Theorem: MeasureTheory.hasSum_integral_of_dominated_convergence
File: Mathlib/MeasureTheory/Integral/DominatedConvergence.lean

Proof steps:
state_before: α : Type u_1
E : Type u_2
G : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace ℝ G
f✝ g : α → E
m : MeasurableSpace α
μ : Measure α
ι : Type u_4
inst✝ : Countable ι
F : ι → α → G
f : α → G
bound : ι → α → ℝ
hF_meas : ∀ (n : ι), AEStronglyMeasurable (F n) μ
h_bound : ∀ (n : ι), ∀ᵐ (a : α) ∂μ, ‖F n a‖ ≤ bound n a
bound_summable : ∀ᵐ (a : α) ∂μ, Summable fun n => bound n a
bound_integrable : Integrable (fun a => ∑' (n : ι), bound n a) μ
h_lim : ∀ᵐ (a : α) ∂μ, HasSum (fun n => F n a) (f a)
⊢ HasSum (fun n => ∫ (a : α), F n a ∂μ) (∫ (a : α), f a ∂μ)
state_after: α : Type u_1
E : Type u_2
G : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace ℝ G
f✝ g : α → E
m : MeasurableSpace α
μ : Measure α
ι : Type u_4
inst✝ : Countable ι
F : ι → α → G
f : α → G
bound : ι → α → ℝ
hF_meas : ∀ (n : ι), AEStronglyMeasurable (F n) μ
h_bound : ∀ (n : ι), ∀ᵐ (a : α) ∂μ, ‖F n a‖ ≤ bound n a
bound_summable : ∀ᵐ (a : α) ∂μ, Summable fun n => bound n a
bound_integrable : Integrable (fun a => ∑' (n : ι), bound n a) μ
h_lim : ∀ᵐ (a : α) ∂μ, HasSum (fun n => F n a) (f a)
hb_nonneg : ∀ᵐ (a : α) ∂μ, ∀ (n : ι), 0 ≤ bound n a
⊢ HasSum (fun n => ∫ (a : α), F n a ∂μ) (∫ (a : α), f a ∂μ)
tactic: have hb_nonneg : ∀ᵐ a ∂μ, ∀ n, 0 ≤ bound n a :=
    eventually_countable_forall.2 fun n => (h_bound n).mono fun a => (norm_nonneg _).trans
state_before: α : Type u_1
E : Type u_2
G : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace ℝ G
f✝ g : α → E
m : MeasurableSpace α
μ : Measure α
ι : Type u_4
inst✝ : Countable ι
F : ι → α → G
f : α → G
bound : ι → α → ℝ
hF_meas : ∀ (n : ι), AEStronglyMeasurable (F n) μ
h_bound : ∀ (n : ι), ∀ᵐ (a : α) ∂μ, ‖F n a‖ ≤ bound n a
bound_summable : ∀ᵐ (a : α) ∂μ, Summable fun n => bound n a
bound_integrable : Integrable (fun a => ∑' (n : ι), bound n a) μ
h_lim : ∀ᵐ (a : α) ∂μ, HasSum (fun n => F n a) (f a)
hb_nonneg : ∀ᵐ (a : α) ∂μ, ∀ (n : ι), 0 ≤ bound n a
⊢ HasSum (fun n => ∫ (a : α), F n a ∂μ) (∫ (a : α), f a ∂μ)
state_after: α : Type u_1
E : Type u_2
G : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace ℝ G
f✝ g : α → E
m : MeasurableSpace α
μ : Measure α
ι : Type u_4
inst✝ : Countable ι
F : ι → α → G
f : α → G
bound : ι → α → ℝ
hF_meas : ∀ (n : ι), AEStronglyMeasurable (F n) μ
h_bound : ∀ (n : ι), ∀ᵐ (a : α) ∂μ, ‖F n a‖ ≤ bound n a
bound_summable : ∀ᵐ (a : α) ∂μ, Summable fun n => bound n a
bound_integrable : Integrable (fun a => ∑' (n : ι), bound n a) μ
h_lim : ∀ᵐ (a : α) ∂μ, HasSum (fun n => F n a) (f a)
hb_nonneg : ∀ᵐ (a : α) ∂μ, ∀ (n : ι), 0 ≤ bound n a
hb_le_tsum : ∀ (n : ι), bound n ≤ᶠ[ae μ] fun a => ∑' (n : ι), bound n a
⊢ HasSum (fun n => ∫ (a : α), F n a ∂μ) (∫ (a : α), f a ∂μ)
tactic: have hb_le_tsum : ∀ n, bound n ≤ᵐ[μ] fun a => ∑' n, bound n a := by
    intro n
    filter_upwards [hb_nonneg, bound_summable]
      with _ ha0 ha_sum using le_tsum ha_sum _ fun i _ => ha0 i
state_before: α : Type u_1
E : Type u_2
G : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace ℝ G
f✝ g : α → E
m : MeasurableSpace α
μ : Measure α
ι : Type u_4
inst✝ : Countable ι
F : ι → α → G
f : α → G
bound : ι → α → ℝ
hF_meas : ∀ (n : ι), AEStronglyMeasurable (F n) μ
h_bound : ∀ (n : ι), ∀ᵐ (a : α) ∂μ, ‖F n a‖ ≤ bound n a
bound_summable : ∀ᵐ (a : α) ∂μ, Summable fun n => bound n a
bound_integrable : Integrable (fun a => ∑' (n : ι), bound n a) μ
h_lim : ∀ᵐ (a : α) ∂μ, HasSum (fun n => F n a) (f a)
hb_nonneg : ∀ᵐ (a : α) ∂μ, ∀ (n : ι), 0 ≤ bound n a
hb_le_tsum : ∀ (n : ι), bound n ≤ᶠ[ae μ] fun a => ∑' (n : ι), bound n a
⊢ HasSum (fun n => ∫ (a : α), F n a ∂μ) (∫ (a : α), f a ∂μ)
state_after: α : Type u_1
E : Type u_2
G : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace ℝ G
f✝ g : α → E
m : MeasurableSpace α
μ : Measure α
ι : Type u_4
inst✝ : Countable ι
F : ι → α → G
f : α → G
bound : ι → α → ℝ
hF_meas : ∀ (n : ι), AEStronglyMeasurable (F n) μ
h_bound : ∀ (n : ι), ∀ᵐ (a : α) ∂μ, ‖F n a‖ ≤ bound n a
bound_summable : ∀ᵐ (a : α) ∂μ, Summable fun n => bound n a
bound_integrable : Integrable (fun a => ∑' (n : ι), bound n a) μ
h_lim : ∀ᵐ (a : α) ∂μ, HasSum (fun n => F n a) (f a)
hb_nonneg : ∀ᵐ (a : α) ∂μ, ∀ (n : ι), 0 ≤ bound n a
hb_le_tsum : ∀ (n : ι), bound n ≤ᶠ[ae μ] fun a => ∑' (n : ι), bound n a
hF_integrable : ∀ (n : ι), Integrable (F n) μ
⊢ HasSum (fun n => ∫ (a : α), F n a ∂μ) (∫ (a : α), f a ∂μ)
tactic: have hF_integrable : ∀ n, Integrable (F n) μ := by
    refine fun n => bound_integrable.mono' (hF_meas n) ?_
    exact EventuallyLE.trans (h_bound n) (hb_le_tsum n)
state_before: α : Type u_1
E : Type u_2
G : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace ℝ G
f✝ g : α → E
m : MeasurableSpace α
μ : Measure α
ι : Type u_4
inst✝ : Countable ι
F : ι → α → G
f : α → G
bound : ι → α → ℝ
hF_meas : ∀ (n : ι), AEStronglyMeasurable (F n) μ
h_bound : ∀ (n : ι), ∀ᵐ (a : α) ∂μ, ‖F n a‖ ≤ bound n a
bound_summable : ∀ᵐ (a : α) ∂μ, Summable fun n => bound n a
bound_integrable : Integrable (fun a => ∑' (n : ι), bound n a) μ
h_lim : ∀ᵐ (a : α) ∂μ, HasSum (fun n => F n a) (f a)
hb_nonneg : ∀ᵐ (a : α) ∂μ, ∀ (n : ι), 0 ≤ bound n a
hb_le_tsum : ∀ (n : ι), bound n ≤ᶠ[ae μ] fun a => ∑' (n : ι), bound n a
hF_integrable : ∀ (n : ι), Integrable (F n) μ
⊢ HasSum (fun n => ∫ (a : α), F n a ∂μ) (∫ (a : α), f a ∂μ)
state_after: α : Type u_1
E : Type u_2
G : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace ℝ G
f✝ g : α → E
m : MeasurableSpace α
μ : Measure α
ι : Type u_4
inst✝ : Countable ι
F : ι → α → G
f : α → G
bound : ι → α → ℝ
hF_meas : ∀ (n : ι), AEStronglyMeasurable (F n) μ
h_bound : ∀ (n : ι), ∀ᵐ (a : α) ∂μ, ‖F n a‖ ≤ bound n a
bound_summable : ∀ᵐ (a : α) ∂μ, Summable fun n => bound n a
bound_integrable : Integrable (fun a => ∑' (n : ι), bound n a) μ
h_lim : ∀ᵐ (a : α) ∂μ, HasSum (fun n => F n a) (f a)
hb_nonneg : ∀ᵐ (a : α) ∂μ, ∀ (n : ι), 0 ≤ bound n a
hb_le_tsum : ∀ (n : ι), bound n ≤ᶠ[ae μ] fun a => ∑' (n : ι), bound n a
hF_integrable : ∀ (n : ι), Integrable (F n) μ
⊢ Tendsto (fun s => ∫ (a : α), ∑ i ∈ s, F i a ∂μ) atTop (𝓝 (∫ (a : α), f a ∂μ))
tactic: simp only [HasSum, ← integral_finset_sum _ fun n _ => hF_integrable n]
state_before: α : Type u_1
E : Type u_2
G : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace ℝ G
f✝ g : α → E
m : MeasurableSpace α
μ : Measure α
ι : Type u_4
inst✝ : Countable ι
F : ι → α → G
f : α → G
bound : ι → α → ℝ
hF_meas : ∀ (n : ι), AEStronglyMeasurable (F n) μ
h_bound : ∀ (n : ι), ∀ᵐ (a : α) ∂μ, ‖F n a‖ ≤ bound n a
bound_summable : ∀ᵐ (a : α) ∂μ, Summable fun n => bound n a
bound_integrable : Integrable (fun a => ∑' (n : ι), bound n a) μ
h_lim : ∀ᵐ (a : α) ∂μ, HasSum (fun n => F n a) (f a)
hb_nonneg : ∀ᵐ (a : α) ∂μ, ∀ (n : ι), 0 ≤ bound n a
hb_le_tsum : ∀ (n : ι), bound n ≤ᶠ[ae μ] fun a => ∑' (n : ι), bound n a
hF_integrable : ∀ (n : ι), Integrable (F n) μ
⊢ Tendsto (fun s => ∫ (a : α), ∑ i ∈ s, F i a ∂μ) atTop (𝓝 (∫ (a : α), f a ∂μ))
state_after: case refine_1
α : Type u_1
E : Type u_2
G : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace ℝ G
f✝ g : α → E
m : MeasurableSpace α
μ : Measure α
ι : Type u_4
inst✝ : Countable ι
F : ι → α → G
f : α → G
bound : ι → α → ℝ
hF_meas : ∀ (n : ι), AEStronglyMeasurable (F n) μ
h_bound : ∀ (n : ι), ∀ᵐ (a : α) ∂μ, ‖F n a‖ ≤ bound n a
bound_summable : ∀ᵐ (a : α) ∂μ, Summable fun n => bound n a
bound_integrable : Integrable (fun a => ∑' (n : ι), bound n a) μ
h_lim : ∀ᵐ (a : α) ∂μ, HasSum (fun n => F n a) (f a)
hb_nonneg : ∀ᵐ (a : α) ∂μ, ∀ (n : ι), 0 ≤ bound n a
hb_le_tsum : ∀ (n : ι), bound n ≤ᶠ[ae μ] fun a => ∑' (n : ι), bound n a
hF_integrable : ∀ (n : ι), Integrable (F n) μ
⊢ ∀ᶠ (n : Finset ι) in atTop, AEStronglyMeasurable (fun a => ∑ i ∈ n, F i a) μ

case refine_2
α : Type u_1
E : Type u_2
G : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace ℝ G
f✝ g : α → E
m : MeasurableSpace α
μ : Measure α
ι : Type u_4
inst✝ : Countable ι
F : ι → α → G
f : α → G
bound : ι → α → ℝ
hF_meas : ∀ (n : ι), AEStronglyMeasurable (F n) μ
h_bound : ∀ (n : ι), ∀ᵐ (a : α) ∂μ, ‖F n a‖ ≤ bound n a
bound_summable : ∀ᵐ (a : α) ∂μ, Summable fun n => bound n a
bound_integrable : Integrable (fun a => ∑' (n : ι), bound n a) μ
h_lim : ∀ᵐ (a : α) ∂μ, HasSum (fun n => F n a) (f a)
hb_nonneg : ∀ᵐ (a : α) ∂μ, ∀ (n : ι), 0 ≤ bound n a
hb_le_tsum : ∀ (n : ι), bound n ≤ᶠ[ae μ] fun a => ∑' (n : ι), bound n a
hF_integrable : ∀ (n : ι), Integrable (F n) μ
⊢ ∀ᶠ (n : Finset ι) in atTop, ∀ᵐ (a : α) ∂μ, ‖∑ i ∈ n, F i a‖ ≤ (fun a => ∑' (n : ι), bound n a) a
tactic: refine tendsto_integral_filter_of_dominated_convergence
      (fun a => ∑' n, bound n a) ?_ ?_ bound_integrable h_lim