Theorem: MeasureTheory.integral_finset_sum_measure
File: Mathlib/MeasureTheory/Integral/Bochner.lean

tactic: induction s using Finset.cons_induction_on with
| h₁ => simp
| h₂ h ih =>
  rw [Finset.forall_mem_cons] at hf
  rw [Finset.sum_cons, Finset.sum_cons, ← ih hf.2]
  exact integral_add_measure hf.1 (integrable_finset_sum_measure.2 hf.2)
state_before: α : Type u_2
E : Type u_3
F : Type ?u.1309166
𝕜 : Type ?u.1309169
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace ℝ E
inst✝⁸ : CompleteSpace E
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : SMulCommClass ℝ 𝕜 E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
inst✝² : CompleteSpace F
f✝ g : α → E
m✝ : MeasurableSpace α
μ✝ : Measure α
X : Type ?u.1311860
inst✝¹ : TopologicalSpace X
inst✝ : FirstCountableTopology X
ν : Measure α
ι : Type u_1
m : MeasurableSpace α
f : α → E
μ : ι → Measure α
s : Finset ι
hf : ∀ (i : ι), i ∈ s → Integrable f
⊢ (∫ (a : α), f a ∂∑ i in s, μ i) = ∑ i in s, ∫ (a : α), f a ∂μ i

tactic: simp
state_before: case h₁
α : Type u_2
E : Type u_3
F : Type ?u.1309166
𝕜 : Type ?u.1309169
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace ℝ E
inst✝⁸ : CompleteSpace E
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : SMulCommClass ℝ 𝕜 E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
inst✝² : CompleteSpace F
f✝ g : α → E
m✝ : MeasurableSpace α
μ✝ : Measure α
X : Type ?u.1311860
inst✝¹ : TopologicalSpace X
inst✝ : FirstCountableTopology X
ν : Measure α
ι : Type u_1
m : MeasurableSpace α
f : α → E
μ : ι → Measure α
hf : ∀ (i : ι), i ∈ ∅ → Integrable f
⊢ (∫ (a : α), f a ∂∑ i in ∅, μ i) = ∑ i in ∅, ∫ (a : α), f a ∂μ i

tactic: rw [Finset.forall_mem_cons] at hf
state_before: case h₂
α : Type u_2
E : Type u_3
F : Type ?u.1309166
𝕜 : Type ?u.1309169
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace ℝ E
inst✝⁸ : CompleteSpace E
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : SMulCommClass ℝ 𝕜 E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
inst✝² : CompleteSpace F
f✝ g : α → E
m✝ : MeasurableSpace α
μ✝ : Measure α
X : Type ?u.1311860
inst✝¹ : TopologicalSpace X
inst✝ : FirstCountableTopology X
ν : Measure α
ι : Type u_1
m : MeasurableSpace α
f : α → E
μ : ι → Measure α
a✝ : ι
s✝ : Finset ι
h : ¬a✝ ∈ s✝
ih : (∀ (i : ι), i ∈ s✝ → Integrable f) → (∫ (a : α), f a ∂∑ i in s✝, μ i) = ∑ i in s✝, ∫ (a : α), f a ∂μ i
hf : ∀ (i : ι), i ∈ Finset.cons a✝ s✝ h → Integrable f
⊢ (∫ (a : α), f a ∂∑ i in Finset.cons a✝ s✝ h, μ i) = ∑ i in Finset.cons a✝ s✝ h, ∫ (a : α), f a ∂μ i

tactic: rw [Finset.sum_cons, Finset.sum_cons, ← ih hf.2]
state_before: case h₂
α : Type u_2
E : Type u_3
F : Type ?u.1309166
𝕜 : Type ?u.1309169
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace ℝ E
inst✝⁸ : CompleteSpace E
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : SMulCommClass ℝ 𝕜 E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
inst✝² : CompleteSpace F
f✝ g : α → E
m✝ : MeasurableSpace α
μ✝ : Measure α
X : Type ?u.1311860
inst✝¹ : TopologicalSpace X
inst✝ : FirstCountableTopology X
ν : Measure α
ι : Type u_1
m : MeasurableSpace α
f : α → E
μ : ι → Measure α
a✝ : ι
s✝ : Finset ι
h : ¬a✝ ∈ s✝
ih : (∀ (i : ι), i ∈ s✝ → Integrable f) → (∫ (a : α), f a ∂∑ i in s✝, μ i) = ∑ i in s✝, ∫ (a : α), f a ∂μ i
hf : Integrable f ∧ ∀ (x : ι), x ∈ s✝ → Integrable f
⊢ (∫ (a : α), f a ∂∑ i in Finset.cons a✝ s✝ h, μ i) = ∑ i in Finset.cons a✝ s✝ h, ∫ (a : α), f a ∂μ i

tactic: exact integral_add_measure hf.1 (integrable_finset_sum_measure.2 hf.2)
state_before: case h₂
α : Type u_2
E : Type u_3
F : Type ?u.1309166
𝕜 : Type ?u.1309169
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace ℝ E
inst✝⁸ : CompleteSpace E
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : SMulCommClass ℝ 𝕜 E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
inst✝² : CompleteSpace F
f✝ g : α → E
m✝ : MeasurableSpace α
μ✝ : Measure α
X : Type ?u.1311860
inst✝¹ : TopologicalSpace X
inst✝ : FirstCountableTopology X
ν : Measure α
ι : Type u_1
m : MeasurableSpace α
f : α → E
μ : ι → Measure α
a✝ : ι
s✝ : Finset ι
h : ¬a✝ ∈ s✝
ih : (∀ (i : ι), i ∈ s✝ → Integrable f) → (∫ (a : α), f a ∂∑ i in s✝, μ i) = ∑ i in s✝, ∫ (a : α), f a ∂μ i
hf : Integrable f ∧ ∀ (x : ι), x ∈ s✝ → Integrable f
⊢ (∫ (a : α), f a ∂μ a✝ + ∑ x in s✝, μ x) = (∫ (a : α), f a ∂μ a✝) + ∫ (a : α), f a ∂∑ i in s✝, μ i