| import OAI.Analysis.Daugavet.Enlargement |
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| namespace OAI |
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| noncomputable section |
| open MeasureTheory ProbabilityTheory Set Filter |
| open scoped ENNReal NNReal Topology BigOperators |
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| universe uX u_1 uB |
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| namespace ExactModuli |
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| variable {Ω : Type u_1} [MeasurableSpace Ω] {μ : Measure Ω} [IsProbabilityMeasure μ] |
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| lemma measureDistance_self (f : Lp ℝ 1 μ) : measureDistance μ f f = 0 := by |
| apply le_antisymm _ (measureDistance_nonneg f f) |
| apply le_of_forall_pos_le_add |
| intro ε hε |
| simpa only [zero_add] using (measureDistance_le_of_tail f f hε (by simpa [not_lt_of_ge hε.le] using hε)) |
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|
| omit [IsProbabilityMeasure μ] in |
| lemma measureDistance_symm [IsProbabilityMeasure μ] (f g : Lp ℝ 1 μ) : |
| measureDistance μ f g = measureDistance μ g f := by |
| simp only [measureDistance, abs_sub_comm (f _) (g _)] |
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| lemma measureDistance_triangle (f g k : Lp ℝ 1 μ) : |
| measureDistance μ f k ≤ measureDistance μ f g + measureDistance μ g k := by |
| apply le_of_forall_pos_le_add |
| intro ε hε |
| let a := measureDistance μ f g + ε / 2 |
| let b := measureDistance μ g k + ε / 2 |
| have ha : 0 < a := by dsimp [a]; linarith [measureDistance_nonneg f g] |
| have hb : 0 < b := by dsimp [b]; linarith [measureDistance_nonneg g k] |
| have hfa := tail_lt_of_measureDistance_lt f g (show measureDistance μ f g < a by dsimp [a]; linarith) |
| have hgb := tail_lt_of_measureDistance_lt g k (show measureDistance μ g k < b by dsimp [b]; linarith) |
| have hsub : {ω | a + b < |f ω - k ω|} ⊆ |
| {ω | a < |f ω - g ω|} ∪ {ω | b < |g ω - k ω|} := by |
| intro ω hω |
| by_contra hh |
| simp only [Set.mem_union, Set.mem_ofPred_eq, not_or, not_lt] at hh |
| have ht : |f ω - k ω| ≤ |f ω - g ω| + |g ω - k ω| := abs_sub_le _ _ _ |
| exact (not_lt_of_ge (ht.trans (add_le_add hh.1 hh.2))) hω |
| have htail : μ.real {ω | a + b < |f ω - k ω|} < a + b := |
| (measureReal_mono hsub).trans_lt ((measureReal_union_le _ _).trans_lt (add_lt_add hfa hgb)) |
| have hh := measureDistance_le_of_tail f k (add_pos ha hb) htail |
| dsimp [a,b] at hh |
| linarith |
|
|
| /-- Markov's norm-to-measure estimate, in a form avoiding an unnecessary metric instance. -/ |
| lemma measureDistance_le_of_norm_sub_lt_sq (f g : Lp ℝ 1 μ) {ε : ℝ} (hε : 0 < ε) |
| (hfg : ‖f - g‖ < ε ^ 2) : measureDistance μ f g ≤ ε := by |
| have hi : Integrable (fun ω => |f ω - g ω|) μ := |
| ((L1.integrable_coeFn f).sub (L1.integrable_coeFn g)).abs |
| have hint : (∫ ω, |f ω - g ω| ∂μ) = ‖f - g‖ := by |
| exact (KW.L1_norm_of_ae (f-g) (Lp.coeFn_sub f g)).symm |
| have hm := mul_meas_ge_le_integral_of_nonneg (ae_of_all μ (fun ω => abs_nonneg (f ω-g ω))) hi ε |
| rw [hint] at hm |
| have hle : μ.real {ω | ε ≤ |f ω - g ω|} < ε := by nlinarith |
| apply measureDistance_le_of_tail f g hε |
| exact (measureReal_mono (μ := μ) (show {ω | ε < |f ω-g ω|} ⊆ {ω | ε ≤ |f ω-g ω|} from by |
| intro ω hω |
| change ε < |f ω-g ω| at hω |
| change ε ≤ |f ω-g ω| |
| exact hω.le)).trans_lt hle |
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|
| end ExactModuli |
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|
| namespace ExactModuli.KW |
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|
| lemma normalizeL1_approx {B : Type uB} [MeasurableSpace B] {μ : Measure B} |
| (x y : Lp ℝ 1 μ) (hx : ‖x‖ ≤ 1) : |
| ‖x - normalizeL1 y‖ ≤ 2 * ‖x-y‖ := by |
| by_cases hy : ‖y‖ ≤ 1 |
| · simp only [normalizeL1, max_eq_left hy, inv_one, one_smul] |
| linarith [norm_nonneg (x-y)] |
| have hyp : 0 < ‖y‖ := by linarith |
| have hinv : ‖y‖⁻¹ ≤ 1 := (inv_le_one₀ hyp).mpr (le_of_not_ge hy) |
| have hcost : ‖y - normalizeL1 y‖ = ‖y‖ - 1 := by |
| calc |
| _ = ‖(1 - ‖y‖⁻¹) • y‖ := by |
| rw [normalizeL1, max_eq_right (le_of_not_ge hy), sub_smul, one_smul] |
| _ = (1 - ‖y‖⁻¹) * ‖y‖ := by |
| rw [norm_smul, Real.norm_eq_abs, abs_of_nonneg (sub_nonneg.mpr hinv)] |
| _ = ‖y‖ - 1 := by rw [sub_mul, one_mul, inv_mul_cancel₀ hyp.ne'] |
| calc |
| ‖x - normalizeL1 y‖ ≤ ‖x-y‖ + ‖y-normalizeL1 y‖ := norm_sub_le_norm_sub_add_norm_sub _ _ _ |
| _ ≤ 2 * ‖x-y‖ := by |
| rw [hcost] |
| have ht := norm_sub_norm_le y x |
| rw [norm_sub_rev] at ht |
| linarith |
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|
| lemma sphere_normalize_approx {X : Type uX} [NormedAddCommGroup X] [NormedSpace ℝ X] |
| (x y : X) (hx : ‖x‖ = 1) (hy : y ≠ 0) : |
| ‖x - ‖y‖⁻¹ • y‖ ≤ 2 * ‖x-y‖ := by |
| have hyp : 0 < ‖y‖ := norm_pos_iff.mpr hy |
| have hcost : ‖y - ‖y‖⁻¹ • y‖ = |‖y‖ - 1| := by |
| calc |
| _ = ‖(1 - ‖y‖⁻¹) • y‖ := by rw [sub_smul, one_smul] |
| _ = |(1 - ‖y‖⁻¹) * ‖y‖| := by |
| rw [norm_smul, Real.norm_eq_abs, abs_mul, abs_of_nonneg (norm_nonneg y)] |
| _ = |‖y‖ - 1| := by rw [sub_mul, one_mul, inv_mul_cancel₀ hyp.ne'] |
| calc |
| ‖x - ‖y‖⁻¹ • y‖ ≤ ‖x-y‖ + ‖y-‖y‖⁻¹ • y‖ := norm_sub_le_norm_sub_add_norm_sub _ _ _ |
| _ ≤ 2 * ‖x-y‖ := by |
| rw [hcost, ← hx] |
| have ht := abs_norm_sub_norm_le y x |
| rw [norm_sub_rev] at ht |
| linarith |
|
|
| end ExactModuli.KW |
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| end |
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| end OAI |
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