-- Mathlib/MeasureTheory/Integral/Lebesgue.lean
/-- If a monotone sequence of functions has an upper bound and the sequence of integrals of these
functions tends to the integral of the upper bound, then the sequence of functions converges
almost everywhere to the upper bound. -/
lemma tendsto_of_lintegral_tendsto_of_monotone {α : Type*} {mα : MeasurableSpace α}
    {f : ℕ → α → ℝ≥0∞} {F : α → ℝ≥0∞} {μ : Measure α}
    (hF_meas : AEMeasurable F μ)
    (hf_tendsto : Tendsto (fun i ↦ ∫⁻ a, f i a ∂μ) atTop (𝓝 (∫⁻ a, F a ∂μ)))
    (hf_mono : ∀ᵐ a ∂μ, Monotone (fun i ↦ f i a))
    (h_bound : ∀ᵐ a ∂μ, ∀ i, f i a ≤ F a) (h_int_finite : ∫⁻ a, F a ∂μ ≠ ∞) :
    ∀ᵐ a ∂μ, Tendsto (fun i ↦ f i a) atTop (𝓝 (F a))

Proof state:
α : Type u_5
mα : MeasurableSpace α
f : ℕ → α → ℝ≥0∞
F : α → ℝ≥0∞
μ : Measure α
hF_meas : AEMeasurable F μ
hf_tendsto : Tendsto (fun i => ∫⁻ (a : α), f i a ∂μ) atTop (𝓝 (∫⁻ (a : α), F a ∂μ))
hf_mono : ∀ᵐ (a : α) ∂μ, Monotone fun i => f i a
h_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), f i a ≤ F a
h_int_finite : ∫⁻ (a : α), F a ∂μ ≠ ⊤
this : ∀ (n : ℕ), ∃ g, Measurable g ∧ g ≤ f n ∧ ∫⁻ (a : α), f n a ∂μ = ∫⁻ (a : α), g a ∂μ
⊢ ∀ᵐ (a : α) ∂μ, Tendsto (fun i => f i a) atTop (𝓝 (F a))

Tactic:
choose g gmeas gf hg using this

Result:
α : Type u_5
mα : MeasurableSpace α
f : ℕ → α → ℝ≥0∞
F : α → ℝ≥0∞
μ : Measure α
hF_meas : AEMeasurable F μ
hf_tendsto : Tendsto (fun i => ∫⁻ (a : α), f i a ∂μ) atTop (𝓝 (∫⁻ (a : α), F a ∂μ))
hf_mono : ∀ᵐ (a : α) ∂μ, Monotone fun i => f i a
h_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), f i a ≤ F a
h_int_finite : ∫⁻ (a : α), F a ∂μ ≠ ⊤
g : ℕ → α → ℝ≥0∞
gmeas : ∀ (n : ℕ), Measurable (g n)
gf : ∀ (n : ℕ), g n ≤ f n
hg : ∀ (n : ℕ), ∫⁻ (a : α), f n a ∂μ = ∫⁻ (a : α), g n a ∂μ
⊢ ∀ᵐ (a : α) ∂μ, Tendsto (fun i => f i a) atTop (𝓝 (F a))

Proof state:
case a
α : Type u_5
mα : MeasurableSpace α
f : ℕ → α → ℝ≥0∞
F : α → ℝ≥0∞
μ : Measure α
hF_meas : AEMeasurable F μ
hf_tendsto : Tendsto (fun i => ∫⁻ (a : α), f i a ∂μ) atTop (𝓝 (∫⁻ (a : α), F a ∂μ))
hf_mono : ∀ᵐ (a : α) ∂μ, Monotone fun i => f i a
h_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), f i a ≤ F a
h_int_finite : ∫⁻ (a : α), F a ∂μ ≠ ⊤
g : ℕ → α → ℝ≥0∞
gmeas : ∀ (n : ℕ), Measurable (g n)
gf : ∀ (n : ℕ), g n ≤ f n
hg : ∀ (n : ℕ), ∫⁻ (a : α), f n a ∂μ = ∫⁻ (a : α), g n a ∂μ
g' : ℕ → α → ℝ≥0∞ := fun t => Nat.rec (motive := fun x => α → ℝ≥0∞) (g 0) (fun n I x => g (n + 1) x ⊔ I x) t
M : ∀ (n : ℕ), Measurable (g' n)
I : ∀ (n : ℕ) (x : α), g n x ≤ g' n x
I' : ∀ᵐ (x : α) ∂μ, ∀ (n : ℕ), g' n x ≤ f n x
n : ℕ
⊢ ∫⁻ (x : α), g' n x ∂μ ≤ ∫⁻ (x : α), f n x ∂μ

Tactic:
apply lintegral_mono_ae

Result:
case a.h
α : Type u_5
mα : MeasurableSpace α
f : ℕ → α → ℝ≥0∞
F : α → ℝ≥0∞
μ : Measure α
hF_meas : AEMeasurable F μ
hf_tendsto : Tendsto (fun i => ∫⁻ (a : α), f i a ∂μ) atTop (𝓝 (∫⁻ (a : α), F a ∂μ))
hf_mono : ∀ᵐ (a : α) ∂μ, Monotone fun i => f i a
h_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), f i a ≤ F a
h_int_finite : ∫⁻ (a : α), F a ∂μ ≠ ⊤
g : ℕ → α → ℝ≥0∞
gmeas : ∀ (n : ℕ), Measurable (g n)
gf : ∀ (n : ℕ), g n ≤ f n
hg : ∀ (n : ℕ), ∫⁻ (a : α), f n a ∂μ = ∫⁻ (a : α), g n a ∂μ
g' : ℕ → α → ℝ≥0∞ := fun t => Nat.rec (motive := fun x => α → ℝ≥0∞) (g 0) (fun n I x => g (n + 1) x ⊔ I x) t
M : ∀ (n : ℕ), Measurable (g' n)
I : ∀ (n : ℕ) (x : α), g n x ≤ g' n x
I' : ∀ᵐ (x : α) ∂μ, ∀ (n : ℕ), g' n x ≤ f n x
n : ℕ
⊢ ∀ᵐ (a : α) ∂μ, g' n a ≤ f n a

Proof state:
case a
α : Type u_5
mα : MeasurableSpace α
f : ℕ → α → ℝ≥0∞
F : α → ℝ≥0∞
μ : Measure α
hF_meas : AEMeasurable F μ
hf_tendsto : Tendsto (fun i => ∫⁻ (a : α), f i a ∂μ) atTop (𝓝 (∫⁻ (a : α), F a ∂μ))
hf_mono : ∀ᵐ (a : α) ∂μ, Monotone fun i => f i a
h_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), f i a ≤ F a
h_int_finite : ∫⁻ (a : α), F a ∂μ ≠ ⊤
g : ℕ → α → ℝ≥0∞
gmeas : ∀ (n : ℕ), Measurable (g n)
gf : ∀ (n : ℕ), g n ≤ f n
hg : ∀ (n : ℕ), ∫⁻ (a : α), f n a ∂μ = ∫⁻ (a : α), g n a ∂μ
g' : ℕ → α → ℝ≥0∞ := fun t => Nat.rec (motive := fun x => α → ℝ≥0∞) (g 0) (fun n I x => g (n + 1) x ⊔ I x) t
M : ∀ (n : ℕ), Measurable (g' n)
I : ∀ (n : ℕ) (x : α), g n x ≤ g' n x
I' : ∀ᵐ (x : α) ∂μ, ∀ (n : ℕ), g' n x ≤ f n x
n : ℕ
⊢ ∫⁻ (x : α), f n x ∂μ ≤ ∫⁻ (x : α), g' n x ∂μ

Tactic:
rw [hg n]

Result:
case a
α : Type u_5
mα : MeasurableSpace α
f : ℕ → α → ℝ≥0∞
F : α → ℝ≥0∞
μ : Measure α
hF_meas : AEMeasurable F μ
hf_tendsto : Tendsto (fun i => ∫⁻ (a : α), f i a ∂μ) atTop (𝓝 (∫⁻ (a : α), F a ∂μ))
hf_mono : ∀ᵐ (a : α) ∂μ, Monotone fun i => f i a
h_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), f i a ≤ F a
h_int_finite : ∫⁻ (a : α), F a ∂μ ≠ ⊤
g : ℕ → α → ℝ≥0∞
gmeas : ∀ (n : ℕ), Measurable (g n)
gf : ∀ (n : ℕ), g n ≤ f n
hg : ∀ (n : ℕ), ∫⁻ (a : α), f n a ∂μ = ∫⁻ (a : α), g n a ∂μ
g' : ℕ → α → ℝ≥0∞ := fun t => Nat.rec (motive := fun x => α → ℝ≥0∞) (g 0) (fun n I x => g (n + 1) x ⊔ I x) t
M : ∀ (n : ℕ), Measurable (g' n)
I : ∀ (n : ℕ) (x : α), g n x ≤ g' n x
I' : ∀ᵐ (x : α) ∂μ, ∀ (n : ℕ), g' n x ≤ f n x
n : ℕ
⊢ ∫⁻ (a : α), g n a ∂μ ≤ ∫⁻ (x : α), g' n x ∂μ

Proof state:
α : Type u_5
mα : MeasurableSpace α
f : ℕ → α → ℝ≥0∞
F : α → ℝ≥0∞
μ : Measure α
hF_meas : AEMeasurable F μ
hf_tendsto : Tendsto (fun i => ∫⁻ (a : α), f i a ∂μ) atTop (𝓝 (∫⁻ (a : α), F a ∂μ))
hf_mono : ∀ᵐ (a : α) ∂μ, Monotone fun i => f i a
h_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), f i a ≤ F a
h_int_finite : ∫⁻ (a : α), F a ∂μ ≠ ⊤
g : ℕ → α → ℝ≥0∞
gmeas : ∀ (n : ℕ), Measurable (g n)
gf : ∀ (n : ℕ), g n ≤ f n
hg : ∀ (n : ℕ), ∫⁻ (a : α), f n a ∂μ = ∫⁻ (a : α), g n a ∂μ
g' : ℕ → α → ℝ≥0∞ := fun t => Nat.rec (motive := fun x => α → ℝ≥0∞) (g 0) (fun n I x => g (n + 1) x ⊔ I x) t
M : ∀ (n : ℕ), Measurable (g' n)
I : ∀ (n : ℕ) (x : α), g n x ≤ g' n x
I' : ∀ᵐ (x : α) ∂μ, ∀ (n : ℕ), g' n x ≤ f n x
Int_eq : ∀ (n : ℕ), ∫⁻ (x : α), g' n x ∂μ = ∫⁻ (x : α), f n x ∂μ
⊢ Tendsto (fun i => ∫⁻ (a : α), g' i a ∂μ) atTop (𝓝 (∫⁻ (a : α), F a ∂μ))

Tactic:
simp_rw [Int_eq]

Result:
α : Type u_5
mα : MeasurableSpace α
f : ℕ → α → ℝ≥0∞
F : α → ℝ≥0∞
μ : Measure α
hF_meas : AEMeasurable F μ
hf_tendsto : Tendsto (fun i => ∫⁻ (a : α), f i a ∂μ) atTop (𝓝 (∫⁻ (a : α), F a ∂μ))
hf_mono : ∀ᵐ (a : α) ∂μ, Monotone fun i => f i a
h_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), f i a ≤ F a
h_int_finite : ∫⁻ (a : α), F a ∂μ ≠ ⊤
g : ℕ → α → ℝ≥0∞
gmeas : ∀ (n : ℕ), Measurable (g n)
gf : ∀ (n : ℕ), g n ≤ f n
hg : ∀ (n : ℕ), ∫⁻ (a : α), f n a ∂μ = ∫⁻ (a : α), g n a ∂μ
g' : ℕ → α → ℝ≥0∞ := fun t => Nat.rec (motive := fun x => α → ℝ≥0∞) (g 0) (fun n I x => g (n + 1) x ⊔ I x) t
M : ∀ (n : ℕ), Measurable (g' n)
I : ∀ (n : ℕ) (x : α), g n x ≤ g' n x
I' : ∀ᵐ (x : α) ∂μ, ∀ (n : ℕ), g' n x ≤ f n x
Int_eq : ∀ (n : ℕ), ∫⁻ (x : α), g' n x ∂μ = ∫⁻ (x : α), f n x ∂μ
⊢ Tendsto (fun i => ∫⁻ (x : α), f i x ∂μ) atTop (𝓝 (∫⁻ (a : α), F a ∂μ))