/-- The derivative of a function on a measurable set is almost everywhere measurable on this set
with respect to Lebesgue measure. Note that, in general, it is not genuinely measurable there,
as `f'` is not unique (but only on a set of measure `0`, as the argument shows). -/
theorem aemeasurable_fderivWithin (hs : MeasurableSet s)
    (hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) : AEMeasurable f' (μ.restrict s) := by
  /- It suffices to show that `f'` can be uniformly approximated by a measurable function.
    Fix `ε > 0`. Thanks to `exists_partition_approximatesLinearOn_of_hasFDerivWithinAt`, one
    can find a countable measurable partition of `s` into sets `s ∩ t n` on which `f` is well
    approximated by linear maps `A n`. On almost all of `s ∩ t n`, it follows from
    `ApproximatesLinearOn.norm_fderiv_sub_le` that `f'` is uniformly approximated by `A n`, which
    gives the conclusion. -/
  -- fix a precision `ε`
  refine' aemeasurable_of_unif_approx fun ε εpos => _
  let δ : ℝ≥0 := ⟨ε, le_of_lt εpos⟩
  have δpos : 0 < δ := εpos
  -- partition `s` into sets `s ∩ t n` on which `f` is approximated by linear maps `A n`.
  obtain ⟨t, A, t_disj, t_meas, t_cover, ht, _⟩ :
    ∃ (t : ℕ → Set E) (A : ℕ → E →L[ℝ] E),
      Pairwise (Disjoint on t) ∧
        (∀ n : ℕ, MeasurableSet (t n)) ∧
          (s ⊆ ⋃ n : ℕ, t n) ∧
            (∀ n : ℕ, ApproximatesLinearOn f (A n) (s ∩ t n) δ) ∧
              (s.Nonempty → ∀ n, ∃ y ∈ s, A n = f' y) :=
    exists_partition_approximatesLinearOn_of_hasFDerivWithinAt f s f' hf' (fun _ => δ) fun _ =>
      δpos.ne'
  -- define a measurable function `g` which coincides with `A n` on `t n`.
  obtain ⟨g, g_meas, hg⟩ :
      ∃ g : E → E →L[ℝ] E, Measurable g ∧ ∀ (n : ℕ) (x : E), x ∈ t n → g x = A n :=
    exists_measurable_piecewise t t_meas (fun n _ => A n) (fun n => measurable_const) <|
      t_disj.mono fun i j h => by simp only [h.inter_eq, eqOn_empty]
  refine' ⟨g, g_meas.aemeasurable, _⟩
  -- reduce to checking that `f'` and `g` are close on almost all of `s ∩ t n`, for all `n`.
  suffices H : ∀ᵐ x : E ∂sum fun n => μ.restrict (s ∩ t n), dist (g x) (f' x) ≤ ε
  · have : μ.restrict s ≤ sum fun n => μ.restrict (s ∩ t n) := by
      have : s = ⋃ n, s ∩ t n := by
        rw [← inter_iUnion]
        exact Subset.antisymm (subset_inter Subset.rfl t_cover) (inter_subset_left _ _)
      conv_lhs => rw [this]
      exact restrict_iUnion_le
    exact ae_mono this H
  -- fix such an `n`.
  refine' ae_sum_iff.2 fun n => _
  -- on almost all `s ∩ t n`, `f' x` is close to `A n` thanks to
  -- `ApproximatesLinearOn.norm_fderiv_sub_le`.
  have E₁ : ∀ᵐ x : E ∂μ.restrict (s ∩ t n), ‖f' x - A n‖₊ ≤ δ :=
    (ht n).norm_fderiv_sub_le μ (hs.inter (t_meas n)) f' fun x hx =>
      (hf' x hx.1).mono (inter_subset_left _ _)
  -- moreover, `g x` is equal to `A n` there.
  

Proof state:
case H
E : Type u_1
F : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
s : Set E
f : E → E
f' : E → E →L[ℝ] E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : IsAddHaarMeasure μ
hs : MeasurableSet s
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
ε : ℝ
εpos : ε > 0
δ : ℝ≥0 := { val := ε, property := (_ : 0 ≤ ε) }
δpos : 0 < δ
t : ℕ → Set E
A : ℕ → E →L[ℝ] E
t_disj : Pairwise (Disjoint on t)
t_meas : ∀ (n : ℕ), MeasurableSet (t n)
t_cover : s ⊆ ⋃ n, t n
ht : ∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) δ
right✝ : Set.Nonempty s → ∀ (n : ℕ), ∃ y ∈ s, A n = f' y
g : E → E →L[ℝ] E
g_meas : Measurable g
hg : ∀ (n : ℕ), ∀ x ∈ t n, g x = A n
n : ℕ
E₁ : ∀ᵐ (x : E) ∂Measure.restrict μ (s ∩ t n), ‖f' x - A n‖₊ ≤ δ
⊢ ∀ᵐ (x : E) ∂Measure.restrict μ (s ∩ t n), dist (g x) (f' x) ≤ ε

Next tactic:
have E₂ : ∀ᵐ x : E ∂μ.restrict (s ∩ t n), g x = A n := by
    suffices H : ∀ᵐ x : E ∂μ.restrict (t n), g x = A n
    exact ae_mono (restrict_mono (inter_subset_right _ _) le_rfl) H
    filter_upwards [ae_restrict_mem (t_meas n)]
    exact hg n