Theorem: MeasureTheory.UnifIntegrable.add
File: Mathlib/MeasureTheory/Function/UniformIntegrable.lean

tactic: intro ε hε
state_before: α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f g : ι → α → β
p : ℝ≥0∞
hf : UnifIntegrable f p μ
hg : UnifIntegrable g p μ
hp : 1 ≤ p
hf_meas : ∀ (i : ι), AEStronglyMeasurable (f i) μ
hg_meas : ∀ (i : ι), AEStronglyMeasurable (g i) μ
⊢ UnifIntegrable (f + g) p μ

tactic: have hε2 : 0 < ε / 2 := half_pos hε
state_before: α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f g : ι → α → β
p : ℝ≥0∞
hf : UnifIntegrable f p μ
hg : UnifIntegrable g p μ
hp : 1 ≤ p
hf_meas : ∀ (i : ι), AEStronglyMeasurable (f i) μ
hg_meas : ∀ (i : ι), AEStronglyMeasurable (g i) μ
ε : ℝ
hε : 0 < ε
⊢ ∃ δ x,
    ∀ (i : ι) (s : Set α),
      MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (Set.indicator s ((f + g) i)) p μ ≤ ENNReal.ofReal ε

tactic: obtain ⟨δ₁, hδ₁_pos, hfδ₁⟩ := hf hε2
state_before: α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f g : ι → α → β
p : ℝ≥0∞
hf : UnifIntegrable f p μ
hg : UnifIntegrable g p μ
hp : 1 ≤ p
hf_meas : ∀ (i : ι), AEStronglyMeasurable (f i) μ
hg_meas : ∀ (i : ι), AEStronglyMeasurable (g i) μ
ε : ℝ
hε : 0 < ε
hε2 : 0 < ε / 2
⊢ ∃ δ x,
    ∀ (i : ι) (s : Set α),
      MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (Set.indicator s ((f + g) i)) p μ ≤ ENNReal.ofReal ε

tactic: obtain ⟨δ₂, hδ₂_pos, hgδ₂⟩ := hg hε2
state_before: case intro.intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f g : ι → α → β
p : ℝ≥0∞
hf : UnifIntegrable f p μ
hg : UnifIntegrable g p μ
hp : 1 ≤ p
hf_meas : ∀ (i : ι), AEStronglyMeasurable (f i) μ
hg_meas : ∀ (i : ι), AEStronglyMeasurable (g i) μ
ε : ℝ
hε : 0 < ε
hε2 : 0 < ε / 2
δ₁ : ℝ
hδ₁_pos : 0 < δ₁
hfδ₁ :
  ∀ (i : ι) (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₁ → snorm (Set.indicator s (f i)) p μ ≤ ENNReal.ofReal (ε / 2)
⊢ ∃ δ x,
    ∀ (i : ι) (s : Set α),
      MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (Set.indicator s ((f + g) i)) p μ ≤ ENNReal.ofReal ε

tactic: refine' ⟨min δ₁ δ₂, lt_min hδ₁_pos hδ₂_pos, fun i s hs hμs => _⟩
state_before: case intro.intro.intro.intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f g : ι → α → β
p : ℝ≥0∞
hf : UnifIntegrable f p μ
hg : UnifIntegrable g p μ
hp : 1 ≤ p
hf_meas : ∀ (i : ι), AEStronglyMeasurable (f i) μ
hg_meas : ∀ (i : ι), AEStronglyMeasurable (g i) μ
ε : ℝ
hε : 0 < ε
hε2 : 0 < ε / 2
δ₁ : ℝ
hδ₁_pos : 0 < δ₁
hfδ₁ :
  ∀ (i : ι) (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₁ → snorm (Set.indicator s (f i)) p μ ≤ ENNReal.ofReal (ε / 2)
δ₂ : ℝ
hδ₂_pos : 0 < δ₂
hgδ₂ :
  ∀ (i : ι) (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₂ → snorm (Set.indicator s (g i)) p μ ≤ ENNReal.ofReal (ε / 2)
⊢ ∃ δ x,
    ∀ (i : ι) (s : Set α),
      MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (Set.indicator s ((f + g) i)) p μ ≤ ENNReal.ofReal ε

tactic: simp_rw [Pi.add_apply, Set.indicator_add']
state_before: case intro.intro.intro.intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f g : ι → α → β
p : ℝ≥0∞
hf : UnifIntegrable f p μ
hg : UnifIntegrable g p μ
hp : 1 ≤ p
hf_meas : ∀ (i : ι), AEStronglyMeasurable (f i) μ
hg_meas : ∀ (i : ι), AEStronglyMeasurable (g i) μ
ε : ℝ
hε : 0 < ε
hε2 : 0 < ε / 2
δ₁ : ℝ
hδ₁_pos : 0 < δ₁
hfδ₁ :
  ∀ (i : ι) (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₁ → snorm (Set.indicator s (f i)) p μ ≤ ENNReal.ofReal (ε / 2)
δ₂ : ℝ
hδ₂_pos : 0 < δ₂
hgδ₂ :
  ∀ (i : ι) (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₂ → snorm (Set.indicator s (g i)) p μ ≤ ENNReal.ofReal (ε / 2)
i : ι
s : Set α
hs : MeasurableSet s
hμs : ↑↑μ s ≤ ENNReal.ofReal (min δ₁ δ₂)
⊢ snorm (Set.indicator s ((f + g) i)) p μ ≤ ENNReal.ofReal ε

tactic: refine' (snorm_add_le ((hf_meas i).indicator hs) ((hg_meas i).indicator hs) hp).trans _
state_before: case intro.intro.intro.intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f g : ι → α → β
p : ℝ≥0∞
hf : UnifIntegrable f p μ
hg : UnifIntegrable g p μ
hp : 1 ≤ p
hf_meas : ∀ (i : ι), AEStronglyMeasurable (f i) μ
hg_meas : ∀ (i : ι), AEStronglyMeasurable (g i) μ
ε : ℝ
hε : 0 < ε
hε2 : 0 < ε / 2
δ₁ : ℝ
hδ₁_pos : 0 < δ₁
hfδ₁ :
  ∀ (i : ι) (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₁ → snorm (Set.indicator s (f i)) p μ ≤ ENNReal.ofReal (ε / 2)
δ₂ : ℝ
hδ₂_pos : 0 < δ₂
hgδ₂ :
  ∀ (i : ι) (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₂ → snorm (Set.indicator s (g i)) p μ ≤ ENNReal.ofReal (ε / 2)
i : ι
s : Set α
hs : MeasurableSet s
hμs : ↑↑μ s ≤ ENNReal.ofReal (min δ₁ δ₂)
⊢ snorm (Set.indicator s (f i) + Set.indicator s (g i)) p μ ≤ ENNReal.ofReal ε

tactic: have hε_halves : ENNReal.ofReal ε = ENNReal.ofReal (ε / 2) + ENNReal.ofReal (ε / 2) := by
  rw [← ENNReal.ofReal_add hε2.le hε2.le, add_halves]
state_before: case intro.intro.intro.intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f g : ι → α → β
p : ℝ≥0∞
hf : UnifIntegrable f p μ
hg : UnifIntegrable g p μ
hp : 1 ≤ p
hf_meas : ∀ (i : ι), AEStronglyMeasurable (f i) μ
hg_meas : ∀ (i : ι), AEStronglyMeasurable (g i) μ
ε : ℝ
hε : 0 < ε
hε2 : 0 < ε / 2
δ₁ : ℝ
hδ₁_pos : 0 < δ₁
hfδ₁ :
  ∀ (i : ι) (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₁ → snorm (Set.indicator s (f i)) p μ ≤ ENNReal.ofReal (ε / 2)
δ₂ : ℝ
hδ₂_pos : 0 < δ₂
hgδ₂ :
  ∀ (i : ι) (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₂ → snorm (Set.indicator s (g i)) p μ ≤ ENNReal.ofReal (ε / 2)
i : ι
s : Set α
hs : MeasurableSet s
hμs : ↑↑μ s ≤ ENNReal.ofReal (min δ₁ δ₂)
⊢ snorm (Set.indicator s (f i)) p μ + snorm (Set.indicator s (g i)) p μ ≤ ENNReal.ofReal ε

tactic: rw [hε_halves]
state_before: case intro.intro.intro.intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f g : ι → α → β
p : ℝ≥0∞
hf : UnifIntegrable f p μ
hg : UnifIntegrable g p μ
hp : 1 ≤ p
hf_meas : ∀ (i : ι), AEStronglyMeasurable (f i) μ
hg_meas : ∀ (i : ι), AEStronglyMeasurable (g i) μ
ε : ℝ
hε : 0 < ε
hε2 : 0 < ε / 2
δ₁ : ℝ
hδ₁_pos : 0 < δ₁
hfδ₁ :
  ∀ (i : ι) (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₁ → snorm (Set.indicator s (f i)) p μ ≤ ENNReal.ofReal (ε / 2)
δ₂ : ℝ
hδ₂_pos : 0 < δ₂
hgδ₂ :
  ∀ (i : ι) (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₂ → snorm (Set.indicator s (g i)) p μ ≤ ENNReal.ofReal (ε / 2)
i : ι
s : Set α
hs : MeasurableSet s
hμs : ↑↑μ s ≤ ENNReal.ofReal (min δ₁ δ₂)
hε_halves : ENNReal.ofReal ε = ENNReal.ofReal (ε / 2) + ENNReal.ofReal (ε / 2)
⊢ snorm (Set.indicator s (f i)) p μ + snorm (Set.indicator s (g i)) p μ ≤ ENNReal.ofReal ε

tactic: exact add_le_add (hfδ₁ i s hs (hμs.trans (ENNReal.ofReal_le_ofReal (min_le_left _ _))))
  (hgδ₂ i s hs (hμs.trans (ENNReal.ofReal_le_ofReal (min_le_right _ _))))
state_before: case intro.intro.intro.intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f g : ι → α → β
p : ℝ≥0∞
hf : UnifIntegrable f p μ
hg : UnifIntegrable g p μ
hp : 1 ≤ p
hf_meas : ∀ (i : ι), AEStronglyMeasurable (f i) μ
hg_meas : ∀ (i : ι), AEStronglyMeasurable (g i) μ
ε : ℝ
hε : 0 < ε
hε2 : 0 < ε / 2
δ₁ : ℝ
hδ₁_pos : 0 < δ₁
hfδ₁ :
  ∀ (i : ι) (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₁ → snorm (Set.indicator s (f i)) p μ ≤ ENNReal.ofReal (ε / 2)
δ₂ : ℝ
hδ₂_pos : 0 < δ₂
hgδ₂ :
  ∀ (i : ι) (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₂ → snorm (Set.indicator s (g i)) p μ ≤ ENNReal.ofReal (ε / 2)
i : ι
s : Set α
hs : MeasurableSet s
hμs : ↑↑μ s ≤ ENNReal.ofReal (min δ₁ δ₂)
hε_halves : ENNReal.ofReal ε = ENNReal.ofReal (ε / 2) + ENNReal.ofReal (ε / 2)
⊢ snorm (Set.indicator s (f i)) p μ + snorm (Set.indicator s (g i)) p μ ≤
    ENNReal.ofReal (ε / 2) + ENNReal.ofReal (ε / 2)

tactic: rw [← ENNReal.ofReal_add hε2.le hε2.le, add_halves]
state_before: α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f g : ι → α → β
p : ℝ≥0∞
hf : UnifIntegrable f p μ
hg : UnifIntegrable g p μ
hp : 1 ≤ p
hf_meas : ∀ (i : ι), AEStronglyMeasurable (f i) μ
hg_meas : ∀ (i : ι), AEStronglyMeasurable (g i) μ
ε : ℝ
hε : 0 < ε
hε2 : 0 < ε / 2
δ₁ : ℝ
hδ₁_pos : 0 < δ₁
hfδ₁ :
  ∀ (i : ι) (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₁ → snorm (Set.indicator s (f i)) p μ ≤ ENNReal.ofReal (ε / 2)
δ₂ : ℝ
hδ₂_pos : 0 < δ₂
hgδ₂ :
  ∀ (i : ι) (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₂ → snorm (Set.indicator s (g i)) p μ ≤ ENNReal.ofReal (ε / 2)
i : ι
s : Set α
hs : MeasurableSet s
hμs : ↑↑μ s ≤ ENNReal.ofReal (min δ₁ δ₂)
⊢ ENNReal.ofReal ε = ENNReal.ofReal (ε / 2) + ENNReal.ofReal (ε / 2)