1) hmargin hr hr1 |
|
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|
|
| theorem HasLocalMassBound.affine_null {ν : Measure E} {a : ℝ} |
| (hν : HasLocalMassBound ν a) (hsupport : ν.support ⊆ Metric.closedBall 0 1) |
| (L : AffineSubspace ℝ E) [Nonempty L] |
| (hmargin : (Module.finrank ℝ L.direction : ℝ) + 1 ≤ a) : ν L = 0 := by |
| let C : ℝ := (3 : ℝ) ^ Module.finrank ℝ L.direction * (4 : ℝ) ^ a |
| have hC : 0 ≤ C := by dsimp [C]; positivity |
| have hb (n : ℕ) : ν L ≤ ENNReal.ofReal (C * (1 / 2 : ℝ) ^ n) := by |
| have hn : 0 < (1 / 2 : ℝ) ^ n := by positivity |
| have hn1 : (1 / 2 : ℝ) ^ n ≤ 1 := pow_le_one₀ (by norm_num) (by norm_num) |
| calc |
| ν L ≤ ν {t | Metric.infDist t L ≤ (1 / 2 : ℝ) ^ n} := by |
| apply measure_mono |
| intro t ht |
| simp only [mem_ofPred_eq, Metric.infDist_zero_of_mem ht] |
| exact hn.le |
| _ ≤ ENNReal.ofReal C * ENNReal.ofReal (((1 / 2 : ℝ) ^ n) ^ 1) := |
| hν.affine_tube_pow hsupport L 1 (by simpa using hmargin) hn hn1 |
| _ = _ := by rw [pow_one, ENNReal.ofReal_mul hC] |
| have hlim : Tendsto (fun n : ℕ => ENNReal.ofReal (C * (1 / 2 : ℝ) ^ n)) |
| atTop (𝓝 0) := by |
| convert ENNReal.tendsto_ofReal ((tendsto_const_nhds (x := C)).mul |
| (tendsto_pow_atTop_nhds_zero_of_lt_one (by norm_num : (0 : ℝ) ≤ 1 / 2) |
| (by norm_num : (1 / 2 : ℝ) < 1))) using 1 |
| simp |
| exact le_antisymm (ge_of_tendsto' hlim hb) (by positivity) |
|
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| end AffineTubes |
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| end CurrentProjection |
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| end |
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| end OAI |
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