-- Mathlib/MeasureTheory/Integral/Bochner.lean
theorem hasSum_integral_measure {ι} {m : MeasurableSpace α} {f : α → G} {μ : ι → Measure α}
    (hf : Integrable f (Measure.sum μ)) :
    HasSum (fun i => ∫ a, f a ∂μ i) (∫ a, f a ∂Measure.sum μ)

Proof state:
case intro.intro
α : Type u_1
G : Type u_5
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
ι : Type u_7
m : MeasurableSpace α
f : α → G
μ : ι → Measure α
hfi : ∀ (i : ι), Integrable f (μ i)
ε : ℝ≥0
ε0 : 0 < ↑ε
hf_lt : ∫⁻ (x : α), ‖f x‖ₑ ∂Measure.sum μ < ⊤
hmem : ∀ᶠ (y : ℝ≥0∞) in 𝓝 (∫⁻ (x : α), ‖f x‖ₑ ∂Measure.sum μ), ∫⁻ (x : α), ‖f x‖ₑ ∂Measure.sum μ < y + ↑ε
s : Finset ι
hs : ∫⁻ (x : α), ‖f x‖ₑ ∂Measure.sum μ < ∑ b ∈ s, (fun i => ∫⁻ (a : α), ‖f a‖ₑ ∂μ i) b + ↑ε
ν : Measure α
hf : Integrable f (∑ i ∈ s, μ i) ∧ Integrable f ν
hν : ∑ i ∈ s, μ i + ν = Measure.sum μ
⊢ ∫⁻ (x : α), ‖f x‖ₑ ∂ν < ↑ε

Tactic:
rw [← hν, lintegral_add_measure, lintegral_finset_sum_measure] at hs

Result:
case intro.intro
α : Type u_1
G : Type u_5
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
ι : Type u_7
m : MeasurableSpace α
f : α → G
μ : ι → Measure α
hfi : ∀ (i : ι), Integrable f (μ i)
ε : ℝ≥0
ε0 : 0 < ↑ε
hf_lt : ∫⁻ (x : α), ‖f x‖ₑ ∂Measure.sum μ < ⊤
hmem : ∀ᶠ (y : ℝ≥0∞) in 𝓝 (∫⁻ (x : α), ‖f x‖ₑ ∂Measure.sum μ), ∫⁻ (x : α), ‖f x‖ₑ ∂Measure.sum μ < y + ↑ε
s : Finset ι
ν : Measure α
hs : ∑ i ∈ s, ∫⁻ (a : α), ‖f a‖ₑ ∂μ i + ∫⁻ (a : α), ‖f a‖ₑ ∂ν < ∑ b ∈ s, (fun i => ∫⁻ (a : α), ‖f a‖ₑ ∂μ i) b + ↑ε
hf : Integrable f (∑ i ∈ s, μ i) ∧ Integrable f ν
hν : ∑ i ∈ s, μ i + ν = Measure.sum μ
⊢ ∫⁻ (x : α), ‖f x‖ₑ ∂ν < ↑ε

Proof state:
α : Type u_1
G : Type u_5
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
ι : Type u_7
m : MeasurableSpace α
f : α → G
μ : ι → Measure α
hf : Integrable f (Measure.sum μ)
hfi : ∀ (i : ι), Integrable f (μ i)
⊢ Tendsto (fun s => ∫ (a : α), f a ∂∑ i ∈ s, μ i) atTop (𝓝 (∫ (a : α), f a ∂Measure.sum μ))

Tactic:
refine Metric.nhds_basis_ball.tendsto_right_iff.mpr fun ε ε0 => ?_

Result:
α : Type u_1
G : Type u_5
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
ι : Type u_7
m : MeasurableSpace α
f : α → G
μ : ι → Measure α
hf : Integrable f (Measure.sum μ)
hfi : ∀ (i : ι), Integrable f (μ i)
ε : ℝ
ε0 : 0 < ε
⊢ ∀ᶠ (x : Finset ι) in atTop, ∫ (a : α), f a ∂∑ i ∈ x, μ i ∈ Metric.ball (∫ (a : α), f a ∂Measure.sum μ) ε

Proof state:
case intro
α : Type u_1
G : Type u_5
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
ι : Type u_7
m : MeasurableSpace α
f : α → G
μ : ι → Measure α
hf : Integrable f (Measure.sum μ)
hfi : ∀ (i : ι), Integrable f (μ i)
ε : ℝ≥0
ε0 : 0 < ↑ε
hf_lt : ∫⁻ (x : α), ‖f x‖ₑ ∂Measure.sum μ < ⊤
hmem : ∀ᶠ (y : ℝ≥0∞) in 𝓝 (∫⁻ (x : α), ‖f x‖ₑ ∂Measure.sum μ), ∫⁻ (x : α), ‖f x‖ₑ ∂Measure.sum μ < y + ↑ε
⊢ ∀ᶠ (x : Finset ι) in atTop, ∫ (a : α), f a ∂∑ i ∈ x, μ i ∈ Metric.ball (∫ (a : α), f a ∂Measure.sum μ) ↑ε

Tactic:
refine ((hasSum_lintegral_measure (fun x => ‖f x‖ₑ) μ).eventually hmem).mono fun s hs => ?_

Result:
case intro
α : Type u_1
G : Type u_5
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
ι : Type u_7
m : MeasurableSpace α
f : α → G
μ : ι → Measure α
hf : Integrable f (Measure.sum μ)
hfi : ∀ (i : ι), Integrable f (μ i)
ε : ℝ≥0
ε0 : 0 < ↑ε
hf_lt : ∫⁻ (x : α), ‖f x‖ₑ ∂Measure.sum μ < ⊤
hmem : ∀ᶠ (y : ℝ≥0∞) in 𝓝 (∫⁻ (x : α), ‖f x‖ₑ ∂Measure.sum μ), ∫⁻ (x : α), ‖f x‖ₑ ∂Measure.sum μ < y + ↑ε
s : Finset ι
hs : ∫⁻ (x : α), ‖f x‖ₑ ∂Measure.sum μ < ∑ b ∈ s, (fun i => ∫⁻ (a : α), ‖f a‖ₑ ∂μ i) b + ↑ε
⊢ ∫ (a : α), f a ∂∑ i ∈ s, μ i ∈ Metric.ball (∫ (a : α), f a ∂Measure.sum μ) ↑ε