Theorem: MeasureTheory.SimpleFunc.tendsto_approxOn_Lp_eLpNorm
File: Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean

Proof steps:
state_before: 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 : eLpNorm (fun x => f x - y₀) p μ < ⊤
⊢ Tendsto (fun n => eLpNorm (↑(approxOn f hf s y₀ h₀ n) - f) p μ) atTop (𝓝 0)
state_after: case pos
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 : eLpNorm (fun x => f x - y₀) p μ < ⊤
hp_zero : p = 0
⊢ Tendsto (fun n => eLpNorm (↑(approxOn f hf s y₀ h₀ n) - f) p μ) atTop (𝓝 0)

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 : eLpNorm (fun x => f x - y₀) p μ < ⊤
hp_zero : ¬p = 0
⊢ Tendsto (fun n => eLpNorm (↑(approxOn f hf s y₀ h₀ n) - f) p μ) atTop (𝓝 0)
tactic: by_cases hp_zero : p = 0
state_before: 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 : eLpNorm (fun x => f x - y₀) p μ < ⊤
hp_zero : ¬p = 0
⊢ Tendsto (fun n => eLpNorm (↑(approxOn f hf s y₀ h₀ n) - f) p μ) atTop (𝓝 0)
state_after: 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 : eLpNorm (fun x => f x - y₀) p μ < ⊤
hp_zero : ¬p = 0
hp : 0 < p.toReal
⊢ Tendsto (fun n => eLpNorm (↑(approxOn f hf s y₀ h₀ n) - f) p μ) atTop (𝓝 0)
tactic: have hp : 0 < p.toReal := toReal_pos hp_zero hp_ne_top
state_before: 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 : eLpNorm (fun x => f x - y₀) p μ < ⊤
hp_zero : ¬p = 0
hp : 0 < p.toReal
⊢ Tendsto (fun n => eLpNorm (↑(approxOn f hf s y₀ h₀ n) - f) p μ) atTop (𝓝 0)
state_after: 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 : eLpNorm (fun x => f x - y₀) p μ < ⊤
hp_zero : ¬p = 0
hp : 0 < p.toReal
⊢ Tendsto (fun n => ∫⁻ (x : β), ↑‖↑(approxOn f hf s y₀ h₀ n) x - f x‖₊ ^ p.toReal ∂μ) atTop (𝓝 0)
tactic: 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 [eLpNorm_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)]
state_before: 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 : eLpNorm (fun x => f x - y₀) p μ < ⊤
hp_zero : ¬p = 0
hp : 0 < p.toReal
⊢ Tendsto (fun n => ∫⁻ (x : β), ↑‖↑(approxOn f hf s y₀ h₀ n) x - f x‖₊ ^ p.toReal ∂μ) atTop (𝓝 0)
state_after: 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 : eLpNorm (fun x => f x - y₀) p μ < ⊤
hp_zero : ¬p = 0
hp : 0 < p.toReal
hF_meas : ∀ (n : ℕ), Measurable fun x => ↑‖↑(approxOn f hf s y₀ h₀ n) x - f x‖₊ ^ p.toReal
⊢ Tendsto (fun n => ∫⁻ (x : β), ↑‖↑(approxOn f hf s y₀ h₀ n) x - f x‖₊ ^ p.toReal ∂μ) atTop (𝓝 0)
tactic: 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
state_before: 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 : eLpNorm (fun x => f x - y₀) p μ < ⊤
hp_zero : ¬p = 0
hp : 0 < p.toReal
hF_meas : ∀ (n : ℕ), Measurable fun x => ↑‖↑(approxOn f hf s y₀ h₀ n) x - f x‖₊ ^ p.toReal
⊢ Tendsto (fun n => ∫⁻ (x : β), ↑‖↑(approxOn f hf s y₀ h₀ n) x - f x‖₊ ^ p.toReal ∂μ) atTop (𝓝 0)
state_after: 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 : eLpNorm (fun x => f x - y₀) p μ < ⊤
hp_zero : ¬p = 0
hp : 0 < p.toReal
hF_meas : ∀ (n : ℕ), Measurable fun x => ↑‖↑(approxOn f hf s y₀ h₀ n) x - f x‖₊ ^ p.toReal
h_bound :
  ∀ (n : ℕ), (fun x => ↑‖↑(approxOn f hf s y₀ h₀ n) x - f x‖₊ ^ p.toReal) ≤ᶠ[ae μ] fun x => ↑‖f x - y₀‖₊ ^ p.toReal
⊢ Tendsto (fun n => ∫⁻ (x : β), ↑‖↑(approxOn f hf s y₀ h₀ n) x - f x‖₊ ^ p.toReal ∂μ) atTop (𝓝 0)
tactic: 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
state_before: 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 : eLpNorm (fun x => f x - y₀) p μ < ⊤
hp_zero : ¬p = 0
hp : 0 < p.toReal
hF_meas : ∀ (n : ℕ), Measurable fun x => ↑‖↑(approxOn f hf s y₀ h₀ n) x - f x‖₊ ^ p.toReal
h_bound :
  ∀ (n : ℕ), (fun x => ↑‖↑(approxOn f hf s y₀ h₀ n) x - f x‖₊ ^ p.toReal) ≤ᶠ[ae μ] fun x => ↑‖f x - y₀‖₊ ^ p.toReal
⊢ Tendsto (fun n => ∫⁻ (x : β), ↑‖↑(approxOn f hf s y₀ h₀ n) x - f x‖₊ ^ p.toReal ∂μ) atTop (𝓝 0)
state_after: 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 : eLpNorm (fun x => f x - y₀) p μ < ⊤
hp_zero : ¬p = 0
hp : 0 < p.toReal
hF_meas : ∀ (n : ℕ), Measurable fun x => ↑‖↑(approxOn f hf s y₀ h₀ n) x - f x‖₊ ^ p.toReal
h_bound :
  ∀ (n : ℕ), (fun x => ↑‖↑(approxOn f hf s y₀ h₀ n) x - f x‖₊ ^ p.toReal) ≤ᶠ[ae μ] fun x => ↑‖f x - y₀‖₊ ^ p.toReal
h_fin : ∫⁻ (a : β), ↑‖f a - y₀‖₊ ^ p.toReal ∂μ ≠ ⊤
⊢ Tendsto (fun n => ∫⁻ (x : β), ↑‖↑(approxOn f hf s y₀ h₀ n) x - f x‖₊ ^ p.toReal ∂μ) atTop (𝓝 0)
tactic: have h_fin : (∫⁻ a : β, (‖f a - y₀‖₊ : ℝ≥0∞) ^ p.toReal ∂μ) ≠ ⊤ :=
    (lintegral_rpow_nnnorm_lt_top_of_eLpNorm_lt_top hp_zero hp_ne_top hi).ne
state_before: 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 : eLpNorm (fun x => f x - y₀) p μ < ⊤
hp_zero : ¬p = 0
hp : 0 < p.toReal
hF_meas : ∀ (n : ℕ), Measurable fun x => ↑‖↑(approxOn f hf s y₀ h₀ n) x - f x‖₊ ^ p.toReal
h_bound :
  ∀ (n : ℕ), (fun x => ↑‖↑(approxOn f hf s y₀ h₀ n) x - f x‖₊ ^ p.toReal) ≤ᶠ[ae μ] fun x => ↑‖f x - y₀‖₊ ^ p.toReal
h_fin : ∫⁻ (a : β), ↑‖f a - y₀‖₊ ^ p.toReal ∂μ ≠ ⊤
⊢ Tendsto (fun n => ∫⁻ (x : β), ↑‖↑(approxOn f hf s y₀ h₀ n) x - f x‖₊ ^ p.toReal ∂μ) atTop (𝓝 0)
state_after: 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 : eLpNorm (fun x => f x - y₀) p μ < ⊤
hp_zero : ¬p = 0
hp : 0 < p.toReal
hF_meas : ∀ (n : ℕ), Measurable fun x => ↑‖↑(approxOn f hf s y₀ h₀ n) x - f x‖₊ ^ p.toReal
h_bound :
  ∀ (n : ℕ), (fun x => ↑‖↑(approxOn f hf s y₀ h₀ n) x - f x‖₊ ^ p.toReal) ≤ᶠ[ae μ] fun x => ↑‖f x - y₀‖₊ ^ p.toReal
h_fin : ∫⁻ (a : β), ↑‖f a - y₀‖₊ ^ p.toReal ∂μ ≠ ⊤
h_lim : ∀ᵐ (a : β) ∂μ, Tendsto (fun n => ↑‖↑(approxOn f hf s y₀ h₀ n) a - f a‖₊ ^ p.toReal) atTop (𝓝 0)
⊢ Tendsto (fun n => ∫⁻ (x : β), ↑‖↑(approxOn f hf s y₀ h₀ n) x - f x‖₊ ^ p.toReal ∂μ) atTop (𝓝 0)
tactic: 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]
state_before: 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 : eLpNorm (fun x => f x - y₀) p μ < ⊤
hp_zero : ¬p = 0
hp : 0 < p.toReal
hF_meas : ∀ (n : ℕ), Measurable fun x => ↑‖↑(approxOn f hf s y₀ h₀ n) x - f x‖₊ ^ p.toReal
h_bound :
  ∀ (n : ℕ), (fun x => ↑‖↑(approxOn f hf s y₀ h₀ n) x - f x‖₊ ^ p.toReal) ≤ᶠ[ae μ] fun x => ↑‖f x - y₀‖₊ ^ p.toReal
h_fin : ∫⁻ (a : β), ↑‖f a - y₀‖₊ ^ p.toReal ∂μ ≠ ⊤
h_lim : ∀ᵐ (a : β) ∂μ, Tendsto (fun n => ↑‖↑(approxOn f hf s y₀ h₀ n) a - f a‖₊ ^ p.toReal) atTop (𝓝 0)
⊢ Tendsto (fun n => ∫⁻ (x : β), ↑‖↑(approxOn f hf s y₀ h₀ n) x - f x‖₊ ^ p.toReal ∂μ) atTop (𝓝 0)
state_after: no goals
tactic: simpa using tendsto_lintegral_of_dominated_convergence _ hF_meas h_bound h_fin h_lim