/-- The indicator function of a measurable set is McShane integrable with respect to any
locally-finite measure. -/
theorem hasIntegralIndicatorConst (l : IntegrationParams) (hl : l.bRiemann = false)
    {s : Set (ι → ℝ)} (hs : MeasurableSet s) (I : Box ι) (y : E) (μ : Measure (ι → ℝ))
    [IsLocallyFiniteMeasure μ] :
    HasIntegral.{u, v, v} I l (s.indicator fun _ => y) μ.toBoxAdditive.toSMul
      ((μ (s ∩ I)).toReal • y) := by
  refine' HasIntegral.of_mul ‖y‖ fun ε ε0 => _
  lift ε to ℝ≥0 using ε0.le; rw [NNReal.coe_pos] at ε0
  /- First we choose a closed set `F ⊆ s ∩ I.Icc` and an open set `U ⊇ s` such that
    both `(s ∩ I.Icc) \ F` and `U \ s` have measure less than `ε`. -/
  have A : μ (s ∩ Box.Icc I) ≠ ∞ :=
    ((measure_mono <| Set.inter_subset_right _ _).trans_lt (I.measure_Icc_lt_top μ)).ne
  have B : μ (s ∩ I) ≠ ∞ :=
    ((measure_mono <| Set.inter_subset_right _ _).trans_lt (I.measure_coe_lt_top μ)).ne
  obtain ⟨F, hFs, hFc, hμF⟩ : ∃ F, F ⊆ s ∩ Box.Icc I ∧ IsClosed F ∧ μ ((s ∩ Box.Icc I) \ F) < ε :=
    (hs.inter I.measurableSet_Icc).exists_isClosed_diff_lt A (ENNReal.coe_pos.2 ε0).ne'
  obtain ⟨U, hsU, hUo, hUt, hμU⟩ :
      ∃ U, s ∩ Box.Icc I ⊆ U ∧ IsOpen U ∧ μ U < ∞ ∧ μ (U \ (s ∩ Box.Icc I)) < ε :=
    (hs.inter I.measurableSet_Icc).exists_isOpen_diff_lt A (ENNReal.coe_pos.2 ε0).ne'
  /- Then we choose `r` so that `closed_ball x (r x) ⊆ U` whenever `x ∈ s ∩ I.Icc` and
    `closed_ball x (r x)` is disjoint with `F` otherwise. -/
  have : ∀ x ∈ s ∩ Box.Icc I, ∃ r : Ioi (0 : ℝ), closedBall x r ⊆ U := fun x hx => by
    rcases nhds_basis_closedBall.mem_iff.1 (hUo.mem_nhds <| hsU hx) with ⟨r, hr₀, hr⟩
    exact ⟨⟨r, hr₀⟩, hr⟩
  choose! rs hrsU using this
  have : ∀ x ∈ Box.Icc I \ s, ∃ r : Ioi (0 : ℝ), closedBall x r ⊆ Fᶜ := fun x hx => by
    obtain ⟨r, hr₀, hr⟩ :=
      nhds_basis_closedBall.mem_iff.1 (hFc.isOpen_compl.mem_nhds fun hx' => hx.2 (hFs hx').1)
    exact ⟨⟨r, hr₀⟩, hr⟩
  choose! rs' hrs'F using this
  set r : (ι → ℝ) → Ioi (0 : ℝ) := s.piecewise rs rs'
  refine' ⟨fun _ => r, fun c => l.rCond_of_bRiemann_eq_false hl, fun c π hπ hπp => _⟩; rw [mul_comm]
  /- Then the union of boxes `J ∈ π` such that `π.tag ∈ s` includes `F` and is included by `U`,
    hence its measure is `ε`-close to the measure of `s`. -/
  dsimp [integralSum]
  simp only [mem_closedBall, dist_eq_norm, ← indicator_const_smul_apply,
    sum_indicator_eq_sum_filter, ← sum_smul, ← sub_smul, norm_smul, Real.norm_eq_abs, ←
    Prepartition.filter_boxes, ← Prepartition.measure_iUnion_toReal]
  gcongr
  set t := (π.filter (π.tag · ∈ s)).iUnion
  change abs ((μ t).toReal - (μ (s ∩ I)).toReal) ≤ ε
  have htU : t ⊆ U ∩ I := by
    simp only [TaggedPrepartition.iUnion_def, iUnion_subset_iff, TaggedPrepartition.mem_filter,
      and_imp]
    refine' fun J hJ hJs x hx => ⟨hrsU _ ⟨hJs, π.tag_mem_Icc J⟩ _, π.le_of_mem' J hJ hx⟩
    simpa only [s.piecewise_eq_of_mem _ _ hJs] using hπ.1 J hJ (Box.coe_subset_Icc hx)
  refine' abs_sub_le_iff.2 ⟨_, _⟩
  · refine' (ENNReal.le_toReal_sub B).trans (ENNReal.toReal_le_coe_of_le_coe _)
    refine' (tsub_le_tsub (measure_mono htU) le_rfl).trans (le_measure_diff.trans _)
    refine' (measure_mono fun x hx => _).trans hμU.le
    exact ⟨hx.1.1, fun hx' => hx.2 ⟨hx'.1, hx.1.2⟩⟩
  · have hμt : μ t ≠ ∞ := ((measure_mono (htU.trans (inter_subset_left _ _))).trans_lt hUt).ne
    refine' (ENNReal.le_toReal_sub hμt).trans (ENNReal.toReal_le_coe_of_le_coe _)
    refine' le_measure_diff.trans ((measure_mono _).trans hμF.le)
    rintro x ⟨⟨hxs, hxI⟩, hxt⟩
    refine' ⟨⟨hxs, Box.coe_subset_Icc hxI⟩, fun hxF => hxt _⟩
    simp only [TaggedPrepartition.iUnion_def, TaggedPrepartition.mem_filter, Set.mem_iUnion,
      exists_prop]
    rcases hπp x hxI with ⟨J, hJπ, hxJ⟩
    refine' ⟨J, ⟨hJπ, _⟩, hxJ⟩
    contrapose hxF
    refine' hrs'F _ ⟨π.tag_mem_Icc J, hxF⟩ _
    

Proof state:
case intro.intro.intro.intro.intro.intro.intro.intro.h.refine'_2.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
hl : l.bRiemann = false
s : Set (ι → ℝ)
hs : MeasurableSet s
I : Box ι
y : E
μ : Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
ε : ℝ≥0
ε0 : 0 < ε
A : ↑↑μ (s ∩ Box.Icc I) ≠ ⊤
B : ↑↑μ (s ∩ ↑I) ≠ ⊤
F : Set (ι → ℝ)
hFs : F ⊆ s ∩ Box.Icc I
hFc : IsClosed F
hμF : ↑↑μ ((s ∩ Box.Icc I) \ F) < ↑ε
U : Set (ι → ℝ)
hsU : s ∩ Box.Icc I ⊆ U
hUo : IsOpen U
hUt : ↑↑μ U < ⊤
hμU : ↑↑μ (U \ (s ∩ Box.Icc I)) < ↑ε
rs : (ι → ℝ) → ↑(Set.Ioi 0)
hrsU : ∀ x ∈ s ∩ Box.Icc I, closedBall x ↑(rs x) ⊆ U
rs' : (ι → ℝ) → ↑(Set.Ioi 0)
hrs'F : ∀ x ∈ Box.Icc I \ s, closedBall x ↑(rs' x) ⊆ Fᶜ
r : (ι → ℝ) → ↑(Set.Ioi 0) := Set.piecewise s rs rs'
c : ℝ≥0
π : TaggedPrepartition I
hπ : IntegrationParams.MemBaseSet l I c ((fun x => r) c) π
hπp : TaggedPrepartition.IsPartition π
t : Set (ι → ℝ) := TaggedPrepartition.iUnion (TaggedPrepartition.filter π fun x => TaggedPrepartition.tag π x ∈ s)
htU : t ⊆ U ∩ ↑I
hμt : ↑↑μ t ≠ ⊤
x : ι → ℝ
hxt : x ∉ t
hxs : x ∈ s
hxI : x ∈ ↑I
J : Box ι
hJπ : J ∈ π.toPrepartition
hxJ : x ∈ J
hxF : TaggedPrepartition.tag π J ∉ s
⊢ x ∈ closedBall (TaggedPrepartition.tag π J) ↑(rs' (TaggedPrepartition.tag π J))

Next tactic:
simpa only [s.piecewise_eq_of_not_mem _ _ hxF] using hπ.1 J hJπ (Box.coe_subset_Icc hxJ)