Theorem: MeasureTheory.integral_finset_biUnion
File: Mathlib/MeasureTheory/Integral/SetIntegral.lean

tactic: induction' t using Finset.induction_on with a t hat IH hs h's
state_before: X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
inst✝² : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f g : X → E
s✝ t✝ : Set X
μ ν : Measure X
l l' : Filter X
ι : Type u_5
t : Finset ι
s : ι → Set X
hs : ∀ i ∈ t, MeasurableSet (s i)
h's : (↑t).Pairwise (Disjoint on s)
hf : ∀ i ∈ t, IntegrableOn f (s i) μ
⊢ ∫ (x : X) in ⋃ i ∈ t, s i, f x ∂μ = ∑ i ∈ t, ∫ (x : X) in s i, f x ∂μ
state_after: case empty
X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
inst✝² : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f g : X → E
s✝ t : Set X
μ ν : Measure X
l l' : Filter X
ι : Type u_5
s : ι → Set X
hs : ∀ i ∈ ∅, MeasurableSet (s i)
h's : (↑∅).Pairwise (Disjoint on s)
hf : ∀ i ∈ ∅, IntegrableOn f (s i) μ
⊢ ∫ (x : X) in ⋃ i ∈ ∅, s i, f x ∂μ = ∑ i ∈ ∅, ∫ (x : X) in s i, f x ∂μ

case insert
X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
inst✝² : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f g : X → E
s✝ t✝ : Set X
μ ν : Measure X
l l' : Filter X
ι : Type u_5
s : ι → Set X
a : ι
t : Finset ι
hat : a ∉ t
IH :
  (∀ i ∈ t, MeasurableSet (s i)) →
    (↑t).Pairwise (Disjoint on s) →
      (∀ i ∈ t, IntegrableOn f (s i) μ) → ∫ (x : X) in ⋃ i ∈ t, s i, f x ∂μ = ∑ i ∈ t, ∫ (x : X) in s i, f x ∂μ
hs : ∀ i ∈ insert a t, MeasurableSet (s i)
h's : (↑(insert a t)).Pairwise (Disjoint on s)
hf : ∀ i ∈ insert a t, IntegrableOn f (s i) μ
⊢ ∫ (x : X) in ⋃ i ∈ insert a t, s i, f x ∂μ = ∑ i ∈ insert a t, ∫ (x : X) in s i, f x ∂μ

tactic: simp
state_before: case empty
X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
inst✝² : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f g : X → E
s✝ t : Set X
μ ν : Measure X
l l' : Filter X
ι : Type u_5
s : ι → Set X
hs : ∀ i ∈ ∅, MeasurableSet (s i)
h's : (↑∅).Pairwise (Disjoint on s)
hf : ∀ i ∈ ∅, IntegrableOn f (s i) μ
⊢ ∫ (x : X) in ⋃ i ∈ ∅, s i, f x ∂μ = ∑ i ∈ ∅, ∫ (x : X) in s i, f x ∂μ
state_after: no goals

tactic: simp only [Finset.coe_insert, Finset.forall_mem_insert, Set.pairwise_insert,
  Finset.set_biUnion_insert] at hs hf h's ⊢
state_before: case insert
X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
inst✝² : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f g : X → E
s✝ t✝ : Set X
μ ν : Measure X
l l' : Filter X
ι : Type u_5
s : ι → Set X
a : ι
t : Finset ι
hat : a ∉ t
IH :
  (∀ i ∈ t, MeasurableSet (s i)) →
    (↑t).Pairwise (Disjoint on s) →
      (∀ i ∈ t, IntegrableOn f (s i) μ) → ∫ (x : X) in ⋃ i ∈ t, s i, f x ∂μ = ∑ i ∈ t, ∫ (x : X) in s i, f x ∂μ
hs : ∀ i ∈ insert a t, MeasurableSet (s i)
h's : (↑(insert a t)).Pairwise (Disjoint on s)
hf : ∀ i ∈ insert a t, IntegrableOn f (s i) μ
⊢ ∫ (x : X) in ⋃ i ∈ insert a t, s i, f x ∂μ = ∑ i ∈ insert a t, ∫ (x : X) in s i, f x ∂μ
state_after: case insert
X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
inst✝² : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f g : X → E
s✝ t✝ : Set X
μ ν : Measure X
l l' : Filter X
ι : Type u_5
s : ι → Set X
a : ι
t : Finset ι
hat : a ∉ t
IH :
  (∀ i ∈ t, MeasurableSet (s i)) →
    (↑t).Pairwise (Disjoint on s) →
      (∀ i ∈ t, IntegrableOn f (s i) μ) → ∫ (x : X) in ⋃ i ∈ t, s i, f x ∂μ = ∑ i ∈ t, ∫ (x : X) in s i, f x ∂μ
hs : MeasurableSet (s a) ∧ ∀ x ∈ t, MeasurableSet (s x)
hf : IntegrableOn f (s a) μ ∧ ∀ x ∈ t, IntegrableOn f (s x) μ
h's : (↑t).Pairwise (Disjoint on s) ∧ ∀ b ∈ ↑t, a ≠ b → (Disjoint on s) a b ∧ (Disjoint on s) b a
⊢ ∫ (x : X) in s a ∪ ⋃ x ∈ t, s x, f x ∂μ = ∑ i ∈ insert a t, ∫ (x : X) in s i, f x ∂μ

tactic: rw [integral_union _ _ hf.1 (integrableOn_finset_iUnion.2 hf.2)]
state_before: case insert
X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
inst✝² : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f g : X → E
s✝ t✝ : Set X
μ ν : Measure X
l l' : Filter X
ι : Type u_5
s : ι → Set X
a : ι
t : Finset ι
hat : a ∉ t
IH :
  (∀ i ∈ t, MeasurableSet (s i)) →
    (↑t).Pairwise (Disjoint on s) →
      (∀ i ∈ t, IntegrableOn f (s i) μ) → ∫ (x : X) in ⋃ i ∈ t, s i, f x ∂μ = ∑ i ∈ t, ∫ (x : X) in s i, f x ∂μ
hs : MeasurableSet (s a) ∧ ∀ x ∈ t, MeasurableSet (s x)
hf : IntegrableOn f (s a) μ ∧ ∀ x ∈ t, IntegrableOn f (s x) μ
h's : (↑t).Pairwise (Disjoint on s) ∧ ∀ b ∈ ↑t, a ≠ b → (Disjoint on s) a b ∧ (Disjoint on s) b a
⊢ ∫ (x : X) in s a ∪ ⋃ x ∈ t, s x, f x ∂μ = ∑ i ∈ insert a t, ∫ (x : X) in s i, f x ∂μ
state_after: case insert
X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
inst✝² : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f g : X → E
s✝ t✝ : Set X
μ ν : Measure X
l l' : Filter X
ι : Type u_5
s : ι → Set X
a : ι
t : Finset ι
hat : a ∉ t
IH :
  (∀ i ∈ t, MeasurableSet (s i)) →
    (↑t).Pairwise (Disjoint on s) →
      (∀ i ∈ t, IntegrableOn f (s i) μ) → ∫ (x : X) in ⋃ i ∈ t, s i, f x ∂μ = ∑ i ∈ t, ∫ (x : X) in s i, f x ∂μ
hs : MeasurableSet (s a) ∧ ∀ x ∈ t, MeasurableSet (s x)
hf : IntegrableOn f (s a) μ ∧ ∀ x ∈ t, IntegrableOn f (s x) μ
h's : (↑t).Pairwise (Disjoint on s) ∧ ∀ b ∈ ↑t, a ≠ b → (Disjoint on s) a b ∧ (Disjoint on s) b a
⊢ ∫ (x : X) in s a, f x ∂μ + ∫ (x : X) in ⋃ i ∈ t, s i, f x ∂μ = ∑ i ∈ insert a t, ∫ (x : X) in s i, f x ∂μ

X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
inst✝² : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f g : X → E
s✝ t✝ : Set X
μ ν : Measure X
l l' : Filter X
ι : Type u_5
s : ι → Set X
a : ι
t : Finset ι
hat : a ∉ t
IH :
  (∀ i ∈ t, MeasurableSet (s i)) →
    (↑t).Pairwise (Disjoint on s) →
      (∀ i ∈ t, IntegrableOn f (s i) μ) → ∫ (x : X) in ⋃ i ∈ t, s i, f x ∂μ = ∑ i ∈ t, ∫ (x : X) in s i, f x ∂μ
hs : MeasurableSet (s a) ∧ ∀ x ∈ t, MeasurableSet (s x)
hf : IntegrableOn f (s a) μ ∧ ∀ x ∈ t, IntegrableOn f (s x) μ
h's : (↑t).Pairwise (Disjoint on s) ∧ ∀ b ∈ ↑t, a ≠ b → (Disjoint on s) a b ∧ (Disjoint on s) b a
⊢ Disjoint (s a) (⋃ i ∈ t, s i)

X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
inst✝² : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f g : X → E
s✝ t✝ : Set X
μ ν : Measure X
l l' : Filter X
ι : Type u_5
s : ι → Set X
a : ι
t : Finset ι
hat : a ∉ t
IH :
  (∀ i ∈ t, MeasurableSet (s i)) →
    (↑t).Pairwise (Disjoint on s) →
      (∀ i ∈ t, IntegrableOn f (s i) μ) → ∫ (x : X) in ⋃ i ∈ t, s i, f x ∂μ = ∑ i ∈ t, ∫ (x : X) in s i, f x ∂μ
hs : MeasurableSet (s a) ∧ ∀ x ∈ t, MeasurableSet (s x)
hf : IntegrableOn f (s a) μ ∧ ∀ x ∈ t, IntegrableOn f (s x) μ
h's : (↑t).Pairwise (Disjoint on s) ∧ ∀ b ∈ ↑t, a ≠ b → (Disjoint on s) a b ∧ (Disjoint on s) b a
⊢ MeasurableSet (⋃ i ∈ t, s i)

tactic: rw [Finset.sum_insert hat, IH hs.2 h's.1 hf.2]
state_before: case insert
X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
inst✝² : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f g : X → E
s✝ t✝ : Set X
μ ν : Measure X
l l' : Filter X
ι : Type u_5
s : ι → Set X
a : ι
t : Finset ι
hat : a ∉ t
IH :
  (∀ i ∈ t, MeasurableSet (s i)) →
    (↑t).Pairwise (Disjoint on s) →
      (∀ i ∈ t, IntegrableOn f (s i) μ) → ∫ (x : X) in ⋃ i ∈ t, s i, f x ∂μ = ∑ i ∈ t, ∫ (x : X) in s i, f x ∂μ
hs : MeasurableSet (s a) ∧ ∀ x ∈ t, MeasurableSet (s x)
hf : IntegrableOn f (s a) μ ∧ ∀ x ∈ t, IntegrableOn f (s x) μ
h's : (↑t).Pairwise (Disjoint on s) ∧ ∀ b ∈ ↑t, a ≠ b → (Disjoint on s) a b ∧ (Disjoint on s) b a
⊢ ∫ (x : X) in s a, f x ∂μ + ∫ (x : X) in ⋃ i ∈ t, s i, f x ∂μ = ∑ i ∈ insert a t, ∫ (x : X) in s i, f x ∂μ
state_after: no goals

tactic: simp only [disjoint_iUnion_right]
state_before: X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
inst✝² : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f g : X → E
s✝ t✝ : Set X
μ ν : Measure X
l l' : Filter X
ι : Type u_5
s : ι → Set X
a : ι
t : Finset ι
hat : a ∉ t
IH :
  (∀ i ∈ t, MeasurableSet (s i)) →
    (↑t).Pairwise (Disjoint on s) →
      (∀ i ∈ t, IntegrableOn f (s i) μ) → ∫ (x : X) in ⋃ i ∈ t, s i, f x ∂μ = ∑ i ∈ t, ∫ (x : X) in s i, f x ∂μ
hs : MeasurableSet (s a) ∧ ∀ x ∈ t, MeasurableSet (s x)
hf : IntegrableOn f (s a) μ ∧ ∀ x ∈ t, IntegrableOn f (s x) μ
h's : (↑t).Pairwise (Disjoint on s) ∧ ∀ b ∈ ↑t, a ≠ b → (Disjoint on s) a b ∧ (Disjoint on s) b a
⊢ Disjoint (s a) (⋃ i ∈ t, s i)
state_after: X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
inst✝² : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f g : X → E
s✝ t✝ : Set X
μ ν : Measure X
l l' : Filter X
ι : Type u_5
s : ι → Set X
a : ι
t : Finset ι
hat : a ∉ t
IH :
  (∀ i ∈ t, MeasurableSet (s i)) →
    (↑t).Pairwise (Disjoint on s) →
      (∀ i ∈ t, IntegrableOn f (s i) μ) → ∫ (x : X) in ⋃ i ∈ t, s i, f x ∂μ = ∑ i ∈ t, ∫ (x : X) in s i, f x ∂μ
hs : MeasurableSet (s a) ∧ ∀ x ∈ t, MeasurableSet (s x)
hf : IntegrableOn f (s a) μ ∧ ∀ x ∈ t, IntegrableOn f (s x) μ
h's : (↑t).Pairwise (Disjoint on s) ∧ ∀ b ∈ ↑t, a ≠ b → (Disjoint on s) a b ∧ (Disjoint on s) b a
⊢ ∀ i ∈ t, Disjoint (s a) (s i)

tactic: exact fun i hi => (h's.2 i hi (ne_of_mem_of_not_mem hi hat).symm).1
state_before: X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
inst✝² : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f g : X → E
s✝ t✝ : Set X
μ ν : Measure X
l l' : Filter X
ι : Type u_5
s : ι → Set X
a : ι
t : Finset ι
hat : a ∉ t
IH :
  (∀ i ∈ t, MeasurableSet (s i)) →
    (↑t).Pairwise (Disjoint on s) →
      (∀ i ∈ t, IntegrableOn f (s i) μ) → ∫ (x : X) in ⋃ i ∈ t, s i, f x ∂μ = ∑ i ∈ t, ∫ (x : X) in s i, f x ∂μ
hs : MeasurableSet (s a) ∧ ∀ x ∈ t, MeasurableSet (s x)
hf : IntegrableOn f (s a) μ ∧ ∀ x ∈ t, IntegrableOn f (s x) μ
h's : (↑t).Pairwise (Disjoint on s) ∧ ∀ b ∈ ↑t, a ≠ b → (Disjoint on s) a b ∧ (Disjoint on s) b a
⊢ ∀ i ∈ t, Disjoint (s a) (s i)
state_after: no goals

tactic: exact Finset.measurableSet_biUnion _ hs.2
state_before: X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
inst✝² : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f g : X → E
s✝ t✝ : Set X
μ ν : Measure X
l l' : Filter X
ι : Type u_5
s : ι → Set X
a : ι
t : Finset ι
hat : a ∉ t
IH :
  (∀ i ∈ t, MeasurableSet (s i)) →
    (↑t).Pairwise (Disjoint on s) →
      (∀ i ∈ t, IntegrableOn f (s i) μ) → ∫ (x : X) in ⋃ i ∈ t, s i, f x ∂μ = ∑ i ∈ t, ∫ (x : X) in s i, f x ∂μ
hs : MeasurableSet (s a) ∧ ∀ x ∈ t, MeasurableSet (s x)
hf : IntegrableOn f (s a) μ ∧ ∀ x ∈ t, IntegrableOn f (s x) μ
h's : (↑t).Pairwise (Disjoint on s) ∧ ∀ b ∈ ↑t, a ≠ b → (Disjoint on s) a b ∧ (Disjoint on s) b a
⊢ MeasurableSet (⋃ i ∈ t, s i)
state_after: no goals