-- Mathlib/MeasureTheory/Function/UniformIntegrable.lean
theorem tendsto_Lp_finite_of_tendsto_ae_of_meas [IsFiniteMeasure μ] (hp : 1 ≤ p) (hp' : p ≠ ∞)
    {f : ℕ → α → β} {g : α → β} (hf : ∀ n, StronglyMeasurable (f n)) (hg : StronglyMeasurable g)
    (hg' : MemLp g p μ) (hui : UnifIntegrable f p μ)
    (hfg : ∀ᵐ x ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))) :
    Tendsto (fun n => eLpNorm (f n - g) p μ) atTop (𝓝 0)

Proof state:
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ℕ → α → β
g : α → β
hf : ∀ (n : ℕ), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hg' : MemLp g p μ
hui : UnifIntegrable f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
ε : ℝ≥0∞
hε : ε > 0
h : ε < ⊤
hμ : ¬μ = 0
hε' : 0 < ε.toReal / 3
hdivp : 0 ≤ 1 / p.toReal
hpow : 0 < measureUnivNNReal μ ^ (1 / p.toReal)
δ₁ : ℝ
hδ₁ : 0 < δ₁
heLpNorm₁ :
  ∀ (i : ℕ) (s : Set α),
    MeasurableSet s → μ s ≤ ENNReal.ofReal δ₁ → eLpNorm (s.indicator (f i)) p μ ≤ ENNReal.ofReal (ε.toReal / 3)
δ₂ : ℝ
hδ₂ : 0 < δ₂
heLpNorm₂ :
  ∀ (s : Set α), MeasurableSet s → μ s ≤ ENNReal.ofReal δ₂ → eLpNorm (s.indicator g) p μ ≤ ENNReal.ofReal (ε.toReal / 3)
t : Set α
htm : MeasurableSet t
ht₁ : μ t ≤ ENNReal.ofReal (δ₁ ⊓ δ₂)
N : ℕ
hN : ∀ b ≥ N, ∀ x ∈ tᶜ, dist (g x) (f b x) < ε.toReal / (3 * ↑(measureUnivNNReal μ) ^ (1 / p.toReal))
n : ℕ
hn : n ≥ N
hnf : eLpNorm (t.indicator (f n)) p μ ≤ ENNReal.ofReal (ε.toReal / 3)
hng : eLpNorm (t.indicator g) p μ ≤ ENNReal.ofReal (ε.toReal / 3)
hlt : eLpNorm (tᶜ.indicator (f n - g)) p μ ≤ ENNReal.ofReal (ε.toReal / 3)
⊢ ε / ENNReal.ofReal 3 = ε / 3

Tactic:
simp

Result:
no goals

Proof state:
case neg
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ℕ → α → β
g : α → β
hf : ∀ (n : ℕ), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hg' : MemLp g p μ
hui : UnifIntegrable f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
ε : ℝ≥0∞
hε : ε > 0
h : ε < ⊤
hμ : ¬μ = 0
hε' : 0 < ε.toReal / 3
⊢ ∃ N, ∀ n ≥ N, eLpNorm (f n - g) p μ ≤ ε

Tactic:
have hdivp : 0 ≤ 1 / p.toReal := by positivity

Result:
case neg
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ℕ → α → β
g : α → β
hf : ∀ (n : ℕ), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hg' : MemLp g p μ
hui : UnifIntegrable f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
ε : ℝ≥0∞
hε : ε > 0
h : ε < ⊤
hμ : ¬μ = 0
hε' : 0 < ε.toReal / 3
hdivp : 0 ≤ 1 / p.toReal
⊢ ∃ N, ∀ n ≥ N, eLpNorm (f n - g) p μ ≤ ε

Proof state:
case neg.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ℕ → α → β
g : α → β
hf : ∀ (n : ℕ), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hg' : MemLp g p μ
hui : UnifIntegrable f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
ε : ℝ≥0∞
hε : ε > 0
h : ε < ⊤
hμ : ¬μ = 0
hε' : 0 < ε.toReal / 3
hdivp : 0 ≤ 1 / p.toReal
hpow : 0 < measureUnivNNReal μ ^ (1 / p.toReal)
δ₁ : ℝ
hδ₁ : 0 < δ₁
heLpNorm₁ :
  ∀ (i : ℕ) (s : Set α),
    MeasurableSet s → μ s ≤ ENNReal.ofReal δ₁ → eLpNorm (s.indicator (f i)) p μ ≤ ENNReal.ofReal (ε.toReal / 3)
δ₂ : ℝ
hδ₂ : 0 < δ₂
heLpNorm₂ :
  ∀ (s : Set α), MeasurableSet s → μ s ≤ ENNReal.ofReal δ₂ → eLpNorm (s.indicator g) p μ ≤ ENNReal.ofReal (ε.toReal / 3)
t : Set α
htm : MeasurableSet t
ht₁ : μ t ≤ ENNReal.ofReal (δ₁ ⊓ δ₂)
ht₂ : ∀ ε > 0, ∀ᶠ (n : ℕ) in atTop, ∀ x ∈ tᶜ, dist (g x) (f n x) < ε
⊢ ∃ N, ∀ n ≥ N, eLpNorm (f n - g) p μ ≤ ε

Tactic:
specialize ht₂ (ε.toReal / (3 * measureUnivNNReal μ ^ (1 / p.toReal)))
  (div_pos (ENNReal.toReal_pos (gt_iff_lt.1 hε).ne.symm h.ne) (mul_pos (by norm_num) hpow))

Result:
case neg.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ℕ → α → β
g : α → β
hf : ∀ (n : ℕ), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hg' : MemLp g p μ
hui : UnifIntegrable f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
ε : ℝ≥0∞
hε : ε > 0
h : ε < ⊤
hμ : ¬μ = 0
hε' : 0 < ε.toReal / 3
hdivp : 0 ≤ 1 / p.toReal
hpow : 0 < measureUnivNNReal μ ^ (1 / p.toReal)
δ₁ : ℝ
hδ₁ : 0 < δ₁
heLpNorm₁ :
  ∀ (i : ℕ) (s : Set α),
    MeasurableSet s → μ s ≤ ENNReal.ofReal δ₁ → eLpNorm (s.indicator (f i)) p μ ≤ ENNReal.ofReal (ε.toReal / 3)
δ₂ : ℝ
hδ₂ : 0 < δ₂
heLpNorm₂ :
  ∀ (s : Set α), MeasurableSet s → μ s ≤ ENNReal.ofReal δ₂ → eLpNorm (s.indicator g) p μ ≤ ENNReal.ofReal (ε.toReal / 3)
t : Set α
htm : MeasurableSet t
ht₁ : μ t ≤ ENNReal.ofReal (δ₁ ⊓ δ₂)
ht₂ : ∀ᶠ (n : ℕ) in atTop, ∀ x ∈ tᶜ, dist (g x) (f n x) < ε.toReal / (3 * ↑(measureUnivNNReal μ) ^ (1 / p.toReal))
⊢ ∃ N, ∀ n ≥ N, eLpNorm (f n - g) p μ ≤ ε

Proof state:
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ℕ → α → β
g : α → β
hf : ∀ (n : ℕ), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hg' : MemLp g p μ
hui : UnifIntegrable f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
ε : ℝ≥0∞
hε : ε > 0
h : ε < ⊤
hμ : ¬μ = 0
hε' : 0 < ε.toReal / 3
hdivp : 0 ≤ 1 / p.toReal
hpow : 0 < measureUnivNNReal μ ^ (1 / p.toReal)
δ₁ : ℝ
hδ₁ : 0 < δ₁
heLpNorm₁ :
  ∀ (i : ℕ) (s : Set α),
    MeasurableSet s → μ s ≤ ENNReal.ofReal δ₁ → eLpNorm (s.indicator (f i)) p μ ≤ ENNReal.ofReal (ε.toReal / 3)
δ₂ : ℝ
hδ₂ : 0 < δ₂
heLpNorm₂ :
  ∀ (s : Set α), MeasurableSet s → μ s ≤ ENNReal.ofReal δ₂ → eLpNorm (s.indicator g) p μ ≤ ENNReal.ofReal (ε.toReal / 3)
t : Set α
htm : MeasurableSet t
ht₁ : μ t ≤ ENNReal.ofReal (δ₁ ⊓ δ₂)
N : ℕ
hN : ∀ b ≥ N, ∀ x ∈ tᶜ, dist (g x) (f b x) < ε.toReal / (3 * ↑(measureUnivNNReal μ) ^ (1 / p.toReal))
n : ℕ
hn : n ≥ N
hnf : eLpNorm (t.indicator (f n)) p μ ≤ ENNReal.ofReal (ε.toReal / 3)
⊢ ENNReal.ofReal (δ₁ ⊓ δ₂) ≤ ENNReal.ofReal δ₂

Tactic:
rw [ENNReal.ofReal_le_ofReal_iff hδ₂.le]

Result:
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ℕ → α → β
g : α → β
hf : ∀ (n : ℕ), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hg' : MemLp g p μ
hui : UnifIntegrable f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
ε : ℝ≥0∞
hε : ε > 0
h : ε < ⊤
hμ : ¬μ = 0
hε' : 0 < ε.toReal / 3
hdivp : 0 ≤ 1 / p.toReal
hpow : 0 < measureUnivNNReal μ ^ (1 / p.toReal)
δ₁ : ℝ
hδ₁ : 0 < δ₁
heLpNorm₁ :
  ∀ (i : ℕ) (s : Set α),
    MeasurableSet s → μ s ≤ ENNReal.ofReal δ₁ → eLpNorm (s.indicator (f i)) p μ ≤ ENNReal.ofReal (ε.toReal / 3)
δ₂ : ℝ
hδ₂ : 0 < δ₂
heLpNorm₂ :
  ∀ (s : Set α), MeasurableSet s → μ s ≤ ENNReal.ofReal δ₂ → eLpNorm (s.indicator g) p μ ≤ ENNReal.ofReal (ε.toReal / 3)
t : Set α
htm : MeasurableSet t
ht₁ : μ t ≤ ENNReal.ofReal (δ₁ ⊓ δ₂)
N : ℕ
hN : ∀ b ≥ N, ∀ x ∈ tᶜ, dist (g x) (f b x) < ε.toReal / (3 * ↑(measureUnivNNReal μ) ^ (1 / p.toReal))
n : ℕ
hn : n ≥ N
hnf : eLpNorm (t.indicator (f n)) p μ ≤ ENNReal.ofReal (ε.toReal / 3)
⊢ δ₁ ⊓ δ₂ ≤ δ₂

Proof state:
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ℕ → α → β
g : α → β
hf : ∀ (n : ℕ), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hg' : MemLp g p μ
hui : UnifIntegrable f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
ε : ℝ≥0∞
hε : ε > 0
h : ε < ⊤
hμ : μ = 0
n : ℕ
x✝ : n ≥ 0
⊢ eLpNorm (f n - g) p μ ≤ ε

Tactic:
simp [hμ]

Result:
no goals