@[simp]
theorem outer_Ioc (a b : ℝ) : f.outer (Ioc a b) = ofReal (f b - f a) := by
  /- It suffices to show that, if `(a, b]` is covered by sets `s i`, then `f b - f a` is bounded
    by `∑ f.length (s i) + ε`. The difficulty is that `f.length` is expressed in terms of half-open
    intervals, while we would like to have a compact interval covered by open intervals to use
    compactness and finite sums, as provided by `length_subadditive_Icc_Ioo`. The trick is to use
    the right-continuity of `f`. If `a'` is close enough to `a` on its right, then `[a', b]` is
    still covered by the sets `s i` and moreover `f b - f a'` is very close to `f b - f a`
    (up to `ε/2`).
    Also, by definition one can cover `s i` by a half-closed interval `(p i, q i]` with `f`-length
    very close to that of `s i` (within a suitably small `ε' i`, say). If one moves `q i` very
    slightly to the right, then the `f`-length will change very little by right continuity, and we
    will get an open interval `(p i, q' i)` covering `s i` with `f (q' i) - f (p i)` within `ε' i`
    of the `f`-length of `s i`. -/
  refine'
    le_antisymm
      (by
        rw [← f.length_Ioc]
        apply outer_le_length)
      (le_iInf₂ fun s hs => ENNReal.le_of_forall_pos_le_add fun ε εpos h => _)
  let δ := ε / 2
  have δpos : 0 < (δ : ℝ≥0∞) := by simpa using εpos.ne'
  rcases ENNReal.exists_pos_sum_of_countable δpos.ne' ℕ with ⟨ε', ε'0, hε⟩
  obtain ⟨a', ha', aa'⟩ : ∃ a', f a' - f a < δ ∧ a < a' := by
    have A : ContinuousWithinAt (fun r => f r - f a) (Ioi a) a := by
      refine' ContinuousWithinAt.sub _ continuousWithinAt_const
      exact (f.right_continuous a).mono Ioi_subset_Ici_self
    have B : f a - f a < δ := by rwa [sub_self, NNReal.coe_pos, ← ENNReal.coe_pos]
    exact (((tendsto_order.1 A).2 _ B).and self_mem_nhdsWithin).exists
  have : ∀ i, ∃ p : ℝ × ℝ, s i ⊆ Ioo p.1 p.2 ∧
      (ofReal (f p.2 - f p.1) : ℝ≥0∞) < f.length (s i) + ε' i := by
    intro i
    have hl :=
      ENNReal.lt_add_right ((ENNReal.le_tsum i).trans_lt h).ne (ENNReal.coe_ne_zero.2 (ε'0 i).ne')
    conv at hl =>
      lhs
      rw [length]
    simp only [iInf_lt_iff, exists_prop] at hl
    rcases hl with ⟨p, q', spq, hq'⟩
    have : ContinuousWithinAt (fun r => ofReal (f r - f p)) (Ioi q') q' := by
      apply ENNReal.continuous_ofReal.continuousAt.comp_continuousWithinAt
      refine' ContinuousWithinAt.sub _ continuousWithinAt_const
      exact (f.right_continuous q').mono Ioi_subset_Ici_self
    rcases (((tendsto_order.1 this).2 _ hq').and self_mem_nhdsWithin).exists with ⟨q, hq, q'q⟩
    exact ⟨⟨p, q⟩, spq.trans (Ioc_subset_Ioo_right q'q), hq⟩
  choose g hg using this
  

Proof state:
case intro.intro.intro.intro
f : StieltjesFunction
a b : ℝ
s : ℕ → Set ℝ
hs : Ioc a b ⊆ ⋃ i, s i
ε : ℝ≥0
εpos : 0 < ε
h : ∑' (i : ℕ), length f (s i) < ⊤
δ : ℝ≥0 := ε / 2
δpos : 0 < ↑δ
ε' : ℕ → ℝ≥0
ε'0 : ∀ (i : ℕ), 0 < ε' i
hε : ∑' (i : ℕ), ↑(ε' i) < ↑δ
a' : ℝ
ha' : ↑f a' - ↑f a < ↑δ
aa' : a < a'
g : ℕ → ℝ × ℝ
hg : ∀ (i : ℕ), s i ⊆ Ioo (g i).1 (g i).2 ∧ ofReal (↑f (g i).2 - ↑f (g i).1) < length f (s i) + ↑(ε' i)
⊢ ofReal (↑f b - ↑f a) ≤ ∑' (i : ℕ), length f (s i) + ↑ε

Next tactic:
have I_subset : Icc a' b ⊆ ⋃ i, Ioo (g i).1 (g i).2 :=
    calc
      Icc a' b ⊆ Ioc a b := fun x hx => ⟨aa'.trans_le hx.1, hx.2⟩
      _ ⊆ ⋃ i, s i := hs
      _ ⊆ ⋃ i, Ioo (g i).1 (g i).2 := iUnion_mono fun i => (hg i).1