Module: Mathlib.MeasureTheory.Function.UniformIntegrable

Goal state before:
pp: 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 (min δ₁ δ₂)
N n : ℕ
hN : ∀ x ∈ tᶜ, dist (g x) (f n x) < ε.toReal / (3 * ↑(measureUnivNNReal μ) ^ (1 / p.toReal))
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)
this✝ : 0 ≤ ε.toReal / (3 * ↑(measureUnivNNReal μ) ^ (1 / p.toReal))
this :
  eLpNorm (tᶜ.indicator (f n - g)) p μ ≤
    ENNReal.ofReal (ε.toReal / (3 * ↑(measureUnivNNReal μ) ^ (1 / p.toReal))) * μ tᶜ ^ (1 / p.toReal)
⊢ ENNReal.ofReal (ε.toReal / 3 * (1 / ↑(measureUnivNNReal μ) ^ (1 / p.toReal) * (μ tᶜ).toReal ^ (1 / p.toReal))) ≤
    ENNReal.ofReal (ε.toReal / 3)

pp: 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 (min δ₁ δ₂)
N n : ℕ
hN : ∀ x ∈ tᶜ, dist (g x) (f n x) < ε.toReal / (3 * ↑(measureUnivNNReal μ) ^ (1 / p.toReal))
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)
this✝ : 0 ≤ ε.toReal / (3 * ↑(measureUnivNNReal μ) ^ (1 / p.toReal))
this :
  eLpNorm (tᶜ.indicator (f n - g)) p μ ≤
    ENNReal.ofReal (ε.toReal / (3 * ↑(measureUnivNNReal μ) ^ (1 / p.toReal))) * μ tᶜ ^ (1 / p.toReal)
⊢ 0 ≤ ε.toReal / 3 * (1 / ↑(measureUnivNNReal μ) ^ (1 / p.toReal))

Tactic:
· refine ENNReal.ofReal_le_ofReal (mul_le_of_le_one_right hε'.le ?_)
      rw [mul_comm, mul_one_div, div_le_one]
      · gcongr
        refine (ENNReal.toReal_le_of_le_ofReal (measureUnivNNReal_pos hμ).le ?_)
        rw [ENNReal.ofReal_coe_nnreal, coe_measureUnivNNReal]
        exact measure_mono (Set.subset_univ _)
      · exact Real.rpow_pos_of_pos (measureUnivNNReal_pos hμ) _

Goal state after:
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 (min δ₁ δ₂)
N n : ℕ
hN : ∀ x ∈ tᶜ, dist (g x) (f n x) < ε.toReal / (3 * ↑(measureUnivNNReal μ) ^ (1 / p.toReal))
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)
this✝ : 0 ≤ ε.toReal / (3 * ↑(measureUnivNNReal μ) ^ (1 / p.toReal))
this :
  eLpNorm (tᶜ.indicator (f n - g)) p μ ≤
    ENNReal.ofReal (ε.toReal / (3 * ↑(measureUnivNNReal μ) ^ (1 / p.toReal))) * μ tᶜ ^ (1 / p.toReal)
⊢ 0 ≤ ε.toReal / 3 * (1 / ↑(measureUnivNNReal μ) ^ (1 / p.toReal))