/-- Given a simple function `f` with values in `ℝ≥0`, there exists an upper 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_upperSemicontinuous_le_lintegral_le (f : α →ₛ ℝ≥0)
    (int_f : (∫⁻ x, f x ∂μ) ≠ ∞) {ε : ℝ≥0∞} (ε0 : ε ≠ 0) :
    ∃ g : α → ℝ≥0, (∀ x, g x ≤ f x) ∧ UpperSemicontinuous g ∧
      (∫⁻ x, f x ∂μ) ≤ (∫⁻ x, g x ∂μ) + ε := by
  induction' f using MeasureTheory.SimpleFunc.induction with c s hs f₁ f₂ _ h₁ h₂ generalizing ε
  · by_cases hc : c = 0
    · refine' ⟨fun _ => 0, _, upperSemicontinuous_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]
      · classical
        simp only [hc, Set.indicator_zero', lintegral_const, zero_mul, Pi.zero_apply,
          SimpleFunc.const_zero, zero_add, zero_le', SimpleFunc.coe_zero,
          Set.piecewise_eq_indicator, ENNReal.coe_zero, SimpleFunc.coe_piecewise, zero_le]
    have μs_lt_top : μ s < ∞ := by
      classical
      simpa only [hs, hc, lt_top_iff_ne_top, true_and_iff, SimpleFunc.coe_const, or_false_iff,
        lintegral_const, ENNReal.coe_indicator, Set.univ_inter, ENNReal.coe_ne_top,
        Measure.restrict_apply MeasurableSet.univ, ENNReal.mul_eq_top, SimpleFunc.const_zero,
        Function.const_apply, lintegral_indicator, ENNReal.coe_eq_zero, Ne.def, not_false_iff,
        SimpleFunc.coe_zero, Set.piecewise_eq_indicator, SimpleFunc.coe_piecewise,
        false_and_iff] using int_f
    have : (0 : ℝ≥0∞) < ε / c := ENNReal.div_pos_iff.2 ⟨ε0, ENNReal.coe_ne_top⟩
    obtain ⟨F, Fs, F_closed, μF⟩ : ∃ (F : _), F ⊆ s ∧ IsClosed F ∧ μ s < μ F + ε / c :=
      hs.exists_isClosed_lt_add μs_lt_top.ne this.ne'
    refine'
      ⟨Set.indicator F fun _ => c, fun x => _, F_closed.upperSemicontinuous_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 Fs (fun x => zero_le _) _
    · suffices (c : ℝ≥0∞) * μ s ≤ c * μ F + ε by
        classical
        simpa only [hs, F_closed.measurableSet, SimpleFunc.coe_const, Function.const_apply,
          lintegral_const, ENNReal.coe_indicator, Set.univ_inter, MeasurableSet.univ,
          SimpleFunc.const_zero, lintegral_indicator, SimpleFunc.coe_zero,
          Set.piecewise_eq_indicator, SimpleFunc.coe_piecewise, Measure.restrict_apply]
      calc
        (c : ℝ≥0∞) * μ s ≤ c * (μ F + ε / c) := mul_le_mul_left' μF.le _
        _ = c * μ F + ε := by
          simp_rw [mul_add]
          rw [ENNReal.mul_div_cancel' _ ENNReal.coe_ne_top]
          simpa using hc
  · have A : ((∫⁻ x : α, f₁ x ∂μ) + ∫⁻ x : α, f₂ x ∂μ) ≠ ⊤ := by
      rwa [← lintegral_add_left f₁.measurable.coe_nnreal_ennreal]
    rcases h₁ (ENNReal.add_ne_top.1 A).1 (ENNReal.half_pos ε0).ne' with
      ⟨g₁, f₁_le_g₁, g₁cont, g₁int⟩
    rcases h₂ (ENNReal.add_ne_top.1 A).2 (ENNReal.half_pos ε0).ne' with
      ⟨g₂, f₂_le_g₂, g₂cont, g₂int⟩
    refine'
      ⟨fun x => g₁ x + g₂ x, fun x => add_le_add (f₁_le_g₁ x) (f₂_le_g₂ x), g₁cont.add g₂cont, _⟩
    simp only [SimpleFunc.coe_add, ENNReal.coe_add, Pi.add_apply]
    

Proof state:
case h_add.intro.intro.intro.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 :
  ∫⁻ (x : α), ↑(↑f₁ x) ∂μ ≠ ⊤ →
    ∀ {ε : ℝ≥0∞},
      ε ≠ 0 →
        ∃ g, (∀ (x : α), g x ≤ ↑f₁ x) ∧ UpperSemicontinuous g ∧ ∫⁻ (x : α), ↑(↑f₁ x) ∂μ ≤ ∫⁻ (x : α), ↑(g x) ∂μ + ε
h₂ :
  ∫⁻ (x : α), ↑(↑f₂ x) ∂μ ≠ ⊤ →
    ∀ {ε : ℝ≥0∞},
      ε ≠ 0 →
        ∃ g, (∀ (x : α), g x ≤ ↑f₂ x) ∧ UpperSemicontinuous g ∧ ∫⁻ (x : α), ↑(↑f₂ x) ∂μ ≤ ∫⁻ (x : α), ↑(g x) ∂μ + ε
int_f : ∫⁻ (x : α), ↑(↑(f₁ + f₂) x) ∂μ ≠ ⊤
ε : ℝ≥0∞
ε0 : ε ≠ 0
A : ∫⁻ (x : α), ↑(↑f₁ x) ∂μ + ∫⁻ (x : α), ↑(↑f₂ x) ∂μ ≠ ⊤
g₁ : α → ℝ≥0
f₁_le_g₁ : ∀ (x : α), g₁ x ≤ ↑f₁ x
g₁cont : UpperSemicontinuous g₁
g₁int : ∫⁻ (x : α), ↑(↑f₁ x) ∂μ ≤ ∫⁻ (x : α), ↑(g₁ x) ∂μ + ε / 2
g₂ : α → ℝ≥0
f₂_le_g₂ : ∀ (x : α), g₂ x ≤ ↑f₂ x
g₂cont : UpperSemicontinuous g₂
g₂int : ∫⁻ (x : α), ↑(↑f₂ x) ∂μ ≤ ∫⁻ (x : α), ↑(g₂ x) ∂μ + ε / 2
⊢ ∫⁻ (x : α), ↑(↑f₁ x) + ↑(↑f₂ x) ∂μ ≤ ∫⁻ (x : α), ↑(g₁ x) + ↑(g₂ x) ∂μ + ε

Next tactic:
rw [lintegral_add_left f₁.measurable.coe_nnreal_ennreal,
      lintegral_add_left g₁cont.measurable.coe_nnreal_ennreal]