/-- Lebesgue dominated convergence theorem provides sufficient conditions under which almost
  everywhere convergence of a sequence of functions implies the convergence of their image by
  `setToFun`.
  We could weaken the condition `bound_integrable` to require `HasFiniteIntegral bound μ` instead
  (i.e. not requiring that `bound` is measurable), but in all applications proving integrability
  is easier. -/
theorem tendsto_setToFun_of_dominated_convergence (hT : DominatedFinMeasAdditive μ T C)
    {fs : ℕ → α → E} {f : α → E} (bound : α → ℝ)
    (fs_measurable : ∀ n, AEStronglyMeasurable (fs n) μ) (bound_integrable : Integrable bound μ)
    (h_bound : ∀ n, ∀ᵐ a ∂μ, ‖fs n a‖ ≤ bound a)
    (h_lim : ∀ᵐ a ∂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) := by
  -- `f` is a.e.-measurable, since it is the a.e.-pointwise limit of a.e.-measurable functions.
  have f_measurable : AEStronglyMeasurable f μ :=
    aestronglyMeasurable_of_tendsto_ae _ fs_measurable h_lim
  -- all functions we consider are integrable
  have fs_int : ∀ n, Integrable (fs n) μ := fun n =>
    bound_integrable.mono' (fs_measurable n) (h_bound _)
  have f_int : Integrable f μ :=
    ⟨f_measurable,
      hasFiniteIntegral_of_dominated_convergence bound_integrable.hasFiniteIntegral h_bound
        h_lim⟩
  -- it suffices to prove the result for the corresponding L1 functions
  suffices
    Tendsto (fun n => L1.setToL1 hT ((fs_int n).toL1 (fs n))) atTop
      (𝓝 (L1.setToL1 hT (f_int.toL1 f))) by
    convert this with n
    · exact setToFun_eq hT (fs_int n)
    · exact setToFun_eq hT f_int
  -- the convergence of setToL1 follows from the convergence of the L1 functions
  refine' L1.tendsto_setToL1 hT _ _ _
  -- up to some rewriting, what we need to prove is `h_lim`
  rw [tendsto_iff_norm_sub_tendsto_zero]
  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)
  convert lintegral_norm_tendsto_zero with n
  rw [L1.norm_def]
  congr 1
  refine' lintegral_congr_ae _
  rw [← Integrable.toL1_sub]
  refine' ((fs_int n).sub f_int).coeFn_toL1.mono fun x hx => _
  dsimp only
  

Proof state:
case h.e'_3.h.e_a
α : Type u_1
E : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
𝕜 : Type u_6
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‖

Next tactic:
rw [hx, ofReal_norm_eq_coe_nnnorm, Pi.sub_apply]