/-- Given a simple function `f` with values in `ℝ≥0`, there exists a lower semicontinuous
function `g ≥ f` with integral arbitrarily close to that of `f`. Formulation in terms of
`lintegral`.
Auxiliary lemma for Vitali-Carathéodory theorem `exists_lt_lower_semicontinuous_integral_lt`. -/
theorem SimpleFunc.exists_le_lowerSemicontinuous_lintegral_ge (f : α →ₛ ℝ≥0) {ε : ℝ≥0∞}
    (ε0 : ε ≠ 0) :
    ∃ g : α → ℝ≥0, (∀ x, f x ≤ g x) ∧ LowerSemicontinuous g ∧
      (∫⁻ x, g x ∂μ) ≤ (∫⁻ x, f x ∂μ) + ε := by
  induction' f using MeasureTheory.SimpleFunc.induction with c s hs f₁ f₂ _ h₁ h₂ generalizing ε
  · let f := SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0)
    by_cases h : ∫⁻ x, f x ∂μ = ⊤
    · refine'
        ⟨fun _ => c, fun x => _, lowerSemicontinuous_const, by
          simp only [_root_.top_add, le_top, h]⟩
      simp only [SimpleFunc.coe_const, SimpleFunc.const_zero, SimpleFunc.coe_zero,
        Set.piecewise_eq_indicator, SimpleFunc.coe_piecewise]
      exact Set.indicator_le_self _ _ _
    by_cases hc : c = 0
    · refine' ⟨fun _ => 0, _, lowerSemicontinuous_const, _⟩
      · classical
        simp only [hc, Set.indicator_zero', Pi.zero_apply, SimpleFunc.const_zero, imp_true_iff,
          eq_self_iff_true, SimpleFunc.coe_zero, Set.piecewise_eq_indicator,
          SimpleFunc.coe_piecewise, le_zero_iff]
      · simp only [lintegral_const, zero_mul, zero_le, ENNReal.coe_zero]
    have : μ s < μ s + ε / c := by
      have : (0 : ℝ≥0∞) < ε / c := ENNReal.div_pos_iff.2 ⟨ε0, ENNReal.coe_ne_top⟩
      simpa using ENNReal.add_lt_add_left ?aux this
    case aux =>
      classical
      simpa [hs, hc, lt_top_iff_ne_top, true_and_iff, SimpleFunc.coe_const,
        Function.const_apply, lintegral_const, ENNReal.coe_indicator, Set.univ_inter,
        ENNReal.coe_ne_top, MeasurableSet.univ, ENNReal.mul_eq_top, SimpleFunc.const_zero,
        or_false_iff, lintegral_indicator, ENNReal.coe_eq_zero, Ne.def, not_false_iff,
        SimpleFunc.coe_zero, Set.piecewise_eq_indicator, SimpleFunc.coe_piecewise, false_and_iff,
        restrict_apply] using h
    obtain ⟨u, su, u_open, μu⟩ : ∃ (u : _), u ⊇ s ∧ IsOpen u ∧ μ u < μ s + ε / c :=
      s.exists_isOpen_lt_of_lt _ this
    refine'
      ⟨Set.indicator u fun _ => c, fun x => _, u_open.lowerSemicontinuous_indicator (zero_le _), _⟩
    · simp only [SimpleFunc.coe_const, SimpleFunc.const_zero, SimpleFunc.coe_zero,
        Set.piecewise_eq_indicator, SimpleFunc.coe_piecewise]
      exact Set.indicator_le_indicator_of_subset su (fun x => zero_le _) _
    · suffices (c : ℝ≥0∞) * μ u ≤ c * μ s + ε by
        classical
        simpa only [ENNReal.coe_indicator, u_open.measurableSet, lintegral_indicator,
          lintegral_const, MeasurableSet.univ, Measure.restrict_apply, Set.univ_inter, const_zero,
          coe_piecewise, coe_const, coe_zero, Set.piecewise_eq_indicator, Function.const_apply, hs]
      calc
        (c : ℝ≥0∞) * μ u ≤ c * (μ s + ε / c) := mul_le_mul_left' μu.le _
        _ = c * μ s + ε := by
          simp_rw [mul_add]
          rw [ENNReal.mul_div_cancel' _ ENNReal.coe_ne_top]
          simpa using hc

  · rcases h₁ (ENNReal.half_pos ε0).ne' with ⟨g₁, f₁_le_g₁, g₁cont, g₁int⟩
    

Proof state:
case h_add.intro.intro.intro
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : MeasurableSpace α
inst✝¹ : BorelSpace α
μ : Measure α
inst✝ : WeaklyRegular μ
f₁ f₂ : α →ₛ ℝ≥0
a✝ : Disjoint (Function.support ↑f₁) (Function.support ↑f₂)
h₁ :
  ∀ {ε : ℝ≥0∞},
    ε ≠ 0 → ∃ g, (∀ (x : α), ↑f₁ x ≤ g x) ∧ LowerSemicontinuous g ∧ ∫⁻ (x : α), ↑(g x) ∂μ ≤ ∫⁻ (x : α), ↑(↑f₁ x) ∂μ + ε
h₂ :
  ∀ {ε : ℝ≥0∞},
    ε ≠ 0 → ∃ g, (∀ (x : α), ↑f₂ x ≤ g x) ∧ LowerSemicontinuous g ∧ ∫⁻ (x : α), ↑(g x) ∂μ ≤ ∫⁻ (x : α), ↑(↑f₂ x) ∂μ + ε
ε : ℝ≥0∞
ε0 : ε ≠ 0
g₁ : α → ℝ≥0
f₁_le_g₁ : ∀ (x : α), ↑f₁ x ≤ g₁ x
g₁cont : LowerSemicontinuous g₁
g₁int : ∫⁻ (x : α), ↑(g₁ x) ∂μ ≤ ∫⁻ (x : α), ↑(↑f₁ x) ∂μ + ε / 2
⊢ ∃ g,
    (∀ (x : α), ↑(f₁ + f₂) x ≤ g x) ∧ LowerSemicontinuous g ∧ ∫⁻ (x : α), ↑(g x) ∂μ ≤ ∫⁻ (x : α), ↑(↑(f₁ + f₂) x) ∂μ + ε

Next tactic:
rcases h₂ (ENNReal.half_pos ε0).ne' with ⟨g₂, f₂_le_g₂, g₂cont, g₂int⟩