/-- A Stieltjes function is almost everywhere differentiable, with derivative equal to the
Radon-Nikodym derivative of the associated Stieltjes measure with respect to Lebesgue. -/
theorem StieltjesFunction.ae_hasDerivAt (f : StieltjesFunction) :
    ∀ᵐ x, HasDerivAt f (rnDeriv f.measure volume x).toReal x := by
  /- Denote by `μ` the Stieltjes measure associated to `f`.
    The general theorem `VitaliFamily.ae_tendsto_rnDeriv` ensures that `μ [x, y] / (y - x)` tends
    to the Radon-Nikodym derivative as `y` tends to `x` from the right. As `μ [x,y] = f y - f (x^-)`
    and `f (x^-) = f x` almost everywhere, this gives differentiability on the right.
    On the left, `μ [y, x] / (x - y)` again tends to the Radon-Nikodym derivative.
    As `μ [y, x] = f x - f (y^-)`, this is not exactly the right result, so one uses a sandwiching
    argument to deduce the convergence for `(f x - f y) / (x - y)`. -/
  filter_upwards [VitaliFamily.ae_tendsto_rnDeriv (vitaliFamily (volume : Measure ℝ) 1) f.measure,
    rnDeriv_lt_top f.measure volume, f.countable_leftLim_ne.ae_not_mem volume] with x hx h'x h''x
  -- Limit on the right, following from differentiation of measures
  have L1 :
    Tendsto (fun y => (f y - f x) / (y - x)) (𝓝[>] x) (𝓝 (rnDeriv f.measure volume x).toReal) := by
    apply Tendsto.congr' _
      ((ENNReal.tendsto_toReal h'x.ne).comp (hx.comp (Real.tendsto_Icc_vitaliFamily_right x)))
    filter_upwards [self_mem_nhdsWithin]
    rintro y (hxy : x < y)
    simp only [comp_apply, StieltjesFunction.measure_Icc, Real.volume_Icc, Classical.not_not.1 h''x]
    rw [← ENNReal.ofReal_div_of_pos (sub_pos.2 hxy), ENNReal.toReal_ofReal]
    exact div_nonneg (sub_nonneg.2 (f.mono hxy.le)) (sub_pos.2 hxy).le
  -- Limit on the left, following from differentiation of measures. Its form is not exactly the one
  -- we need, due to the appearance of a left limit.
  have L2 : Tendsto (fun y => (leftLim f y - f x) / (y - x)) (𝓝[<] x)
      (𝓝 (rnDeriv f.measure volume x).toReal) := by
    apply Tendsto.congr' _
      ((ENNReal.tendsto_toReal h'x.ne).comp (hx.comp (Real.tendsto_Icc_vitaliFamily_left x)))
    filter_upwards [self_mem_nhdsWithin]
    rintro y (hxy : y < x)
    simp only [comp_apply, StieltjesFunction.measure_Icc, Real.volume_Icc]
    rw [← ENNReal.ofReal_div_of_pos (sub_pos.2 hxy), ENNReal.toReal_ofReal, ← neg_neg (y - x),
      div_neg, neg_div', neg_sub, neg_sub]
    exact div_nonneg (sub_nonneg.2 (f.mono.leftLim_le hxy.le)) (sub_pos.2 hxy).le
  -- Shifting a little bit the limit on the left, by `(y - x)^2`.
  have L3 : Tendsto (fun y => (leftLim f (y + 1 * (y - x) ^ 2) - f x) / (y - x)) (𝓝[<] x)
      (𝓝 (rnDeriv f.measure volume x).toReal) := by
    apply tendsto_apply_add_mul_sq_div_sub (nhds_left'_le_nhds_ne x) L2
    apply tendsto_nhdsWithin_of_tendsto_nhds_of_eventually_within
    · apply Tendsto.mono_left _ nhdsWithin_le_nhds
      have : Tendsto (fun y : ℝ => y + ↑1 * (y - x) ^ 2) (𝓝 x) (𝓝 (x + ↑1 * (x - x) ^ 2)) :=
        tendsto_id.add (((tendsto_id.sub_const x).pow 2).const_mul ↑1)
      simpa using this
    · have : Ioo (x - 1) x ∈ 𝓝[<] x := by
        apply Ioo_mem_nhdsWithin_Iio; exact ⟨by linarith, le_refl _⟩
      filter_upwards [this]
      rintro y ⟨hy : x - 1 < y, h'y : y < x⟩
      rw [mem_Iio]
      norm_num; nlinarith
  -- Deduce the correct limit on the left, by sandwiching.
  have L4 :
    Tendsto (fun y => (f y - f x) / (y - x)) (𝓝[<] x) (𝓝 (rnDeriv f.measure volume x).toReal) := by
    apply tendsto_of_tendsto_of_tendsto_of_le_of_le' L3 L2
    · filter_upwards [self_mem_nhdsWithin]
      rintro y (hy : y < x)
      refine' div_le_div_of_nonpos_of_le (by linarith) ((sub_le_sub_iff_right _).2 _)
      

Proof state:
case h
f : StieltjesFunction
x : ℝ
hx :
  Tendsto (fun a => ↑↑(StieltjesFunction.measure f) a / ↑↑volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)
    (𝓝 (rnDeriv (StieltjesFunction.measure f) volume x))
h'x : rnDeriv (StieltjesFunction.measure f) volume x < ⊤
h''x : ¬leftLim (↑f) x ≠ ↑f x
L1 :
  Tendsto (fun y => (↑f y - ↑f x) / (y - x)) (𝓝[>] x)
    (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))
L2 :
  Tendsto (fun y => (leftLim (↑f) y - ↑f x) / (y - x)) (𝓝[<] x)
    (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))
L3 :
  Tendsto (fun y => (leftLim (↑f) (y + 1 * (y - x) ^ 2) - ↑f x) / (y - x)) (𝓝[<] x)
    (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))
y : ℝ
hy : y < x
⊢ ↑f y ≤ leftLim (↑f) (y + 1 * (y - x) ^ 2)

Next tactic:
apply f.mono.le_leftLim