Theorem: MeasureTheory.tendsto_setToFun_of_dominated_convergence
File: Mathlib/MeasureTheory/Integral/SetToL1.lean

tactic: have f_measurable : AEStronglyMeasurable f μ :=
  aestronglyMeasurable_of_tendsto_ae _ fs_measurable h_lim
state_before: α : Type u_1
E : Type u_2
F : Type u_3
F' : Type ?u.1702608
G : Type ?u.1702611
𝕜 : Type ?u.1702614
p : ℝ≥0∞
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace ℝ F'
inst✝¹ : NormedAddCommGroup G
m : MeasurableSpace α
μ : Measure α
inst✝ : CompleteSpace F
T T' T'' : Set α → E →L[ℝ] F
C C' C'' : ℝ
f✝ g : α → E
hT : DominatedFinMeasAdditive μ T C
fs : ℕ → α → E
f : α → E
bound : α → ℝ
fs_measurable : ∀ (n : ℕ), AEStronglyMeasurable (fs n) μ
bound_integrable : Integrable bound
h_bound : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, ‖fs n a‖ ≤ bound a
h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n => fs n a) atTop (𝓝 (f a))
⊢ Tendsto (fun n => setToFun μ T hT (fs n)) atTop (𝓝 (setToFun μ T hT f))

tactic: have fs_int : ∀ n, Integrable (fs n) μ := fun n =>
  bound_integrable.mono' (fs_measurable n) (h_bound _)
state_before: α : Type u_1
E : Type u_2
F : Type u_3
F' : Type ?u.1702608
G : Type ?u.1702611
𝕜 : Type ?u.1702614
p : ℝ≥0∞
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace ℝ F'
inst✝¹ : NormedAddCommGroup G
m : MeasurableSpace α
μ : Measure α
inst✝ : CompleteSpace F
T T' T'' : Set α → E →L[ℝ] F
C C' C'' : ℝ
f✝ g : α → E
hT : DominatedFinMeasAdditive μ T C
fs : ℕ → α → E
f : α → E
bound : α → ℝ
fs_measurable : ∀ (n : ℕ), AEStronglyMeasurable (fs n) μ
bound_integrable : Integrable bound
h_bound : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, ‖fs n a‖ ≤ bound a
h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n => fs n a) atTop (𝓝 (f a))
f_measurable : AEStronglyMeasurable f μ
⊢ Tendsto (fun n => setToFun μ T hT (fs n)) atTop (𝓝 (setToFun μ T hT f))

tactic: have f_int : Integrable f μ :=
  ⟨f_measurable,
    hasFiniteIntegral_of_dominated_convergence bound_integrable.hasFiniteIntegral h_bound
      h_lim⟩
state_before: α : Type u_1
E : Type u_2
F : Type u_3
F' : Type ?u.1702608
G : Type ?u.1702611
𝕜 : Type ?u.1702614
p : ℝ≥0∞
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace ℝ F'
inst✝¹ : NormedAddCommGroup G
m : MeasurableSpace α
μ : Measure α
inst✝ : CompleteSpace F
T T' T'' : Set α → E →L[ℝ] F
C C' C'' : ℝ
f✝ g : α → E
hT : DominatedFinMeasAdditive μ T C
fs : ℕ → α → E
f : α → E
bound : α → ℝ
fs_measurable : ∀ (n : ℕ), AEStronglyMeasurable (fs n) μ
bound_integrable : Integrable bound
h_bound : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, ‖fs n a‖ ≤ bound a
h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n => fs n a) atTop (𝓝 (f a))
f_measurable : AEStronglyMeasurable f μ
fs_int : ∀ (n : ℕ), Integrable (fs n)
⊢ Tendsto (fun n => setToFun μ T hT (fs n)) atTop (𝓝 (setToFun μ T hT f))

tactic: refine' L1.tendsto_setToL1 hT _ _ _
state_before: α : Type u_1
E : Type u_2
F : Type u_3
F' : Type ?u.1702608
G : Type ?u.1702611
𝕜 : Type ?u.1702614
p : ℝ≥0∞
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace ℝ F'
inst✝¹ : NormedAddCommGroup G
m : MeasurableSpace α
μ : Measure α
inst✝ : CompleteSpace F
T T' T'' : Set α → E →L[ℝ] F
C C' C'' : ℝ
f✝ g : α → E
hT : DominatedFinMeasAdditive μ T C
fs : ℕ → α → E
f : α → E
bound : α → ℝ
fs_measurable : ∀ (n : ℕ), AEStronglyMeasurable (fs n) μ
bound_integrable : Integrable bound
h_bound : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, ‖fs n a‖ ≤ bound a
h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n => fs n a) atTop (𝓝 (f a))
f_measurable : AEStronglyMeasurable f μ
fs_int : ∀ (n : ℕ), Integrable (fs n)
f_int : Integrable f
⊢ Tendsto (fun n => ↑(L1.setToL1 hT) (Integrable.toL1 (fs n) (_ : Integrable (fs n)))) atTop
    (𝓝 (↑(L1.setToL1 hT) (Integrable.toL1 f f_int)))

tactic: rw [tendsto_iff_norm_tendsto_zero]
state_before: α : Type u_1
E : Type u_2
F : Type u_3
F' : Type ?u.1702608
G : Type ?u.1702611
𝕜 : Type ?u.1702614
p : ℝ≥0∞
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace ℝ F'
inst✝¹ : NormedAddCommGroup G
m : MeasurableSpace α
μ : Measure α
inst✝ : CompleteSpace F
T T' T'' : Set α → E →L[ℝ] F
C C' C'' : ℝ
f✝ g : α → E
hT : DominatedFinMeasAdditive μ T C
fs : ℕ → α → E
f : α → E
bound : α → ℝ
fs_measurable : ∀ (n : ℕ), AEStronglyMeasurable (fs n) μ
bound_integrable : Integrable bound
h_bound : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, ‖fs n a‖ ≤ bound a
h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n => fs n a) atTop (𝓝 (f a))
f_measurable : AEStronglyMeasurable f μ
fs_int : ∀ (n : ℕ), Integrable (fs n)
f_int : Integrable f
⊢ Tendsto (fun n => Integrable.toL1 (fs n) (_ : Integrable (fs n))) atTop (𝓝 (Integrable.toL1 f f_int))

tactic: have lintegral_norm_tendsto_zero :
  Tendsto (fun n => ENNReal.toReal <| ∫⁻ a, ENNReal.ofReal ‖fs n a - f a‖ ∂μ) atTop (𝓝 0) :=
  (tendsto_toReal zero_ne_top).comp
    (tendsto_lintegral_norm_of_dominated_convergence fs_measurable
      bound_integrable.hasFiniteIntegral h_bound h_lim)
state_before: α : Type u_1
E : Type u_2
F : Type u_3
F' : Type ?u.1702608
G : Type ?u.1702611
𝕜 : Type ?u.1702614
p : ℝ≥0∞
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace ℝ F'
inst✝¹ : NormedAddCommGroup G
m : MeasurableSpace α
μ : Measure α
inst✝ : CompleteSpace F
T T' T'' : Set α → E →L[ℝ] F
C C' C'' : ℝ
f✝ g : α → E
hT : DominatedFinMeasAdditive μ T C
fs : ℕ → α → E
f : α → E
bound : α → ℝ
fs_measurable : ∀ (n : ℕ), AEStronglyMeasurable (fs n) μ
bound_integrable : Integrable bound
h_bound : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, ‖fs n a‖ ≤ bound a
h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n => fs n a) atTop (𝓝 (f a))
f_measurable : AEStronglyMeasurable f μ
fs_int : ∀ (n : ℕ), Integrable (fs n)
f_int : Integrable f
⊢ Tendsto (fun e => ‖Integrable.toL1 (fs e) (_ : Integrable (fs e)) - Integrable.toL1 f f_int‖) atTop (𝓝 0)

tactic: convert lintegral_norm_tendsto_zero with n
state_before: α : Type u_1
E : Type u_2
F : Type u_3
F' : Type ?u.1702608
G : Type ?u.1702611
𝕜 : Type ?u.1702614
p : ℝ≥0∞
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace ℝ F'
inst✝¹ : NormedAddCommGroup G
m : MeasurableSpace α
μ : Measure α
inst✝ : CompleteSpace F
T T' T'' : Set α → E →L[ℝ] F
C C' C'' : ℝ
f✝ g : α → E
hT : DominatedFinMeasAdditive μ T C
fs : ℕ → α → E
f : α → E
bound : α → ℝ
fs_measurable : ∀ (n : ℕ), AEStronglyMeasurable (fs n) μ
bound_integrable : Integrable bound
h_bound : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, ‖fs n a‖ ≤ bound a
h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n => fs n a) atTop (𝓝 (f a))
f_measurable : AEStronglyMeasurable f μ
fs_int : ∀ (n : ℕ), Integrable (fs n)
f_int : Integrable f
lintegral_norm_tendsto_zero :
  Tendsto (fun n => ENNReal.toReal (∫⁻ (a : α), ENNReal.ofReal ‖fs n a - f a‖ ∂μ)) atTop (𝓝 0)
⊢ Tendsto (fun e => ‖Integrable.toL1 (fs e) (_ : Integrable (fs e)) - Integrable.toL1 f f_int‖) atTop (𝓝 0)

tactic: rw [L1.norm_def]
state_before: case h.e'_3.h
α : Type u_1
E : Type u_2
F : Type u_3
F' : Type ?u.1702608
G : Type ?u.1702611
𝕜 : Type ?u.1702614
p : ℝ≥0∞
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace ℝ F'
inst✝¹ : NormedAddCommGroup G
m : MeasurableSpace α
μ : Measure α
inst✝ : CompleteSpace F
T T' T'' : Set α → E →L[ℝ] F
C C' C'' : ℝ
f✝ g : α → E
hT : DominatedFinMeasAdditive μ T C
fs : ℕ → α → E
f : α → E
bound : α → ℝ
fs_measurable : ∀ (n : ℕ), AEStronglyMeasurable (fs n) μ
bound_integrable : Integrable bound
h_bound : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, ‖fs n a‖ ≤ bound a
h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n => fs n a) atTop (𝓝 (f a))
f_measurable : AEStronglyMeasurable f μ
fs_int : ∀ (n : ℕ), Integrable (fs n)
f_int : Integrable f
lintegral_norm_tendsto_zero :
  Tendsto (fun n => ENNReal.toReal (∫⁻ (a : α), ENNReal.ofReal ‖fs n a - f a‖ ∂μ)) atTop (𝓝 0)
n : ℕ
⊢ ‖Integrable.toL1 (fs n) (_ : Integrable (fs n)) - Integrable.toL1 f f_int‖ =
    ENNReal.toReal (∫⁻ (a : α), ENNReal.ofReal ‖fs n a - f a‖ ∂μ)

tactic: congr 1
state_before: case h.e'_3.h
α : Type u_1
E : Type u_2
F : Type u_3
F' : Type ?u.1702608
G : Type ?u.1702611
𝕜 : Type ?u.1702614
p : ℝ≥0∞
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace ℝ F'
inst✝¹ : NormedAddCommGroup G
m : MeasurableSpace α
μ : Measure α
inst✝ : CompleteSpace F
T T' T'' : Set α → E →L[ℝ] F
C C' C'' : ℝ
f✝ g : α → E
hT : DominatedFinMeasAdditive μ T C
fs : ℕ → α → E
f : α → E
bound : α → ℝ
fs_measurable : ∀ (n : ℕ), AEStronglyMeasurable (fs n) μ
bound_integrable : Integrable bound
h_bound : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, ‖fs n a‖ ≤ bound a
h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n => fs n a) atTop (𝓝 (f a))
f_measurable : AEStronglyMeasurable f μ
fs_int : ∀ (n : ℕ), Integrable (fs n)
f_int : Integrable f
lintegral_norm_tendsto_zero :
  Tendsto (fun n => ENNReal.toReal (∫⁻ (a : α), ENNReal.ofReal ‖fs n a - f a‖ ∂μ)) atTop (𝓝 0)
n : ℕ
⊢ ENNReal.toReal (∫⁻ (a : α), ↑‖↑↑(Integrable.toL1 (fs n) (_ : Integrable (fs n)) - Integrable.toL1 f f_int) a‖₊ ∂μ) =
    ENNReal.toReal (∫⁻ (a : α), ENNReal.ofReal ‖fs n a - f a‖ ∂μ)

tactic: refine' lintegral_congr_ae _
state_before: case h.e'_3.h.e_a
α : Type u_1
E : Type u_2
F : Type u_3
F' : Type ?u.1702608
G : Type ?u.1702611
𝕜 : Type ?u.1702614
p : ℝ≥0∞
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace ℝ F'
inst✝¹ : NormedAddCommGroup G
m : MeasurableSpace α
μ : Measure α
inst✝ : CompleteSpace F
T T' T'' : Set α → E →L[ℝ] F
C C' C'' : ℝ
f✝ g : α → E
hT : DominatedFinMeasAdditive μ T C
fs : ℕ → α → E
f : α → E
bound : α → ℝ
fs_measurable : ∀ (n : ℕ), AEStronglyMeasurable (fs n) μ
bound_integrable : Integrable bound
h_bound : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, ‖fs n a‖ ≤ bound a
h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n => fs n a) atTop (𝓝 (f a))
f_measurable : AEStronglyMeasurable f μ
fs_int : ∀ (n : ℕ), Integrable (fs n)
f_int : Integrable f
lintegral_norm_tendsto_zero :
  Tendsto (fun n => ENNReal.toReal (∫⁻ (a : α), ENNReal.ofReal ‖fs n a - f a‖ ∂μ)) atTop (𝓝 0)
n : ℕ
⊢ (∫⁻ (a : α), ↑‖↑↑(Integrable.toL1 (fs n) (_ : Integrable (fs n)) - Integrable.toL1 f f_int) a‖₊ ∂μ) =
    ∫⁻ (a : α), ENNReal.ofReal ‖fs n a - f a‖ ∂μ

tactic: rw [← Integrable.toL1_sub]
state_before: case h.e'_3.h.e_a
α : Type u_1
E : Type u_2
F : Type u_3
F' : Type ?u.1702608
G : Type ?u.1702611
𝕜 : Type ?u.1702614
p : ℝ≥0∞
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace ℝ F'
inst✝¹ : NormedAddCommGroup G
m : MeasurableSpace α
μ : Measure α
inst✝ : CompleteSpace F
T T' T'' : Set α → E →L[ℝ] F
C C' C'' : ℝ
f✝ g : α → E
hT : DominatedFinMeasAdditive μ T C
fs : ℕ → α → E
f : α → E
bound : α → ℝ
fs_measurable : ∀ (n : ℕ), AEStronglyMeasurable (fs n) μ
bound_integrable : Integrable bound
h_bound : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, ‖fs n a‖ ≤ bound a
h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n => fs n a) atTop (𝓝 (f a))
f_measurable : AEStronglyMeasurable f μ
fs_int : ∀ (n : ℕ), Integrable (fs n)
f_int : Integrable f
lintegral_norm_tendsto_zero :
  Tendsto (fun n => ENNReal.toReal (∫⁻ (a : α), ENNReal.ofReal ‖fs n a - f a‖ ∂μ)) atTop (𝓝 0)
n : ℕ
⊢ (fun a => ↑‖↑↑(Integrable.toL1 (fs n) (_ : Integrable (fs n)) - Integrable.toL1 f f_int) a‖₊) =ᵐ[μ] fun a =>
    ENNReal.ofReal ‖fs n a - f a‖

tactic: refine' ((fs_int n).sub f_int).coeFn_toL1.mono fun x hx => _
state_before: case h.e'_3.h.e_a
α : Type u_1
E : Type u_2
F : Type u_3
F' : Type ?u.1702608
G : Type ?u.1702611
𝕜 : Type ?u.1702614
p : ℝ≥0∞
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace ℝ F'
inst✝¹ : NormedAddCommGroup G
m : MeasurableSpace α
μ : Measure α
inst✝ : CompleteSpace F
T T' T'' : Set α → E →L[ℝ] F
C C' C'' : ℝ
f✝ g : α → E
hT : DominatedFinMeasAdditive μ T C
fs : ℕ → α → E
f : α → E
bound : α → ℝ
fs_measurable : ∀ (n : ℕ), AEStronglyMeasurable (fs n) μ
bound_integrable : Integrable bound
h_bound : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, ‖fs n a‖ ≤ bound a
h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n => fs n a) atTop (𝓝 (f a))
f_measurable : AEStronglyMeasurable f μ
fs_int : ∀ (n : ℕ), Integrable (fs n)
f_int : Integrable f
lintegral_norm_tendsto_zero :
  Tendsto (fun n => ENNReal.toReal (∫⁻ (a : α), ENNReal.ofReal ‖fs n a - f a‖ ∂μ)) atTop (𝓝 0)
n : ℕ
⊢ (fun a => ↑‖↑↑(Integrable.toL1 (fs n - f) (_ : Integrable (fs n - f))) a‖₊) =ᵐ[μ] fun a =>
    ENNReal.ofReal ‖fs n a - f a‖

tactic: dsimp only
state_before: case h.e'_3.h.e_a
α : Type u_1
E : Type u_2
F : Type u_3
F' : Type ?u.1702608
G : Type ?u.1702611
𝕜 : Type ?u.1702614
p : ℝ≥0∞
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace ℝ F'
inst✝¹ : NormedAddCommGroup G
m : MeasurableSpace α
μ : Measure α
inst✝ : CompleteSpace F
T T' T'' : Set α → E →L[ℝ] F
C C' C'' : ℝ
f✝ g : α → E
hT : DominatedFinMeasAdditive μ T C
fs : ℕ → α → E
f : α → E
bound : α → ℝ
fs_measurable : ∀ (n : ℕ), AEStronglyMeasurable (fs n) μ
bound_integrable : Integrable bound
h_bound : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, ‖fs n a‖ ≤ bound a
h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n => fs n a) atTop (𝓝 (f a))
f_measurable : AEStronglyMeasurable f μ
fs_int : ∀ (n : ℕ), Integrable (fs n)
f_int : Integrable f
lintegral_norm_tendsto_zero :
  Tendsto (fun n => ENNReal.toReal (∫⁻ (a : α), ENNReal.ofReal ‖fs n a - f a‖ ∂μ)) atTop (𝓝 0)
n : ℕ
x : α
hx : ↑↑(Integrable.toL1 (fs n - f) (_ : Integrable (fs n - f))) x = (fs n - f) x
⊢ (fun a => ↑‖↑↑(Integrable.toL1 (fs n - f) (_ : Integrable (fs n - f))) a‖₊) x =
    (fun a => ENNReal.ofReal ‖fs n a - f a‖) x

tactic: rw [hx, ofReal_norm_eq_coe_nnnorm, Pi.sub_apply]
state_before: case h.e'_3.h.e_a
α : Type u_1
E : Type u_2
F : Type u_3
F' : Type ?u.1702608
G : Type ?u.1702611
𝕜 : Type ?u.1702614
p : ℝ≥0∞
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace ℝ F'
inst✝¹ : NormedAddCommGroup G
m : MeasurableSpace α
μ : Measure α
inst✝ : CompleteSpace F
T T' T'' : Set α → E →L[ℝ] F
C C' C'' : ℝ
f✝ g : α → E
hT : DominatedFinMeasAdditive μ T C
fs : ℕ → α → E
f : α → E
bound : α → ℝ
fs_measurable : ∀ (n : ℕ), AEStronglyMeasurable (fs n) μ
bound_integrable : Integrable bound
h_bound : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, ‖fs n a‖ ≤ bound a
h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n => fs n a) atTop (𝓝 (f a))
f_measurable : AEStronglyMeasurable f μ
fs_int : ∀ (n : ℕ), Integrable (fs n)
f_int : Integrable f
lintegral_norm_tendsto_zero :
  Tendsto (fun n => ENNReal.toReal (∫⁻ (a : α), ENNReal.ofReal ‖fs n a - f a‖ ∂μ)) atTop (𝓝 0)
n : ℕ
x : α
hx : ↑↑(Integrable.toL1 (fs n - f) (_ : Integrable (fs n - f))) x = (fs n - f) x
⊢ ↑‖↑↑(Integrable.toL1 (fs n - f) (_ : Integrable (fs n - f))) x‖₊ = ENNReal.ofReal ‖fs n x - f x‖

tactic: convert this with n
state_before: α : Type u_1
E : Type u_2
F : Type u_3
F' : Type ?u.1702608
G : Type ?u.1702611
𝕜 : Type ?u.1702614
p : ℝ≥0∞
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace ℝ F'
inst✝¹ : NormedAddCommGroup G
m : MeasurableSpace α
μ : Measure α
inst✝ : CompleteSpace F
T T' T'' : Set α → E →L[ℝ] F
C C' C'' : ℝ
f✝ g : α → E
hT : DominatedFinMeasAdditive μ T C
fs : ℕ → α → E
f : α → E
bound : α → ℝ
fs_measurable : ∀ (n : ℕ), AEStronglyMeasurable (fs n) μ
bound_integrable : Integrable bound
h_bound : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, ‖fs n a‖ ≤ bound a
h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n => fs n a) atTop (𝓝 (f a))
f_measurable : AEStronglyMeasurable f μ
fs_int : ∀ (n : ℕ), Integrable (fs n)
f_int : Integrable f
this :
  Tendsto (fun n => ↑(L1.setToL1 hT) (Integrable.toL1 (fs n) (_ : Integrable (fs n)))) atTop
    (𝓝 (↑(L1.setToL1 hT) (Integrable.toL1 f f_int)))
⊢ Tendsto (fun n => setToFun μ T hT (fs n)) atTop (𝓝 (setToFun μ T hT f))

tactic: exact setToFun_eq hT (fs_int n)
state_before: case h.e'_3.h
α : Type u_1
E : Type u_2
F : Type u_3
F' : Type ?u.1702608
G : Type ?u.1702611
𝕜 : Type ?u.1702614
p : ℝ≥0∞
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace ℝ F'
inst✝¹ : NormedAddCommGroup G
m : MeasurableSpace α
μ : Measure α
inst✝ : CompleteSpace F
T T' T'' : Set α → E →L[ℝ] F
C C' C'' : ℝ
f✝ g : α → E
hT : DominatedFinMeasAdditive μ T C
fs : ℕ → α → E
f : α → E
bound : α → ℝ
fs_measurable : ∀ (n : ℕ), AEStronglyMeasurable (fs n) μ
bound_integrable : Integrable bound
h_bound : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, ‖fs n a‖ ≤ bound a
h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n => fs n a) atTop (𝓝 (f a))
f_measurable : AEStronglyMeasurable f μ
fs_int : ∀ (n : ℕ), Integrable (fs n)
f_int : Integrable f
this :
  Tendsto (fun n => ↑(L1.setToL1 hT) (Integrable.toL1 (fs n) (_ : Integrable (fs n)))) atTop
    (𝓝 (↑(L1.setToL1 hT) (Integrable.toL1 f f_int)))
n : ℕ
⊢ setToFun μ T hT (fs n) = ↑(L1.setToL1 hT) (Integrable.toL1 (fs n) (_ : Integrable (fs n)))

tactic: exact setToFun_eq hT f_int
state_before: case h.e'_5.h.e'_3
α : Type u_1
E : Type u_2
F : Type u_3
F' : Type ?u.1702608
G : Type ?u.1702611
𝕜 : Type ?u.1702614
p : ℝ≥0∞
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace ℝ F'
inst✝¹ : NormedAddCommGroup G
m : MeasurableSpace α
μ : Measure α
inst✝ : CompleteSpace F
T T' T'' : Set α → E →L[ℝ] F
C C' C'' : ℝ
f✝ g : α → E
hT : DominatedFinMeasAdditive μ T C
fs : ℕ → α → E
f : α → E
bound : α → ℝ
fs_measurable : ∀ (n : ℕ), AEStronglyMeasurable (fs n) μ
bound_integrable : Integrable bound
h_bound : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, ‖fs n a‖ ≤ bound a
h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n => fs n a) atTop (𝓝 (f a))
f_measurable : AEStronglyMeasurable f μ
fs_int : ∀ (n : ℕ), Integrable (fs n)
f_int : Integrable f
this :
  Tendsto (fun n => ↑(L1.setToL1 hT) (Integrable.toL1 (fs n) (_ : Integrable (fs n)))) atTop
    (𝓝 (↑(L1.setToL1 hT) (Integrable.toL1 f f_int)))
⊢ setToFun μ T hT f = ↑(L1.setToL1 hT) (Integrable.toL1 f f_int)