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feat(MeasureTheory): absolute continuity preserved by Lipschitz postcomposition #42996

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feat(MeasureTheory): absolute continuity preserved by Lipschitz postcomposition #42996

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23 changes: 23 additions & 0 deletions Mathlib/MeasureTheory/Function/AbsolutelyContinuous.lean
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Expand Up @@ -308,6 +308,29 @@ theorem _root_.LipschitzOnWith.absolutelyContinuousOnInterval {f : ℝ → X} {K
_ < (K + 1) * (ε / (K + 1)) := by gcongr; linarith
_ = ε := by field

/-- If `f` is Lipschitz on a set containing `g '' uIcc a b` and `g` is absolutely continuous on
`uIcc a b`, then `f ∘ g` is absolutely continuous on `uIcc a b`. -/
theorem _root_.LipschitzOnWith.comp_absolutelyContinuousOnInterval
{Y : Type*} [PseudoMetricSpace Y] {f : X → Y} {K : ℝ≥0} {t : Set X}
(hf : LipschitzOnWith K f t) {g : ℝ → X} {a b : ℝ} (hg : MapsTo g (uIcc a b) t)
(h : AbsolutelyContinuousOnInterval g a b) :
AbsolutelyContinuousOnInterval (f ∘ g) a b := by
apply squeeze_zero' ?_ ?_ (by simpa using Tendsto.const_mul (K : ℝ) h)
· exact Filter.Eventually.of_forall <| fun _ ↦ Finset.sum_nonneg fun _ _ ↦ dist_nonneg
rw [eventually_inf_principal]
filter_upwards with (n, I) hnI
rw [Finset.mul_sum]
exact Finset.sum_le_sum fun i hi ↦
hf.dist_le_mul _ (hg (hnI.left i hi).left) _ (hg (hnI.left i hi).right)

/-- If `f` is Lipschitz and `g` is absolutely continuous on `uIcc a b`, then `f ∘ g` is absolutely
continuous on `uIcc a b`. -/
theorem _root_.LipschitzWith.comp_absolutelyContinuousOnInterval
{Y : Type*} [PseudoMetricSpace Y] {f : X → Y} {K : ℝ≥0} (hf : LipschitzWith K f)
{g : ℝ → X} {a b : ℝ} (h : AbsolutelyContinuousOnInterval g a b) :
AbsolutelyContinuousOnInterval (f ∘ g) a b :=
hf.lipschitzOnWith.comp_absolutelyContinuousOnInterval (mapsTo_univ _ _) h

/-- If `f` is `C^1` on `uIcc a b`, then `f` is absolutely continuous on `uIcc a b`. -/
theorem _root_.ContDiffOn.absolutelyContinuousOnInterval {E : Type*} [NormedAddCommGroup E]
[NormedSpace ℝ E] {f : ℝ → E} (hf : ContDiffOn ℝ 1 f (uIcc a b)) :
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