/-- If `f` is a.e. equal to zero on a rectangular box, then it has McShane integral zero on this
box. -/
theorem HasIntegral.of_aeEq_zero {l : IntegrationParams} {I : Box ι} {f : (ι → ℝ) → E}
    {μ : Measure (ι → ℝ)} [IsLocallyFiniteMeasure μ] (hf : f =ᵐ[μ.restrict I] 0)
    (hl : l.bRiemann = false) : HasIntegral.{u, v, v} I l f μ.toBoxAdditive.toSMul 0 := by
  /- Each set `{x | n < ‖f x‖ ≤ n + 1}`, `n : ℕ`, has measure zero. We cover it by an open set of
    measure less than `ε / 2 ^ n / (n + 1)`. Then the norm of the integral sum is less than `ε`. -/
  refine' hasIntegral_iff.2 fun ε ε0 => _
  lift ε to ℝ≥0 using ε0.lt.le; rw [gt_iff_lt, NNReal.coe_pos] at ε0
  rcases NNReal.exists_pos_sum_of_countable ε0.ne' ℕ with ⟨δ, δ0, c, hδc, hcε⟩
  haveI := Fact.mk (I.measure_coe_lt_top μ)
  change μ.restrict I {x | f x ≠ 0} = 0 at hf
  set N : (ι → ℝ) → ℕ := fun x => ⌈‖f x‖⌉₊
  have N0 : ∀ {x}, N x = 0 ↔ f x = 0 := by simp
  have : ∀ n, ∃ U, N ⁻¹' {n} ⊆ U ∧ IsOpen U ∧ μ.restrict I U < δ n / n := fun n ↦ by
    refine (N ⁻¹' {n}).exists_isOpen_lt_of_lt _ ?_
    cases' n with n
    · simpa [ENNReal.div_zero (ENNReal.coe_pos.2 (δ0 _)).ne'] using measure_lt_top (μ.restrict I) _
    · refine' (measure_mono_null _ hf).le.trans_lt _
      · exact fun x hxN hxf => n.succ_ne_zero ((Eq.symm hxN).trans <| N0.2 hxf)
      · simp [(δ0 _).ne']
  choose U hNU hUo hμU using this
  have : ∀ x, ∃ r : Ioi (0 : ℝ), closedBall x r ⊆ U (N x) := fun x => by
    obtain ⟨r, hr₀, hr⟩ := nhds_basis_closedBall.mem_iff.1 ((hUo _).mem_nhds (hNU _ rfl))
    exact ⟨⟨r, hr₀⟩, hr⟩
  choose r hrU using this
  refine' ⟨fun _ => r, fun c => l.rCond_of_bRiemann_eq_false hl, fun c π hπ _ => _⟩
  rw [dist_eq_norm, sub_zero, ← integralSum_fiberwise fun J => N (π.tag J)]
  refine' le_trans _ (NNReal.coe_lt_coe.2 hcε).le
  refine' (norm_sum_le_of_le _ _).trans
    (sum_le_hasSum _ (fun n _ => (δ n).2) (NNReal.hasSum_coe.2 hδc))
  rintro n -
  dsimp [integralSum]
  have : ∀ J ∈ π.filter fun J => N (π.tag J) = n,
      ‖(μ ↑J).toReal • f (π.tag J)‖ ≤ (μ J).toReal * n := fun J hJ ↦ by
    rw [TaggedPrepartition.mem_filter] at hJ
    rw [norm_smul, Real.norm_eq_abs, abs_of_nonneg ENNReal.toReal_nonneg]
    gcongr
    exact hJ.2 ▸ Nat.le_ceil _
  refine' (norm_sum_le_of_le _ this).trans _; clear this
  rw [← sum_mul, ← Prepartition.measure_iUnion_toReal]
  generalize hm : μ (π.filter fun J => N (π.tag J) = n).iUnion = m
  

Proof state:
case intro.intro.intro.intro.intro
ι : Type u
E : Type v
inst✝³ : Fintype ι
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
l : IntegrationParams
I : Box ι
f : (ι → ℝ) → E
μ : Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
hl : l.bRiemann = false
ε : ℝ≥0
ε0 : 0 < ε
δ : ℕ → ℝ≥0
δ0 : ∀ (i : ℕ), 0 < δ i
c✝ : ℝ≥0
hδc : HasSum δ c✝
hcε : c✝ < ε
this : Fact (↑↑μ ↑I < ⊤)
hf : ↑↑(Measure.restrict μ ↑I) {x | f x ≠ 0} = 0
N : (ι → ℝ) → ℕ := fun x => ⌈‖f x‖⌉₊
N0 : ∀ {x : ι → ℝ}, N x = 0 ↔ f x = 0
U : ℕ → Set (ι → ℝ)
hNU : ∀ (n : ℕ), N ⁻¹' {n} ⊆ U n
hUo : ∀ (n : ℕ), IsOpen (U n)
hμU : ∀ (n : ℕ), ↑↑(Measure.restrict μ ↑I) (U n) < ↑(δ n) / ↑n
r : (ι → ℝ) → ↑(Set.Ioi 0)
hrU : ∀ (x : ι → ℝ), closedBall x ↑(r x) ⊆ U (N x)
c : ℝ≥0
π : TaggedPrepartition I
hπ : IntegrationParams.MemBaseSet l I c ((fun x => r) c) π
x✝ : TaggedPrepartition.IsPartition π
n : ℕ
m : ℝ≥0∞
hm : ↑↑μ (TaggedPrepartition.iUnion (TaggedPrepartition.filter π fun J => N (TaggedPrepartition.tag π J) = n)) = m
⊢ ENNReal.toReal m * ↑n ≤ ↑(δ n)

Next tactic:
have : m < δ n / n := by
    simp only [Measure.restrict_apply (hUo _).measurableSet] at hμU
    refine' hm ▸ (measure_mono _).trans_lt (hμU _)
    simp only [Set.subset_def, TaggedPrepartition.mem_iUnion, TaggedPrepartition.mem_filter]
    rintro x ⟨J, ⟨hJ, rfl⟩, hx⟩
    exact ⟨hrU _ (hπ.1 _ hJ (Box.coe_subset_Icc hx)), π.le_of_mem' J hJ hx⟩