/-- **Monotone convergence theorem** -- sometimes called **Beppo-Levi convergence**.
See `lintegral_iSup_directed` for a more general form. -/
theorem lintegral_iSup {f : ℕ → α → ℝ≥0∞} (hf : ∀ n, Measurable (f n)) (h_mono : Monotone f) :
    ∫⁻ a, ⨆ n, f n a ∂μ = ⨆ n, ∫⁻ a, f n a ∂μ := by
  set c : ℝ≥0 → ℝ≥0∞ := (↑)
  set F := fun a : α => ⨆ n, f n a
  have _ : Measurable F := measurable_iSup hf
  refine' le_antisymm _ (iSup_lintegral_le _)
  rw [lintegral_eq_nnreal]
  refine' iSup_le fun s => iSup_le fun hsf => _
  refine' ENNReal.le_of_forall_lt_one_mul_le fun a ha => _
  rcases ENNReal.lt_iff_exists_coe.1 ha with ⟨r, rfl, _⟩
  have ha : r < 1 := ENNReal.coe_lt_coe.1 ha
  let rs := s.map fun a => r * a
  have eq_rs : (const α r : α →ₛ ℝ≥0∞) * map c s = rs.map c := by
    ext1 a
    exact ENNReal.coe_mul.symm
  have eq : ∀ p, rs.map c ⁻¹' {p} = ⋃ n, rs.map c ⁻¹' {p} ∩ { a | p ≤ f n a } := by
    intro p
    rw [← inter_iUnion]; nth_rw 1 [← inter_univ (map c rs ⁻¹' {p})]
    refine' Set.ext fun x => and_congr_right fun hx => true_iff_iff.2 _
    by_cases p_eq : p = 0
    · simp [p_eq]
    simp only [coe_map, mem_preimage, Function.comp_apply, mem_singleton_iff] at hx
    subst hx
    have : r * s x ≠ 0 := by rwa [Ne, ← ENNReal.coe_eq_zero]
    have : s x ≠ 0 := by
      refine' mt _ this
      intro h
      rw [h, mul_zero]
    have : (rs.map c) x < ⨆ n : ℕ, f n x := by
      refine' lt_of_lt_of_le (ENNReal.coe_lt_coe.2 _) (hsf x)
      suffices r * s x < 1 * s x by simpa
      exact mul_lt_mul_of_pos_right ha (pos_iff_ne_zero.2 this)
    rcases lt_iSup_iff.1 this with ⟨i, hi⟩
    exact mem_iUnion.2 ⟨i, le_of_lt hi⟩
  have mono : ∀ r : ℝ≥0∞, Monotone fun n => rs.map c ⁻¹' {r} ∩ { a | r ≤ f n a } := by
    intro r i j h
    refine' inter_subset_inter (Subset.refl _) _
    intro x (hx : r ≤ f i x)
    exact le_trans hx (h_mono h x)
  have h_meas : ∀ n, MeasurableSet { a : α | (⇑(map c rs)) a ≤ f n a } := fun n =>
    measurableSet_le (SimpleFunc.measurable _) (hf n)
  calc
    (r : ℝ≥0∞) * (s.map c).lintegral μ = ∑ r in (rs.map c).range, r * μ (rs.map c ⁻¹' {r}) := by
      rw [← const_mul_lintegral, eq_rs, SimpleFunc.lintegral]
    _ = ∑ r in (rs.map c).range, r * μ (⋃ n, rs.map c ⁻¹' {r} ∩ { a | r ≤ f n a }) := by
      simp only [(eq _).symm]
    _ = ∑ r in (rs.map c).range, ⨆ n, r * μ (rs.map c ⁻¹' {r} ∩ { a | r ≤ f n a }) :=
      (Finset.sum_congr rfl fun x _ => by
        rw [measure_iUnion_eq_iSup (mono x).directed_le, ENNReal.mul_iSup])
    _ = ⨆ n, ∑ r in (rs.map c).range, r * μ (rs.map c ⁻¹' {r} ∩ { a | r ≤ f n a }) := by
      rw [ENNReal.finset_sum_iSup_nat]
      intro p i j h
      exact mul_le_mul_left' (measure_mono <| mono p h) _
    _ ≤ ⨆ n : ℕ, ((rs.map c).restrict { a | (rs.map c) a ≤ f n a }).lintegral μ := by
      gcongr with n
      rw [restrict_lintegral _ (h_meas n)]
      · 

Proof state:
case h
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
f : ℕ → α → ℝ≥0∞
hf : ∀ (n : ℕ), Measurable (f n)
h_mono : Monotone f
c : ℝ≥0 → ℝ≥0∞ := ofNNReal
F : α → ℝ≥0∞ := fun a => ⨆ n, f n a
x✝ : Measurable F
s : α →ₛ ℝ≥0
hsf : ∀ (x : α), ↑(↑s x) ≤ ⨆ n, f n x
r : ℝ≥0
right✝ : ↑r < 1
ha✝ : ↑r < 1
ha : r < 1
rs : α →ₛ ℝ≥0 := SimpleFunc.map (fun a => r * a) s
eq_rs : const α ↑r * SimpleFunc.map c s = SimpleFunc.map c rs
eq : ∀ (p : ℝ≥0∞), ↑(SimpleFunc.map c rs) ⁻¹' {p} = ⋃ n, ↑(SimpleFunc.map c rs) ⁻¹' {p} ∩ {a | p ≤ f n a}
mono : ∀ (r : ℝ≥0∞), Monotone fun n => ↑(SimpleFunc.map c rs) ⁻¹' {r} ∩ {a | r ≤ f n a}
h_meas : ∀ (n : ℕ), MeasurableSet {a | ↑(SimpleFunc.map c rs) a ≤ f n a}
n : ℕ
⊢ ∑ r in SimpleFunc.range (SimpleFunc.map c rs), r * ↑↑μ (↑(SimpleFunc.map c rs) ⁻¹' {r} ∩ {a | r ≤ f n a}) ≤
    ∑ r in SimpleFunc.range (SimpleFunc.map c rs),
      r * ↑↑μ (↑(SimpleFunc.map c rs) ⁻¹' {r} ∩ {a | ↑(SimpleFunc.map c rs) a ≤ f n a})

Next tactic:
refine' le_of_eq (Finset.sum_congr rfl fun r _ => _)