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feat(MeasureTheory): basic lemmas for absolute continuity #42992

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feat(MeasureTheory): basic lemmas for absolute continuity #42992

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16 changes: 16 additions & 0 deletions Mathlib/MeasureTheory/Function/AbsolutelyContinuous.lean
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Expand Up @@ -181,6 +181,22 @@ theorem mono (hf : AbsolutelyContinuousOnInterval f a b) (habcd : uIcc c d ⊆ u
refine le_trans (Filter.map_mono ?_) hf
gcongr; exact disjWithin_mono habcd

theorem congr (hf : AbsolutelyContinuousOnInterval f a b)
(hfg : Set.EqOn f g (uIcc a b)) : AbsolutelyContinuousOnInterval g a b := by
apply hf.congr'
rw [eventuallyEq_inf_principal_iff]
filter_upwards with (n, I) hnI
exact Finset.sum_congr rfl fun i hi ↦ by rw [hfg (hnI.1 i hi).1, hfg (hnI.1 i hi).2]

@[refl, simp]
theorem refl : AbsolutelyContinuousOnInterval f a a := by
apply tendsto_nhds_of_eventually_eq
rw [eventually_inf_principal]
filter_upwards with (n, I) hnI
refine Finset.sum_eq_zero fun i hi ↦ ?_
obtain ⟨h₁, h₂⟩ := hnI.1 i hi
simp_all

variable {f g : ℝ → F}

@[to_fun]
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