-- Mathlib/MeasureTheory/Integral/Bochner.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_integral_tendsto_of_monotone {μ : Measure α} {f : ℕ → α → ℝ} {F : α → ℝ}
    (hf_int : ∀ n, Integrable (f n) μ) (hF_int : Integrable 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))
    (hf_bound : ∀ᵐ a ∂μ, ∀ i, f i a ≤ F a) :
    ∀ᵐ a ∂μ, Tendsto (fun i ↦ f i a) atTop (𝓝 (F a))

Proof state:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : ℕ → α → ℝ
F : α → ℝ
hf_int : ∀ (n : ℕ), Integrable (f n) μ
hF_int : Integrable 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
hf_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), f i a ≤ F a
⊢ ∀ᵐ (a : α) ∂μ, Tendsto (fun i => f i a) atTop (𝓝 (F a))

Tactic:
let f' : ℕ → α → ℝ≥0∞ := fun n a ↦ ENNReal.ofReal (f n a - f 0 a)

Result:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : ℕ → α → ℝ
F : α → ℝ
hf_int : ∀ (n : ℕ), Integrable (f n) μ
hF_int : Integrable 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
hf_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), f i a ≤ F a
f' : ℕ → α → ℝ≥0∞ := fun n a => ENNReal.ofReal (f n a - f 0 a)
⊢ ∀ᵐ (a : α) ∂μ, Tendsto (fun i => f i a) atTop (𝓝 (F a))

Proof state:
case h
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : ℕ → α → ℝ
F : α → ℝ
hf_int : ∀ (n : ℕ), Integrable (f n) μ
hF_int : Integrable 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
hf_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), f i a ≤ F a
f' : ℕ → α → ℝ≥0∞ := fun n a => ENNReal.ofReal (f n a - f 0 a)
F' : α → ℝ≥0∞ := fun a => ENNReal.ofReal (F a - f 0 a)
hf'_int_eq : ∀ (i : ℕ), ∫⁻ (a : α), f' i a ∂μ = ENNReal.ofReal (∫ (a : α), f i a ∂μ - ∫ (a : α), f 0 a ∂μ)
hF'_int_eq : ∫⁻ (a : α), F' a ∂μ = ENNReal.ofReal (∫ (a : α), F a ∂μ - ∫ (a : α), f 0 a ∂μ)
h_tendsto : Tendsto (fun i => ∫⁻ (a : α), f' i a ∂μ) atTop (𝓝 (∫⁻ (a : α), F' a ∂μ))
h_mono : ∀ᵐ (a : α) ∂μ, Monotone fun i => f' i a
a : α
ha_bound : ∀ (i : ℕ), f i a ≤ F a
i : ℕ
⊢ f i a - f 0 a ≤ F a - f 0 a

Tactic:
simp only [tsub_le_iff_right, sub_add_cancel, ha_bound i]

Result:
no goals

Proof state:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : ℕ → α → ℝ
F : α → ℝ
hf_int : ∀ (n : ℕ), Integrable (f n) μ
hF_int : Integrable 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
hf_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), f i a ≤ F a
f' : ℕ → α → ℝ≥0∞ := fun n a => ENNReal.ofReal (f n a - f 0 a)
F' : α → ℝ≥0∞ := fun a => ENNReal.ofReal (F a - f 0 a)
i : ℕ
⊢ ∫⁻ (a : α), ENNReal.ofReal (f i a - f 0 a) ∂μ = ENNReal.ofReal (∫ (a : α), f i a ∂μ - ∫ (a : α), f 0 a ∂μ)

Tactic:
rw [← ofReal_integral_eq_lintegral_ofReal, integral_sub (hf_int i) (hf_int 0)]

Result:
case hfi
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : ℕ → α → ℝ
F : α → ℝ
hf_int : ∀ (n : ℕ), Integrable (f n) μ
hF_int : Integrable 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
hf_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), f i a ≤ F a
f' : ℕ → α → ℝ≥0∞ := fun n a => ENNReal.ofReal (f n a - f 0 a)
F' : α → ℝ≥0∞ := fun a => ENNReal.ofReal (F a - f 0 a)
i : ℕ
⊢ Integrable (fun a => f i a - f 0 a) μ

case f_nn
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : ℕ → α → ℝ
F : α → ℝ
hf_int : ∀ (n : ℕ), Integrable (f n) μ
hF_int : Integrable 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
hf_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), f i a ≤ F a
f' : ℕ → α → ℝ≥0∞ := fun n a => ENNReal.ofReal (f n a - f 0 a)
F' : α → ℝ≥0∞ := fun a => ENNReal.ofReal (F a - f 0 a)
i : ℕ
⊢ 0 ≤ᶠ[ae μ] fun a => f i a - f 0 a

Proof state:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : ℕ → α → ℝ
F : α → ℝ
hf_int : ∀ (n : ℕ), Integrable (f n) μ
hF_int : Integrable 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
hf_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), f i a ≤ F a
f' : ℕ → α → ℝ≥0∞ := fun n a => ENNReal.ofReal (f n a - f 0 a)
F' : α → ℝ≥0∞ := fun a => ENNReal.ofReal (F a - f 0 a)
hf'_int_eq : ∀ (i : ℕ), ∫⁻ (a : α), f' i a ∂μ = ENNReal.ofReal (∫ (a : α), f i a ∂μ - ∫ (a : α), f 0 a ∂μ)
hF'_int_eq : ∫⁻ (a : α), F' a ∂μ = ENNReal.ofReal (∫ (a : α), F a ∂μ - ∫ (a : α), f 0 a ∂μ)
h_tendsto : Tendsto (fun i => ∫⁻ (a : α), f' i a ∂μ) atTop (𝓝 (∫⁻ (a : α), F' a ∂μ))
h_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 : ∀ᵐ (a : α) ∂μ, Tendsto (fun i => f' i a) atTop (𝓝 (F' a))
a : α
ha : Tendsto (fun i => f' i a) atTop (𝓝 (F' a))
ha_mono : Monotone fun i => f i a
ha_bound : ∀ (i : ℕ), f i a ≤ F a
⊢ (fun i => f i a) = fun i => (f' i a).toReal + f 0 a

Tactic:
unfold f'

Result:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : ℕ → α → ℝ
F : α → ℝ
hf_int : ∀ (n : ℕ), Integrable (f n) μ
hF_int : Integrable 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
hf_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), f i a ≤ F a
f' : ℕ → α → ℝ≥0∞ := fun n a => ENNReal.ofReal (f n a - f 0 a)
F' : α → ℝ≥0∞ := fun a => ENNReal.ofReal (F a - f 0 a)
hf'_int_eq : ∀ (i : ℕ), ∫⁻ (a : α), f' i a ∂μ = ENNReal.ofReal (∫ (a : α), f i a ∂μ - ∫ (a : α), f 0 a ∂μ)
hF'_int_eq : ∫⁻ (a : α), F' a ∂μ = ENNReal.ofReal (∫ (a : α), F a ∂μ - ∫ (a : α), f 0 a ∂μ)
h_tendsto : Tendsto (fun i => ∫⁻ (a : α), f' i a ∂μ) atTop (𝓝 (∫⁻ (a : α), F' a ∂μ))
h_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 : ∀ᵐ (a : α) ∂μ, Tendsto (fun i => f' i a) atTop (𝓝 (F' a))
a : α
ha : Tendsto (fun i => f' i a) atTop (𝓝 (F' a))
ha_mono : Monotone fun i => f i a
ha_bound : ∀ (i : ℕ), f i a ≤ F a
⊢ (fun i => f i a) = fun i => (ENNReal.ofReal (f i a - f 0 a)).toReal + f 0 a

Proof state:
case hfi
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : ℕ → α → ℝ
F : α → ℝ
hf_int : ∀ (n : ℕ), Integrable (f n) μ
hF_int : Integrable 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
hf_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), f i a ≤ F a
f' : ℕ → α → ℝ≥0∞ := fun n a => ENNReal.ofReal (f n a - f 0 a)
F' : α → ℝ≥0∞ := fun a => ENNReal.ofReal (F a - f 0 a)
i : ℕ
⊢ Integrable (fun a => f i a - f 0 a) μ

Tactic:
exact (hf_int i).sub (hf_int 0)

Result:
no goals