File size: 2,774 Bytes
f61e547 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 | import Mathlib
namespace OAI
noncomputable section
open Set Filter MeasureTheory
open scoped Topology NNReal
namespace Problem367
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
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
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
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
end Problem367
end
end OAI
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