-- Mathlib/MeasureTheory/Integral/SetToL1.lean
theorem tendsto_setToFun_filter_of_dominated_convergence (hT : DominatedFinMeasAdditive μ T C) {ι}
    {l : Filter ι} [l.IsCountablyGenerated] {fs : ι → α → E} {f : α → E} (bound : α → ℝ)
    (hfs_meas : ∀ᶠ n in l, AEStronglyMeasurable (fs n) μ)
    (h_bound : ∀ᶠ n in l, ∀ᵐ a ∂μ, ‖fs n a‖ ≤ bound a) (bound_integrable : Integrable bound μ)
    (h_lim : ∀ᵐ a ∂μ, Tendsto (fun n => fs n a) l (𝓝 (f a))) :
    Tendsto (fun n => setToFun μ T hT (fs n)) l (𝓝 <| setToFun μ T hT f)

Proof state:
case intro
α : Type u_1
E : Type u_2
F : Type u_3
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
inst✝¹ : CompleteSpace F
T : Set α → E →L[ℝ] F
C : ℝ
hT : DominatedFinMeasAdditive μ T C
ι : Type u_7
l : Filter ι
inst✝ : l.IsCountablyGenerated
fs : ι → α → E
f : α → E
bound : α → ℝ
hfs_meas : ∀ᶠ (n : ι) in l, AEStronglyMeasurable (fs n) μ
h_bound : ∀ᶠ (n : ι) in l, ∀ᵐ (a : α) ∂μ, ‖fs n a‖ ≤ bound a
bound_integrable : Integrable bound μ
h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n => fs n a) l (𝓝 (f a))
x : ℕ → ι
xl : Tendsto x atTop l
hxl : ∀ s ∈ l, ∃ a, ∀ b ≥ a, x b ∈ s
h✝ : {x | (fun n => AEStronglyMeasurable (fs n) μ) x} ∩ {x | (fun n => ∀ᵐ (a : α) ∂μ, ‖fs n a‖ ≤ bound a) x} ∈ l
k : ℕ
h :
  ∀ b ≥ k, x b ∈ {x | (fun n => AEStronglyMeasurable (fs n) μ) x} ∩ {x | (fun n => ∀ᵐ (a : α) ∂μ, ‖fs n a‖ ≤ bound a) x}
⊢ Tendsto (fun n => ((fun n => setToFun μ T hT (fs n)) ∘ x) (n + k)) atTop (𝓝 (setToFun μ T hT f))

Tactic:
refine tendsto_setToFun_of_dominated_convergence hT bound ?_ bound_integrable ?_ ?_

Result:
case intro.refine_1
α : Type u_1
E : Type u_2
F : Type u_3
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
inst✝¹ : CompleteSpace F
T : Set α → E →L[ℝ] F
C : ℝ
hT : DominatedFinMeasAdditive μ T C
ι : Type u_7
l : Filter ι
inst✝ : l.IsCountablyGenerated
fs : ι → α → E
f : α → E
bound : α → ℝ
hfs_meas : ∀ᶠ (n : ι) in l, AEStronglyMeasurable (fs n) μ
h_bound : ∀ᶠ (n : ι) in l, ∀ᵐ (a : α) ∂μ, ‖fs n a‖ ≤ bound a
bound_integrable : Integrable bound μ
h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n => fs n a) l (𝓝 (f a))
x : ℕ → ι
xl : Tendsto x atTop l
hxl : ∀ s ∈ l, ∃ a, ∀ b ≥ a, x b ∈ s
h✝ : {x | (fun n => AEStronglyMeasurable (fs n) μ) x} ∩ {x | (fun n => ∀ᵐ (a : α) ∂μ, ‖fs n a‖ ≤ bound a) x} ∈ l
k : ℕ
h :
  ∀ b ≥ k, x b ∈ {x | (fun n => AEStronglyMeasurable (fs n) μ) x} ∩ {x | (fun n => ∀ᵐ (a : α) ∂μ, ‖fs n a‖ ≤ bound a) x}
⊢ ∀ (n : ℕ), AEStronglyMeasurable (fs (x (n + k))) μ

case intro.refine_2
α : Type u_1
E : Type u_2
F : Type u_3
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
inst✝¹ : CompleteSpace F
T : Set α → E →L[ℝ] F
C : ℝ
hT : DominatedFinMeasAdditive μ T C
ι : Type u_7
l : Filter ι
inst✝ : l.IsCountablyGenerated
fs : ι → α → E
f : α → E
bound : α → ℝ
hfs_meas : ∀ᶠ (n : ι) in l, AEStronglyMeasurable (fs n) μ
h_bound : ∀ᶠ (n : ι) in l, ∀ᵐ (a : α) ∂μ, ‖fs n a‖ ≤ bound a
bound_integrable : Integrable bound μ
h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n => fs n a) l (𝓝 (f a))
x : ℕ → ι
xl : Tendsto x atTop l
hxl : ∀ s ∈ l, ∃ a, ∀ b ≥ a, x b ∈ s
h✝ : {x | (fun n => AEStronglyMeasurable (fs n) μ) x} ∩ {x | (fun n => ∀ᵐ (a : α) ∂μ, ‖fs n a‖ ≤ bound a) x} ∈ l
k : ℕ
h :
  ∀ b ≥ k, x b ∈ {x | (fun n => AEStronglyMeasurable (fs n) μ) x} ∩ {x | (fun n => ∀ᵐ (a : α) ∂μ, ‖fs n a‖ ≤ bound a) x}
⊢ ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, ‖fs (x (n + k)) a‖ ≤ bound a

case intro.refine_3
α : Type u_1
E : Type u_2
F : Type u_3
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
inst✝¹ : CompleteSpace F
T : Set α → E →L[ℝ] F
C : ℝ
hT : DominatedFinMeasAdditive μ T C
ι : Type u_7
l : Filter ι
inst✝ : l.IsCountablyGenerated
fs : ι → α → E
f : α → E
bound : α → ℝ
hfs_meas : ∀ᶠ (n : ι) in l, AEStronglyMeasurable (fs n) μ
h_bound : ∀ᶠ (n : ι) in l, ∀ᵐ (a : α) ∂μ, ‖fs n a‖ ≤ bound a
bound_integrable : Integrable bound μ
h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n => fs n a) l (𝓝 (f a))
x : ℕ → ι
xl : Tendsto x atTop l
hxl : ∀ s ∈ l, ∃ a, ∀ b ≥ a, x b ∈ s
h✝ : {x | (fun n => AEStronglyMeasurable (fs n) μ) x} ∩ {x | (fun n => ∀ᵐ (a : α) ∂μ, ‖fs n a‖ ≤ bound a) x} ∈ l
k : ℕ
h :
  ∀ b ≥ k, x b ∈ {x | (fun n => AEStronglyMeasurable (fs n) μ) x} ∩ {x | (fun n => ∀ᵐ (a : α) ∂μ, ‖fs n a‖ ≤ bound a) x}
⊢ ∀ᵐ (a : α) ∂μ, Tendsto (fun n => fs (x (n + k)) a) atTop (𝓝 (f a))

Proof state:
case intro.refine_2
α : Type u_1
E : Type u_2
F : Type u_3
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
inst✝¹ : CompleteSpace F
T : Set α → E →L[ℝ] F
C : ℝ
hT : DominatedFinMeasAdditive μ T C
ι : Type u_7
l : Filter ι
inst✝ : l.IsCountablyGenerated
fs : ι → α → E
f : α → E
bound : α → ℝ
hfs_meas : ∀ᶠ (n : ι) in l, AEStronglyMeasurable (fs n) μ
h_bound : ∀ᶠ (n : ι) in l, ∀ᵐ (a : α) ∂μ, ‖fs n a‖ ≤ bound a
bound_integrable : Integrable bound μ
h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n => fs n a) l (𝓝 (f a))
x : ℕ → ι
xl : Tendsto x atTop l
hxl : ∀ s ∈ l, ∃ a, ∀ b ≥ a, x b ∈ s
h✝ : {x | (fun n => AEStronglyMeasurable (fs n) μ) x} ∩ {x | (fun n => ∀ᵐ (a : α) ∂μ, ‖fs n a‖ ≤ bound a) x} ∈ l
k : ℕ
h :
  ∀ b ≥ k, x b ∈ {x | (fun n => AEStronglyMeasurable (fs n) μ) x} ∩ {x | (fun n => ∀ᵐ (a : α) ∂μ, ‖fs n a‖ ≤ bound a) x}
⊢ ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, ‖fs (x (n + k)) a‖ ≤ bound a

Tactic:
exact fun n => (h _ (self_le_add_left _ _)).2

Result:
no goals