theorem SimpleFunc.exists_lt_lintegral_simpleFunc_of_lt_lintegral {m : MeasurableSpace α}
    {μ : Measure α} [SigmaFinite μ] {f : α →ₛ ℝ≥0} {L : ℝ≥0∞} (hL : L < ∫⁻ x, f x ∂μ) :
    ∃ g : α →ₛ ℝ≥0, (∀ x, g x ≤ f x) ∧ ∫⁻ x, g x ∂μ < ∞ ∧ L < ∫⁻ x, g x ∂μ := by
  induction' f using MeasureTheory.SimpleFunc.induction with c s hs f₁ f₂ _ h₁ h₂ generalizing L
  · simp only [hs, const_zero, coe_piecewise, coe_const, SimpleFunc.coe_zero, univ_inter,
      piecewise_eq_indicator, lintegral_indicator, lintegral_const, Measure.restrict_apply',
      ENNReal.coe_indicator, Function.const_apply] at hL
    have c_ne_zero : c ≠ 0 := by
      intro hc
      simp only [hc, ENNReal.coe_zero, zero_mul, not_lt_zero] at hL
    have : L / c < μ s := by
      rwa [ENNReal.div_lt_iff, mul_comm]
      · simp only [c_ne_zero, Ne.def, coe_eq_zero, not_false_iff, true_or_iff]
      · simp only [Ne.def, coe_ne_top, not_false_iff, true_or_iff]
    obtain ⟨t, ht, ts, mlt, t_top⟩ :
      ∃ t : Set α, MeasurableSet t ∧ t ⊆ s ∧ L / ↑c < μ t ∧ μ t < ∞ :=
      Measure.exists_subset_measure_lt_top hs this
    refine' ⟨piecewise t ht (const α c) (const α 0), fun x => _, _, _⟩
    · refine indicator_le_indicator_of_subset ts (fun x => ?_) x
      exact zero_le _
    · simp only [ht, const_zero, coe_piecewise, coe_const, SimpleFunc.coe_zero, univ_inter,
        piecewise_eq_indicator, ENNReal.coe_indicator, Function.const_apply, lintegral_indicator,
        lintegral_const, Measure.restrict_apply', ENNReal.mul_lt_top ENNReal.coe_ne_top t_top.ne]
    · simp only [ht, const_zero, coe_piecewise, coe_const, SimpleFunc.coe_zero,
        piecewise_eq_indicator, ENNReal.coe_indicator, Function.const_apply, lintegral_indicator,
        lintegral_const, Measure.restrict_apply', univ_inter]
      rwa [mul_comm, ← ENNReal.div_lt_iff]
      · simp only [c_ne_zero, Ne.def, coe_eq_zero, not_false_iff, true_or_iff]
      · simp only [Ne.def, coe_ne_top, not_false_iff, true_or_iff]
  · replace hL : L < ∫⁻ x, f₁ x ∂μ + ∫⁻ x, f₂ x ∂μ
    · rwa [← lintegral_add_left f₁.measurable.coe_nnreal_ennreal]
    by_cases hf₁ : ∫⁻ x, f₁ x ∂μ = 0
    · simp only [hf₁, zero_add] at hL
      rcases h₂ hL with ⟨g, g_le, g_top, gL⟩
      refine' ⟨g, fun x => (g_le x).trans _, g_top, gL⟩
      simp only [SimpleFunc.coe_add, Pi.add_apply, le_add_iff_nonneg_left, zero_le']
    by_cases hf₂ : ∫⁻ x, f₂ x ∂μ = 0
    · simp only [hf₂, add_zero] at hL
      rcases h₁ hL with ⟨g, g_le, g_top, gL⟩
      refine' ⟨g, fun x => (g_le x).trans _, g_top, gL⟩
      simp only [SimpleFunc.coe_add, Pi.add_apply, le_add_iff_nonneg_right, zero_le']
    obtain ⟨L₁, L₂, hL₁, hL₂, hL⟩ :
      ∃ L₁ L₂ : ℝ≥0∞, (L₁ < ∫⁻ x, f₁ x ∂μ) ∧ (L₂ < ∫⁻ x, f₂ x ∂μ) ∧ L < L₁ + L₂ :=
      ENNReal.exists_lt_add_of_lt_add hL hf₁ hf₂
    rcases h₁ hL₁ with ⟨g₁, g₁_le, g₁_top, hg₁⟩
    rcases h₂ hL₂ with ⟨g₂, g₂_le, g₂_top, hg₂⟩
    refine' ⟨g₁ + g₂, fun x => add_le_add (g₁_le x) (g₂_le x), _, _⟩
    · apply lt_of_le_of_lt _ (add_lt_top.2 ⟨g₁_top, g₂_top⟩)
      rw [← lintegral_add_left g₁.measurable.coe_nnreal_ennreal]
      exact le_rfl
    · apply hL.trans ((ENNReal.add_lt_add hg₁ hg₂).trans_le _)
      rw [← lintegral_add_left g₁.measurable.coe_nnreal_ennreal]
      

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m✝ m0 : MeasurableSpace α
E : Type u_5
inst✝³ : NormedAddCommGroup E
inst✝² : MeasurableSpace E
inst✝¹ : OpensMeasurableSpace E
m : MeasurableSpace α
μ : Measure α
inst✝ : SigmaFinite μ
f₁ f₂ : α →ₛ ℝ≥0
a✝ : Disjoint (support ↑f₁) (support ↑f₂)
h₁ :
  ∀ {L : ℝ≥0∞},
    L < ∫⁻ (x : α), ↑(↑f₁ x) ∂μ →
      ∃ g, (∀ (x : α), ↑g x ≤ ↑f₁ x) ∧ ∫⁻ (x : α), ↑(↑g x) ∂μ < ⊤ ∧ L < ∫⁻ (x : α), ↑(↑g x) ∂μ
h₂ :
  ∀ {L : ℝ≥0∞},
    L < ∫⁻ (x : α), ↑(↑f₂ x) ∂μ →
      ∃ g, (∀ (x : α), ↑g x ≤ ↑f₂ x) ∧ ∫⁻ (x : α), ↑(↑g x) ∂μ < ⊤ ∧ L < ∫⁻ (x : α), ↑(↑g x) ∂μ
L : ℝ≥0∞
hL✝ : L < ∫⁻ (x : α), ↑(↑f₁ x) ∂μ + ∫⁻ (x : α), ↑(↑f₂ x) ∂μ
hf₁ : ¬∫⁻ (x : α), ↑(↑f₁ x) ∂μ = 0
hf₂ : ¬∫⁻ (x : α), ↑(↑f₂ x) ∂μ = 0
L₁ L₂ : ℝ≥0∞
hL₁ : L₁ < ∫⁻ (x : α), ↑(↑f₁ x) ∂μ
hL₂ : L₂ < ∫⁻ (x : α), ↑(↑f₂ x) ∂μ
hL : L < L₁ + L₂
g₁ : α →ₛ ℝ≥0
g₁_le : ∀ (x : α), ↑g₁ x ≤ ↑f₁ x
g₁_top : ∫⁻ (x : α), ↑(↑g₁ x) ∂μ < ⊤
hg₁ : L₁ < ∫⁻ (x : α), ↑(↑g₁ x) ∂μ
g₂ : α →ₛ ℝ≥0
g₂_le : ∀ (x : α), ↑g₂ x ≤ ↑f₂ x
g₂_top : ∫⁻ (x : α), ↑(↑g₂ x) ∂μ < ⊤
hg₂ : L₂ < ∫⁻ (x : α), ↑(↑g₂ x) ∂μ
⊢ ∫⁻ (a : α), ↑(↑g₁ a) + ↑(↑g₂ a) ∂μ ≤ ∫⁻ (x : α), ↑(↑(g₁ + g₂) x) ∂μ

Next tactic:
simp only [coe_add, Pi.add_apply, ENNReal.coe_add, le_rfl]