theorem tendsto_approxOn_Lp_snorm [OpensMeasurableSpace E] {f : β → E} (hf : Measurable f)
    {s : Set E} {y₀ : E} (h₀ : y₀ ∈ s) [SeparableSpace s] (hp_ne_top : p ≠ ∞) {μ : Measure β}
    (hμ : ∀ᵐ x ∂μ, f x ∈ closure s) (hi : snorm (fun x => f x - y₀) p μ < ∞) :
    Tendsto (fun n => snorm (⇑(approxOn f hf s y₀ h₀ n) - f) p μ) atTop (𝓝 0) := by
  by_cases hp_zero : p = 0
  · simpa only [hp_zero, snorm_exponent_zero] using tendsto_const_nhds
  have hp : 0 < p.toReal := toReal_pos hp_zero hp_ne_top
  suffices
      Tendsto (fun n => ∫⁻ x, (‖approxOn f hf s y₀ h₀ n x - f x‖₊ : ℝ≥0∞) ^ p.toReal ∂μ) atTop
        (𝓝 0) by
    simp only [snorm_eq_lintegral_rpow_nnnorm hp_zero hp_ne_top]
    convert continuous_rpow_const.continuousAt.tendsto.comp this
    simp [zero_rpow_of_pos (_root_.inv_pos.mpr hp)]
  -- We simply check the conditions of the Dominated Convergence Theorem:
  -- (1) The function "`p`-th power of distance between `f` and the approximation" is measurable
  have hF_meas :
    ∀ n, Measurable fun x => (‖approxOn f hf s y₀ h₀ n x - f x‖₊ : ℝ≥0∞) ^ p.toReal := by
    simpa only [← edist_eq_coe_nnnorm_sub] using fun n =>
      (approxOn f hf s y₀ h₀ n).measurable_bind (fun y x => edist y (f x) ^ p.toReal) fun y =>
        (measurable_edist_right.comp hf).pow_const p.toReal
  -- (2) The functions "`p`-th power of distance between `f` and the approximation" are uniformly
  -- bounded, at any given point, by `fun x => ‖f x - y₀‖ ^ p.toReal`
  have h_bound :
    ∀ n, (fun x => (‖approxOn f hf s y₀ h₀ n x - f x‖₊ : ℝ≥0∞) ^ p.toReal) ≤ᵐ[μ] fun x =>
        (‖f x - y₀‖₊ : ℝ≥0∞) ^ p.toReal :=
    fun n =>
    eventually_of_forall fun x =>
      rpow_le_rpow (coe_mono (nnnorm_approxOn_le hf h₀ x n)) toReal_nonneg
  -- (3) The bounding function `fun x => ‖f x - y₀‖ ^ p.toReal` has finite integral
  have h_fin : (∫⁻ a : β, (‖f a - y₀‖₊ : ℝ≥0∞) ^ p.toReal ∂μ) ≠ ⊤ :=
    (lintegral_rpow_nnnorm_lt_top_of_snorm_lt_top hp_zero hp_ne_top hi).ne
  -- (4) The functions "`p`-th power of distance between `f` and the approximation" tend pointwise
  -- to zero
  have h_lim :
    ∀ᵐ a : β ∂μ,
      Tendsto (fun n => (‖approxOn f hf s y₀ h₀ n a - f a‖₊ : ℝ≥0∞) ^ p.toReal) atTop (𝓝 0) := by
    filter_upwards [hμ] with a ha
    have : Tendsto (fun n => (approxOn f hf s y₀ h₀ n) a - f a) atTop (𝓝 (f a - f a)) :=
      (tendsto_approxOn hf h₀ ha).sub tendsto_const_nhds
    convert continuous_rpow_const.continuousAt.tendsto.comp (tendsto_coe.mpr this.nnnorm)
    simp [zero_rpow_of_pos hp]
  -- Then we apply the Dominated Convergence Theorem
  

Proof state:
case neg
α : Type u_1
β : Type u_2
ι : Type u_3
E : Type u_4
F : Type u_5
𝕜 : Type u_6
inst✝⁵ : MeasurableSpace β
inst✝⁴ : MeasurableSpace E
inst✝³ : NormedAddCommGroup E
inst✝² : NormedAddCommGroup F
q : ℝ
p : ℝ≥0∞
inst✝¹ : OpensMeasurableSpace E
f : β → E
hf : Measurable f
s : Set E
y₀ : E
h₀ : y₀ ∈ s
inst✝ : SeparableSpace ↑s
hp_ne_top : p ≠ ⊤
μ : Measure β
hμ : ∀ᵐ (x : β) ∂μ, f x ∈ closure s
hi : snorm (fun x => f x - y₀) p μ < ⊤
hp_zero : ¬p = 0
hp : 0 < ENNReal.toReal p
hF_meas : ∀ (n : ℕ), Measurable fun x => ↑‖↑(approxOn f hf s y₀ h₀ n) x - f x‖₊ ^ ENNReal.toReal p
h_bound :
  ∀ (n : ℕ),
    (fun x => ↑‖↑(approxOn f hf s y₀ h₀ n) x - f x‖₊ ^ ENNReal.toReal p) ≤ᵐ[μ] fun x => ↑‖f x - y₀‖₊ ^ ENNReal.toReal p
h_fin : ∫⁻ (a : β), ↑‖f a - y₀‖₊ ^ ENNReal.toReal p ∂μ ≠ ⊤
h_lim : ∀ᵐ (a : β) ∂μ, Tendsto (fun n => ↑‖↑(approxOn f hf s y₀ h₀ n) a - f a‖₊ ^ ENNReal.toReal p) atTop (𝓝 0)
⊢ Tendsto (fun n => ∫⁻ (x : β), ↑‖↑(approxOn f hf s y₀ h₀ n) x - f x‖₊ ^ ENNReal.toReal p ∂μ) atTop (𝓝 0)

Next tactic:
simpa using tendsto_lintegral_of_dominated_convergence _ hF_meas h_bound h_fin h_lim