/-- If the average of a function `f` along a sequence of sets `aₙ` converges to `c` (more precisely,
we require that `⨍ y in a i, ‖f y - c‖ ∂μ` tends to `0`), then the integral of `gₙ • f` also tends
to `c` if `gₙ` is supported in `aₙ`, has integral converging to one and supremum at most `K / μ aₙ`.
-/
theorem tendsto_integral_smul_of_tendsto_average_norm_sub
    {ι : Type*} {a : ι → Set α} {l : Filter ι} {f : α → E} {c : E} {g : ι → α → ℝ} (K : ℝ)
    (hf : Tendsto (fun i ↦ ⨍ y in a i, ‖f y - c‖ ∂μ) l (𝓝 0))
    (f_int : ∀ᶠ i in l, IntegrableOn f (a i) μ)
    (hg : Tendsto (fun i ↦ ∫ y, g i y ∂μ) l (𝓝 1))
    (g_supp : ∀ᶠ i in l, Function.support (g i) ⊆ a i)
    (g_bound : ∀ᶠ i in l, ∀ x, |g i x| ≤ K / (μ (a i)).toReal) :
    Tendsto (fun i ↦ ∫ y, g i y • f y ∂μ) l (𝓝 c) := by
  have g_int : ∀ᶠ i in l, Integrable (g i) μ := by
    filter_upwards [(tendsto_order.1 hg).1 _ zero_lt_one] with i hi
    contrapose hi
    simp only [integral_undef hi, lt_self_iff_false, not_false_eq_true]
  have I : ∀ᶠ i in l, ∫ y, g i y • (f y - c) ∂μ + (∫ y, g i y ∂μ) • c = ∫ y, g i y • f y ∂μ := by
    filter_upwards [f_int, g_int, g_supp, g_bound] with i hif hig hisupp hibound
    rw [← integral_smul_const, ← integral_add]
    · simp only [smul_sub, sub_add_cancel]
    · simp_rw [smul_sub]
      apply Integrable.sub _ (hig.smul_const _)
      have A : Function.support (fun y ↦ g i y • f y) ⊆ a i := by
        apply Subset.trans _ hisupp
        exact Function.support_smul_subset_left _ _
      rw [← integrableOn_iff_integrable_of_support_subset A]
      apply Integrable.smul_of_top_right hif
      exact memℒp_top_of_bound hig.aestronglyMeasurable.restrict
        (K / (μ (a i)).toReal) (eventually_of_forall hibound)
    · exact hig.smul_const _
  have L0 : Tendsto (fun i ↦ ∫ y, g i y • (f y - c) ∂μ) l (𝓝 0) := by
    have := hf.const_mul K
    simp only [mul_zero] at this
    refine' squeeze_zero_norm' _ this
    filter_upwards [g_supp, g_bound, f_int, (tendsto_order.1 hg).1 _ zero_lt_one]
      with i hi h'i h''i hi_int
    have mu_ai : μ (a i) < ∞ := by
      rw [lt_top_iff_ne_top]
      intro h
      simp only [h, ENNReal.top_toReal, _root_.div_zero, abs_nonpos_iff] at h'i
      have : ∫ (y : α), g i y ∂μ = ∫ (y : α), 0 ∂μ := by congr; ext y; exact h'i y
      simp [this] at hi_int
    apply (norm_integral_le_integral_norm _).trans
    simp_rw [average_eq, smul_eq_mul, ← integral_mul_left, norm_smul, ← mul_assoc, ← div_eq_mul_inv]
    

Proof state:
case h
α : Type u_1
E : Type u_2
F : Type u_3
m0 : MeasurableSpace α
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : CompleteSpace E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace F
μ ν : Measure α
s t : Set α
ι : Type u_4
a : ι → Set α
l : Filter ι
f : α → E
c : E
g : ι → α → ℝ
K : ℝ
hf : Tendsto (fun i => ⨍ (y : α) in a i, ‖f y - c‖ ∂μ) l (𝓝 0)
f_int : ∀ᶠ (i : ι) in l, IntegrableOn f (a i)
hg : Tendsto (fun i => ∫ (y : α), g i y ∂μ) l (𝓝 1)
g_supp : ∀ᶠ (i : ι) in l, support (g i) ⊆ a i
g_bound : ∀ᶠ (i : ι) in l, ∀ (x : α), |g i x| ≤ K / ENNReal.toReal (↑↑μ (a i))
g_int : ∀ᶠ (i : ι) in l, Integrable (g i)
I : ∀ᶠ (i : ι) in l, ∫ (y : α), g i y • (f y - c) ∂μ + (∫ (y : α), g i y ∂μ) • c = ∫ (y : α), g i y • f y ∂μ
this : Tendsto (fun k => K * ⨍ (y : α) in a k, ‖f y - c‖ ∂μ) l (𝓝 0)
i : ι
hi : support (g i) ⊆ a i
h'i : ∀ (x : α), |g i x| ≤ K / ENNReal.toReal (↑↑μ (a i))
h''i : IntegrableOn f (a i)
hi_int : 0 < ∫ (y : α), g i y ∂μ
mu_ai : ↑↑μ (a i) < ⊤
⊢ ∫ (a : α), ‖g i a‖ * ‖f a - c‖ ∂μ ≤
    ∫ (a_1 : α) in a i, K / ENNReal.toReal (↑↑(Measure.restrict μ (a i)) univ) * ‖f a_1 - c‖ ∂μ

Next tactic:
have : ∀ x, x ∉ a i → ‖g i x‖ * ‖(f x - c)‖ = 0 := by
      intro x hx
      have : g i x = 0 := by rw [← Function.nmem_support]; exact fun h ↦ hx (hi h)
      simp [this]