| import Mathlib |
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| namespace OAI |
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| noncomputable section |
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| open Set Filter MeasureTheory |
| open scoped Topology NNReal |
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| namespace Problem367 |
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| theorem absolutelyContinuousOnInterval_comp_lipschitzOn |
| {X Y : Type*} [PseudoMetricSpace X] [PseudoMetricSpace Y] |
| {f : ℝ → X} {g : X → Y} {a b : ℝ} {s : Set X} {K : ℝ≥0} |
| (hf : AbsolutelyContinuousOnInterval f a b) |
| (hg : LipschitzOnWith K g s) |
| (hfs : MapsTo f (uIcc a b) s) : |
| AbsolutelyContinuousOnInterval (g ∘ f) a b := by |
| rw [absolutelyContinuousOnInterval_iff] at hf ⊢ |
| intro ε hε |
| obtain ⟨δ, hδ, hfδ⟩ := hf (ε / (K + 1)) (by positivity) |
| refine ⟨δ, hδ, ?_⟩ |
| intro E hE hlen |
| have hsmall := hfδ E hE hlen |
| calc |
| ∑ i ∈ Finset.range E.1, dist ((g ∘ f) (E.2 i).1) ((g ∘ f) (E.2 i).2) |
| ≤ ∑ i ∈ Finset.range E.1, (K : ℝ) * dist (f (E.2 i).1) (f (E.2 i).2) := by |
| apply Finset.sum_le_sum |
| intro i hi |
| exact hg.dist_le_mul _ (hfs (hE.1 i hi).1) _ (hfs (hE.1 i hi).2) |
| _ = (K : ℝ) * ∑ i ∈ Finset.range E.1, dist (f (E.2 i).1) (f (E.2 i).2) := |
| (Finset.mul_sum _ _ _).symm |
| _ ≤ (K : ℝ) * (ε / (K + 1)) := mul_le_mul_of_nonneg_left hsmall.le K.2 |
| _ < ((K : ℝ) + 1) * (ε / (K + 1)) := by |
| apply mul_lt_mul_of_pos_right (by linarith) |
| positivity |
| _ = ε := by field_simp |
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| theorem absolutelyContinuousOnInterval_comp_contDiffOn_Icc |
| {f g : ℝ → ℝ} {a b c d : ℝ} |
| (hf : AbsolutelyContinuousOnInterval f a b) |
| (hg : ContDiffOn ℝ 1 g (Icc c d)) |
| (hfs : MapsTo f (uIcc a b) (Icc c d)) : |
| AbsolutelyContinuousOnInterval (g ∘ f) a b := by |
| obtain ⟨K, hK⟩ := hg.exists_lipschitzOnWith (by decide) (convex_Icc c d) isCompact_Icc |
| exact absolutelyContinuousOnInterval_comp_lipschitzOn hf hK hfs |
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| theorem ac_le_add_integral_of_neg_le_deriv |
| {f k : ℝ → ℝ} {a b : ℝ} (hab : a ≤ b) |
| (hf : AbsolutelyContinuousOnInterval f a b) |
| (hk : IntervalIntegrable k volume a b) |
| (hbound : ∀ᵐ x ∂volume.restrict (Icc a b), -k x ≤ deriv f x) : |
| f a ≤ f b + ∫ x in a..b, k x := by |
| have h := intervalIntegral.integral_mono_ae_restrict hab hk.neg |
| hf.intervalIntegrable_deriv hbound |
| simp only [Pi.neg_apply] at h |
| rw [intervalIntegral.integral_neg, hf.integral_deriv_eq_sub] at h |
| linarith |
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| theorem ac_sub_le_integral_of_deriv_le |
| {f k : ℝ → ℝ} {a b : ℝ} (hab : a ≤ b) |
| (hf : AbsolutelyContinuousOnInterval f a b) |
| (hk : IntervalIntegrable k volume a b) |
| (hbound : ∀ᵐ x ∂volume.restrict (Icc a b), deriv f x ≤ k x) : |
| f b - f a ≤ ∫ x in a..b, k x := by |
| rw [← hf.integral_deriv_eq_sub] |
| exact intervalIntegral.integral_mono_ae_restrict hab hf.intervalIntegrable_deriv hk hbound |
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| end Problem367 |
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| end |
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| end OAI |
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