/-- If a sequence of peak functions `φᵢ` converges uniformly to zero away from a point `x₀`, and
`g` is integrable and continuous at `x₀`, then `∫ φᵢ • g` converges to `x₀`. Auxiliary lemma
where one assumes additionally `g x₀ = 0`. -/
theorem tendsto_set_integral_peak_smul_of_integrableOn_of_continuousWithinAt_aux
    (hs : MeasurableSet s) (hnφ : ∀ᶠ i in l, ∀ x ∈ s, 0 ≤ φ i x)
    (hlφ : ∀ u : Set α, IsOpen u → x₀ ∈ u → TendstoUniformlyOn φ 0 l (s \ u))
    (hiφ : ∀ᶠ i in l, ∫ x in s, φ i x ∂μ = 1) (hmg : IntegrableOn g s μ) (h'g : g x₀ = 0)
    (hcg : ContinuousWithinAt g s x₀) :
    Tendsto (fun i : ι => ∫ x in s, φ i x • g x ∂μ) l (𝓝 0) := by
  refine' Metric.tendsto_nhds.2 fun ε εpos => _
  obtain ⟨δ, hδ, δpos⟩ : ∃ δ, (δ * ∫ x in s, ‖g x‖ ∂μ) + δ < ε ∧ 0 < δ := by
    have A :
      Tendsto (fun δ => (δ * ∫ x in s, ‖g x‖ ∂μ) + δ) (𝓝[>] 0)
        (𝓝 ((0 * ∫ x in s, ‖g x‖ ∂μ) + 0)) := by
      apply Tendsto.mono_left _ nhdsWithin_le_nhds
      exact (tendsto_id.mul tendsto_const_nhds).add tendsto_id
    rw [zero_mul, zero_add] at A
    exact (((tendsto_order.1 A).2 ε εpos).and self_mem_nhdsWithin).exists
  suffices ∀ᶠ i in l, ‖∫ x in s, φ i x • g x ∂μ‖ ≤ (δ * ∫ x in s, ‖g x‖ ∂μ) + δ by
    filter_upwards [this] with i hi
    simp only [dist_zero_right]
    exact hi.trans_lt hδ
  obtain ⟨u, u_open, x₀u, hu⟩ : ∃ u, IsOpen u ∧ x₀ ∈ u ∧ ∀ x ∈ u ∩ s, g x ∈ ball (g x₀) δ
  exact mem_nhdsWithin.1 (hcg (ball_mem_nhds _ δpos))
  filter_upwards [tendstoUniformlyOn_iff.1 (hlφ u u_open x₀u) δ δpos, hiφ, hnφ,
    integrableOn_peak_smul_of_integrableOn_of_continuousWithinAt hs hlφ hiφ hmg hcg] with i hi h'i
    hφpos h''i
  have B : ‖∫ x in s ∩ u, φ i x • g x ∂μ‖ ≤ δ :=
    calc
      ‖∫ x in s ∩ u, φ i x • g x ∂μ‖ ≤ ∫ x in s ∩ u, ‖φ i x • g x‖ ∂μ :=
        norm_integral_le_integral_norm _
      _ ≤ ∫ x in s ∩ u, ‖φ i x‖ * δ ∂μ := by
        refine' set_integral_mono_on _ _ (hs.inter u_open.measurableSet) fun x hx => _
        · exact IntegrableOn.mono_set h''i.norm (inter_subset_left _ _)
        · exact
            IntegrableOn.mono_set ((integrable_of_integral_eq_one h'i).norm.mul_const _)
              (inter_subset_left _ _)
        rw [norm_smul]
        apply mul_le_mul_of_nonneg_left _ (norm_nonneg _)
        rw [inter_comm, h'g] at hu
        exact (mem_ball_zero_iff.1 (hu x hx)).le
      _ ≤ ∫ x in s, ‖φ i x‖ * δ ∂μ := by
        apply set_integral_mono_set
        · exact (integrable_of_integral_eq_one h'i).norm.mul_const _
        · exact eventually_of_forall fun x => mul_nonneg (norm_nonneg _) δpos.le
        · apply eventually_of_forall; exact inter_subset_left s u
      _ = ∫ x in s, φ i x * δ ∂μ := by
        apply set_integral_congr hs fun x hx => ?_
        rw [Real.norm_of_nonneg (hφpos _ hx)]
      _ = δ := by rw [integral_mul_right, h'i, one_mul]
  have C : ‖∫ x in s \ u, φ i x • g x ∂μ‖ ≤ δ * ∫ x in s, ‖g x‖ ∂μ :=
    calc
      ‖∫ x in s \ u, φ i x • g x ∂μ‖ ≤ ∫ x in s \ u, ‖φ i x • g x‖ ∂μ :=
        norm_integral_le_integral_norm _
      _ ≤ ∫ x in s \ u, δ * ‖g x‖ ∂μ := by
        refine' set_integral_mono_on _ _ (hs.diff u_open.measurableSet) fun x hx => _
        · exact IntegrableOn.mono_set h''i.norm (diff_subset _ _)
        · exact IntegrableOn.mono_set (hmg.norm.const_mul _) (diff_subset _ _)
        rw [norm_smul]
        apply mul_le_mul_of_nonneg_right _ (norm_nonneg _)
        simpa only [Pi.zero_apply, dist_zero_left] using (hi x hx).le
      _ ≤ δ * ∫ x in s, ‖g x‖ ∂μ := by
        rw [integral_mul_left]
        apply mul_le_mul_of_nonneg_left (set_integral_mono_set hmg.norm _ _) δpos.le
        · exact eventually_of_forall fun x => norm_nonneg _
        · apply eventually_of_forall; exact diff_subset s u
  

Proof state:
case h
α : Type u_1
E : Type u_2
ι : Type u_3
hm : MeasurableSpace α
μ : Measure α
inst✝⁴ : TopologicalSpace α
inst✝³ : BorelSpace α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
g : α → E
l : Filter ι
x₀ : α
s : Set α
φ : ι → α → ℝ
inst✝ : CompleteSpace E
hs : MeasurableSet s
hnφ : ∀ᶠ (i : ι) in l, ∀ x ∈ s, 0 ≤ φ i x
hlφ : ∀ (u : Set α), IsOpen u → x₀ ∈ u → TendstoUniformlyOn φ 0 l (s \ u)
hiφ : ∀ᶠ (i : ι) in l, ∫ (x : α) in s, φ i x ∂μ = 1
hmg : IntegrableOn g s
h'g : g x₀ = 0
hcg : ContinuousWithinAt g s x₀
ε : ℝ
εpos : ε > 0
δ : ℝ
hδ : δ * ∫ (x : α) in s, ‖g x‖ ∂μ + δ < ε
δpos : 0 < δ
u : Set α
u_open : IsOpen u
x₀u : x₀ ∈ u
hu : ∀ x ∈ u ∩ s, g x ∈ ball (g x₀) δ
i : ι
hi : ∀ x ∈ s \ u, dist (OfNat.ofNat 0 x) (φ i x) < δ
h'i : ∫ (x : α) in s, φ i x ∂μ = 1
hφpos : ∀ x ∈ s, 0 ≤ φ i x
h''i : IntegrableOn (fun x => φ i x • g x) s
B : ‖∫ (x : α) in s ∩ u, φ i x • g x ∂μ‖ ≤ δ
C : ‖∫ (x : α) in s \ u, φ i x • g x ∂μ‖ ≤ δ * ∫ (x : α) in s, ‖g x‖ ∂μ
⊢ ‖∫ (x : α) in s, φ i x • g x ∂μ‖ ≤ δ * ∫ (x : α) in s, ‖g x‖ ∂μ + δ

Next tactic:
calc
    ‖∫ x in s, φ i x • g x ∂μ‖ =
      ‖(∫ x in s \ u, φ i x • g x ∂μ) + ∫ x in s ∩ u, φ i x • g x ∂μ‖ := by
      conv_lhs => rw [← diff_union_inter s u]
      rw [integral_union (disjoint_sdiff_inter _ _) (hs.inter u_open.measurableSet)
          (h''i.mono_set (diff_subset _ _)) (h''i.mono_set (inter_subset_left _ _))]
    _ ≤ ‖∫ x in s \ u, φ i x • g x ∂μ‖ + ‖∫ x in s ∩ u, φ i x • g x ∂μ‖ := (norm_add_le _ _)
    _ ≤ (δ * ∫ x in s, ‖g x‖ ∂μ) + δ := add_le_add C B