Theorem: ProbabilityTheory.kernel.setIntegral_density
File: Mathlib/Probability/Kernel/Disintegration/Density.lean

tactic: have : IsFiniteKernel κ := isFiniteKernel_of_isFiniteKernel_fst (h := isFiniteKernel_of_le hκν)
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : ↥(kernel α (γ × β))
ν : ↥(kernel α γ)
hκν : fst κ ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
A : Set γ
hA : MeasurableSet A
⊢ ∫ (x : γ) in A, density κ ν a x s ∂ν a = ((κ a) (A ×ˢ s)).toReal
state_after: α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : ↥(kernel α (γ × β))
ν : ↥(kernel α γ)
hκν : fst κ ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
A : Set γ
hA : MeasurableSet A
this : IsFiniteKernel κ
⊢ ∫ (x : γ) in A, density κ ν a x s ∂ν a = ((κ a) (A ×ˢ s)).toReal

tactic: have hA' : MeasurableSet[⨆ n, countableFiltration γ n] A := by rwa [iSup_countableFiltration]
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : ↥(kernel α (γ × β))
ν : ↥(kernel α γ)
hκν : fst κ ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
A : Set γ
hA : MeasurableSet A
this : IsFiniteKernel κ
⊢ ∫ (x : γ) in A, density κ ν a x s ∂ν a = ((κ a) (A ×ˢ s)).toReal
state_after: α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : ↥(kernel α (γ × β))
ν : ↥(kernel α γ)
hκν : fst κ ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
A : Set γ
hA : MeasurableSet A
this : IsFiniteKernel κ
hA' : MeasurableSet A
⊢ ∫ (x : γ) in A, density κ ν a x s ∂ν a = ((κ a) (A ×ˢ s)).toReal

tactic: refine induction_on_inter (m := ⨆ n, countableFiltration γ n)
  (C := fun A ↦ ∫ x in A, density κ ν a x s ∂(ν a) = (κ a (A ×ˢ s)).toReal)
  (measurableSpace_iSup_eq (countableFiltration γ)) ?_ ?_ ?_ ?_ ?_ hA'
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : ↥(kernel α (γ × β))
ν : ↥(kernel α γ)
hκν : fst κ ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
A : Set γ
hA : MeasurableSet A
this : IsFiniteKernel κ
hA' : MeasurableSet A
⊢ ∫ (x : γ) in A, density κ ν a x s ∂ν a = ((κ a) (A ×ˢ s)).toReal
state_after: case refine_1
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : ↥(kernel α (γ × β))
ν : ↥(kernel α γ)
hκν : fst κ ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
A : Set γ
hA : MeasurableSet A
this : IsFiniteKernel κ
hA' : MeasurableSet A
⊢ IsPiSystem {s | ∃ n, MeasurableSet s}

case refine_2
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : ↥(kernel α (γ × β))
ν : ↥(kernel α γ)
hκν : fst κ ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
A : Set γ
hA : MeasurableSet A
this : IsFiniteKernel κ
hA' : MeasurableSet A
⊢ (fun A => ∫ (x : γ) in A, density κ ν a x s ∂ν a = ((κ a) (A ×ˢ s)).toReal) ∅

case refine_3
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : ↥(kernel α (γ × β))
ν : ↥(kernel α γ)
hκν : fst κ ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
A : Set γ
hA : MeasurableSet A
this : IsFiniteKernel κ
hA' : MeasurableSet A
⊢ ∀ t ∈ {s | ∃ n, MeasurableSet s}, (fun A => ∫ (x : γ) in A, density κ ν a x s ∂ν a = ((κ a) (A ×ˢ s)).toReal) t

case refine_4
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : ↥(kernel α (γ × β))
ν : ↥(kernel α γ)
hκν : fst κ ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
A : Set γ
hA : MeasurableSet A
this : IsFiniteKernel κ
hA' : MeasurableSet A
⊢ ∀ (t : Set γ),
    MeasurableSet t →
      (fun A => ∫ (x : γ) in A, density κ ν a x s ∂ν a = ((κ a) (A ×ˢ s)).toReal) t →
        (fun A => ∫ (x : γ) in A, density κ ν a x s ∂ν a = ((κ a) (A ×ˢ s)).toReal) tᶜ

case refine_5
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : ↥(kernel α (γ × β))
ν : ↥(kernel α γ)
hκν : fst κ ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
A : Set γ
hA : MeasurableSet A
this : IsFiniteKernel κ
hA' : MeasurableSet A
⊢ ∀ (f : ℕ → Set γ),
    Pairwise (Disjoint on f) →
      (∀ (i : ℕ), MeasurableSet (f i)) →
        (∀ (i : ℕ), (fun A => ∫ (x : γ) in A, density κ ν a x s ∂ν a = ((κ a) (A ×ˢ s)).toReal) (f i)) →
          (fun A => ∫ (x : γ) in A, density κ ν a x s ∂ν a = ((κ a) (A ×ˢ s)).toReal) (⋃ i, f i)

tactic: rwa [iSup_countableFiltration]
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : ↥(kernel α (γ × β))
ν : ↥(kernel α γ)
hκν : fst κ ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
A : Set γ
hA : MeasurableSet A
this : IsFiniteKernel κ
⊢ MeasurableSet A
state_after: no goals

tactic: rintro s ⟨n, hs⟩ t ⟨m, ht⟩ _
state_before: case refine_1
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : ↥(kernel α (γ × β))
ν : ↥(kernel α γ)
hκν : fst κ ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
A : Set γ
hA : MeasurableSet A
this : IsFiniteKernel κ
hA' : MeasurableSet A
⊢ IsPiSystem {s | ∃ n, MeasurableSet s}
state_after: case refine_1.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : ↥(kernel α (γ × β))
ν : ↥(kernel α γ)
hκν : fst κ ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s✝ : Set β
hs✝ : MeasurableSet s✝
A : Set γ
hA : MeasurableSet A
this : IsFiniteKernel κ
hA' : MeasurableSet A
s : Set γ
n : ℕ
hs : MeasurableSet s
t : Set γ
m : ℕ
ht : MeasurableSet t
a✝ : (s ∩ t).Nonempty
⊢ s ∩ t ∈ {s | ∃ n, MeasurableSet s}

tactic: exact ⟨max n m, ((countableFiltration γ).mono (le_max_left n m) _ hs).inter
  ((countableFiltration γ).mono (le_max_right n m) _ ht)⟩
state_before: case refine_1.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : ↥(kernel α (γ × β))
ν : ↥(kernel α γ)
hκν : fst κ ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s✝ : Set β
hs✝ : MeasurableSet s✝
A : Set γ
hA : MeasurableSet A
this : IsFiniteKernel κ
hA' : MeasurableSet A
s : Set γ
n : ℕ
hs : MeasurableSet s
t : Set γ
m : ℕ
ht : MeasurableSet t
a✝ : (s ∩ t).Nonempty
⊢ s ∩ t ∈ {s | ∃ n, MeasurableSet s}
state_after: no goals

tactic: simp
state_before: case refine_2
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : ↥(kernel α (γ × β))
ν : ↥(kernel α γ)
hκν : fst κ ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
A : Set γ
hA : MeasurableSet A
this : IsFiniteKernel κ
hA' : MeasurableSet A
⊢ (fun A => ∫ (x : γ) in A, density κ ν a x s ∂ν a = ((κ a) (A ×ˢ s)).toReal) ∅
state_after: no goals

tactic: intro A ⟨n, hA⟩
state_before: case refine_3
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : ↥(kernel α (γ × β))
ν : ↥(kernel α γ)
hκν : fst κ ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
A : Set γ
hA : MeasurableSet A
this : IsFiniteKernel κ
hA' : MeasurableSet A
⊢ ∀ t ∈ {s | ∃ n, MeasurableSet s}, (fun A => ∫ (x : γ) in A, density κ ν a x s ∂ν a = ((κ a) (A ×ˢ s)).toReal) t
state_after: case refine_3
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : ↥(kernel α (γ × β))
ν : ↥(kernel α γ)
hκν : fst κ ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
A✝ : Set γ
hA✝ : MeasurableSet A✝
this : IsFiniteKernel κ
hA' : MeasurableSet A✝
A : Set γ
n : ℕ
hA : MeasurableSet A
⊢ ∫ (x : γ) in A, density κ ν a x s ∂ν a = ((κ a) (A ×ˢ s)).toReal

tactic: exact setIntegral_density_of_measurableSet hκν n a hs hA
state_before: case refine_3
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : ↥(kernel α (γ × β))
ν : ↥(kernel α γ)
hκν : fst κ ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
A✝ : Set γ
hA✝ : MeasurableSet A✝
this : IsFiniteKernel κ
hA' : MeasurableSet A✝
A : Set γ
n : ℕ
hA : MeasurableSet A
⊢ ∫ (x : γ) in A, density κ ν a x s ∂ν a = ((κ a) (A ×ˢ s)).toReal
state_after: no goals

tactic: intro A hA hA_eq
state_before: case refine_4
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : ↥(kernel α (γ × β))
ν : ↥(kernel α γ)
hκν : fst κ ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
A : Set γ
hA : MeasurableSet A
this : IsFiniteKernel κ
hA' : MeasurableSet A
⊢ ∀ (t : Set γ),
    MeasurableSet t →
      (fun A => ∫ (x : γ) in A, density κ ν a x s ∂ν a = ((κ a) (A ×ˢ s)).toReal) t →
        (fun A => ∫ (x : γ) in A, density κ ν a x s ∂ν a = ((κ a) (A ×ˢ s)).toReal) tᶜ
state_after: case refine_4
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : ↥(kernel α (γ × β))
ν : ↥(kernel α γ)
hκν : fst κ ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
A✝ : Set γ
hA✝ : MeasurableSet A✝
this : IsFiniteKernel κ
hA' : MeasurableSet A✝
A : Set γ
hA : MeasurableSet A
hA_eq : ∫ (x : γ) in A, density κ ν a x s ∂ν a = ((κ a) (A ×ˢ s)).toReal
⊢ ∫ (x : γ) in Aᶜ, density κ ν a x s ∂ν a = ((κ a) (Aᶜ ×ˢ s)).toReal

tactic: have h := integral_add_compl hA (integrable_density hκν a hs)
state_before: case refine_4
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : ↥(kernel α (γ × β))
ν : ↥(kernel α γ)
hκν : fst κ ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
A✝ : Set γ
hA✝ : MeasurableSet A✝
this : IsFiniteKernel κ
hA' : MeasurableSet A✝
A : Set γ
hA : MeasurableSet A
hA_eq : ∫ (x : γ) in A, density κ ν a x s ∂ν a = ((κ a) (A ×ˢ s)).toReal
⊢ ∫ (x : γ) in Aᶜ, density κ ν a x s ∂ν a = ((κ a) (Aᶜ ×ˢ s)).toReal
state_after: case refine_4
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : ↥(kernel α (γ × β))
ν : ↥(kernel α γ)
hκν : fst κ ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
A✝ : Set γ
hA✝ : MeasurableSet A✝
this : IsFiniteKernel κ
hA' : MeasurableSet A✝
A : Set γ
hA : MeasurableSet A
hA_eq : ∫ (x : γ) in A, density κ ν a x s ∂ν a = ((κ a) (A ×ˢ s)).toReal
h : ∫ (x : γ) in A, density κ ν a x s ∂ν a + ∫ (x : γ) in Aᶜ, density κ ν a x s ∂ν a = ∫ (x : γ), density κ ν a x s ∂ν a
⊢ ∫ (x : γ) in Aᶜ, density κ ν a x s ∂ν a = ((κ a) (Aᶜ ×ˢ s)).toReal

tactic: rw [hA_eq, integral_density hκν a hs] at h
state_before: case refine_4
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : ↥(kernel α (γ × β))
ν : ↥(kernel α γ)
hκν : fst κ ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
A✝ : Set γ
hA✝ : MeasurableSet A✝
this : IsFiniteKernel κ
hA' : MeasurableSet A✝
A : Set γ
hA : MeasurableSet A
hA_eq : ∫ (x : γ) in A, density κ ν a x s ∂ν a = ((κ a) (A ×ˢ s)).toReal
h : ∫ (x : γ) in A, density κ ν a x s ∂ν a + ∫ (x : γ) in Aᶜ, density κ ν a x s ∂ν a = ∫ (x : γ), density κ ν a x s ∂ν a
⊢ ∫ (x : γ) in Aᶜ, density κ ν a x s ∂ν a = ((κ a) (Aᶜ ×ˢ s)).toReal
state_after: case refine_4
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : ↥(kernel α (γ × β))
ν : ↥(kernel α γ)
hκν : fst κ ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
A✝ : Set γ
hA✝ : MeasurableSet A✝
this : IsFiniteKernel κ
hA' : MeasurableSet A✝
A : Set γ
hA : MeasurableSet A
hA_eq : ∫ (x : γ) in A, density κ ν a x s ∂ν a = ((κ a) (A ×ˢ s)).toReal
h : ((κ a) (A ×ˢ s)).toReal + ∫ (x : γ) in Aᶜ, density κ ν a x s ∂ν a = ((κ a) (univ ×ˢ s)).toReal
⊢ ∫ (x : γ) in Aᶜ, density κ ν a x s ∂ν a = ((κ a) (Aᶜ ×ˢ s)).toReal

tactic: have : Aᶜ ×ˢ s = univ ×ˢ s \ A ×ˢ s := by
  rw [prod_diff_prod, compl_eq_univ_diff]
  simp
state_before: case refine_4
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : ↥(kernel α (γ × β))
ν : ↥(kernel α γ)
hκν : fst κ ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
A✝ : Set γ
hA✝ : MeasurableSet A✝
this : IsFiniteKernel κ
hA' : MeasurableSet A✝
A : Set γ
hA : MeasurableSet A
hA_eq : ∫ (x : γ) in A, density κ ν a x s ∂ν a = ((κ a) (A ×ˢ s)).toReal
h : ((κ a) (A ×ˢ s)).toReal + ∫ (x : γ) in Aᶜ, density κ ν a x s ∂ν a = ((κ a) (univ ×ˢ s)).toReal
⊢ ∫ (x : γ) in Aᶜ, density κ ν a x s ∂ν a = ((κ a) (Aᶜ ×ˢ s)).toReal
state_after: case refine_4
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : ↥(kernel α (γ × β))
ν : ↥(kernel α γ)
hκν : fst κ ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
A✝ : Set γ
hA✝ : MeasurableSet A✝
this✝ : IsFiniteKernel κ
hA' : MeasurableSet A✝
A : Set γ
hA : MeasurableSet A
hA_eq : ∫ (x : γ) in A, density κ ν a x s ∂ν a = ((κ a) (A ×ˢ s)).toReal
h : ((κ a) (A ×ˢ s)).toReal + ∫ (x : γ) in Aᶜ, density κ ν a x s ∂ν a = ((κ a) (univ ×ˢ s)).toReal
this : Aᶜ ×ˢ s = univ ×ˢ s \ A ×ˢ s
⊢ ∫ (x : γ) in Aᶜ, density κ ν a x s ∂ν a = ((κ a) (Aᶜ ×ˢ s)).toReal

tactic: rw [this, measure_diff (by intro x; simp) (hA.prod hs) (measure_ne_top (κ a) _),
  ENNReal.toReal_sub_of_le (measure_mono (by intro x; simp)) (measure_ne_top _ _)]
state_before: case refine_4
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : ↥(kernel α (γ × β))
ν : ↥(kernel α γ)
hκν : fst κ ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
A✝ : Set γ
hA✝ : MeasurableSet A✝
this✝ : IsFiniteKernel κ
hA' : MeasurableSet A✝
A : Set γ
hA : MeasurableSet A
hA_eq : ∫ (x : γ) in A, density κ ν a x s ∂ν a = ((κ a) (A ×ˢ s)).toReal
h : ((κ a) (A ×ˢ s)).toReal + ∫ (x : γ) in Aᶜ, density κ ν a x s ∂ν a = ((κ a) (univ ×ˢ s)).toReal
this : Aᶜ ×ˢ s = univ ×ˢ s \ A ×ˢ s
⊢ ∫ (x : γ) in Aᶜ, density κ ν a x s ∂ν a = ((κ a) (Aᶜ ×ˢ s)).toReal
state_after: case refine_4
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : ↥(kernel α (γ × β))
ν : ↥(kernel α γ)
hκν : fst κ ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
A✝ : Set γ
hA✝ : MeasurableSet A✝
this✝ : IsFiniteKernel κ
hA' : MeasurableSet A✝
A : Set γ
hA : MeasurableSet A
hA_eq : ∫ (x : γ) in A, density κ ν a x s ∂ν a = ((κ a) (A ×ˢ s)).toReal
h : ((κ a) (A ×ˢ s)).toReal + ∫ (x : γ) in Aᶜ, density κ ν a x s ∂ν a = ((κ a) (univ ×ˢ s)).toReal
this : Aᶜ ×ˢ s = univ ×ˢ s \ A ×ˢ s
⊢ ∫ (x : γ) in Aᶜ, density κ ν a x s ∂ν a = ((κ a) (univ ×ˢ s)).toReal - ((κ a) (A ×ˢ s)).toReal

tactic: rw [eq_tsub_iff_add_eq_of_le, add_comm]
state_before: case refine_4
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : ↥(kernel α (γ × β))
ν : ↥(kernel α γ)
hκν : fst κ ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
A✝ : Set γ
hA✝ : MeasurableSet A✝
this✝ : IsFiniteKernel κ
hA' : MeasurableSet A✝
A : Set γ
hA : MeasurableSet A
hA_eq : ∫ (x : γ) in A, density κ ν a x s ∂ν a = ((κ a) (A ×ˢ s)).toReal
h : ((κ a) (A ×ˢ s)).toReal + ∫ (x : γ) in Aᶜ, density κ ν a x s ∂ν a = ((κ a) (univ ×ˢ s)).toReal
this : Aᶜ ×ˢ s = univ ×ˢ s \ A ×ˢ s
⊢ ∫ (x : γ) in Aᶜ, density κ ν a x s ∂ν a = ((κ a) (univ ×ˢ s)).toReal - ((κ a) (A ×ˢ s)).toReal
state_after: case refine_4
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : ↥(kernel α (γ × β))
ν : ↥(kernel α γ)
hκν : fst κ ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
A✝ : Set γ
hA✝ : MeasurableSet A✝
this✝ : IsFiniteKernel κ
hA' : MeasurableSet A✝
A : Set γ
hA : MeasurableSet A
hA_eq : ∫ (x : γ) in A, density κ ν a x s ∂ν a = ((κ a) (A ×ˢ s)).toReal
h : ((κ a) (A ×ˢ s)).toReal + ∫ (x : γ) in Aᶜ, density κ ν a x s ∂ν a = ((κ a) (univ ×ˢ s)).toReal
this : Aᶜ ×ˢ s = univ ×ˢ s \ A ×ˢ s
⊢ ((κ a) (A ×ˢ s)).toReal + ∫ (x : γ) in Aᶜ, density κ ν a x s ∂ν a = ((κ a) (univ ×ˢ s)).toReal

case refine_4
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : ↥(kernel α (γ × β))
ν : ↥(kernel α γ)
hκν : fst κ ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
A✝ : Set γ
hA✝ : MeasurableSet A✝
this✝ : IsFiniteKernel κ
hA' : MeasurableSet A✝
A : Set γ
hA : MeasurableSet A
hA_eq : ∫ (x : γ) in A, density κ ν a x s ∂ν a = ((κ a) (A ×ˢ s)).toReal
h : ((κ a) (A ×ˢ s)).toReal + ∫ (x : γ) in Aᶜ, density κ ν a x s ∂ν a = ((κ a) (univ ×ˢ s)).toReal
this : Aᶜ ×ˢ s = univ ×ˢ s \ A ×ˢ s
⊢ ((κ a) (A ×ˢ s)).toReal ≤ ((κ a) (univ ×ˢ s)).toReal

tactic: rw [prod_diff_prod, compl_eq_univ_diff]
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : ↥(kernel α (γ × β))
ν : ↥(kernel α γ)
hκν : fst κ ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
A✝ : Set γ
hA✝ : MeasurableSet A✝
this : IsFiniteKernel κ
hA' : MeasurableSet A✝
A : Set γ
hA : MeasurableSet A
hA_eq : ∫ (x : γ) in A, density κ ν a x s ∂ν a = ((κ a) (A ×ˢ s)).toReal
h : ((κ a) (A ×ˢ s)).toReal + ∫ (x : γ) in Aᶜ, density κ ν a x s ∂ν a = ((κ a) (univ ×ˢ s)).toReal
⊢ Aᶜ ×ˢ s = univ ×ˢ s \ A ×ˢ s
state_after: α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : ↥(kernel α (γ × β))
ν : ↥(kernel α γ)
hκν : fst κ ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
A✝ : Set γ
hA✝ : MeasurableSet A✝
this : IsFiniteKernel κ
hA' : MeasurableSet A✝
A : Set γ
hA : MeasurableSet A
hA_eq : ∫ (x : γ) in A, density κ ν a x s ∂ν a = ((κ a) (A ×ˢ s)).toReal
h : ((κ a) (A ×ˢ s)).toReal + ∫ (x : γ) in Aᶜ, density κ ν a x s ∂ν a = ((κ a) (univ ×ˢ s)).toReal
⊢ (univ \ A) ×ˢ s = univ ×ˢ (s \ s) ∪ (univ \ A) ×ˢ s

tactic: simp
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : ↥(kernel α (γ × β))
ν : ↥(kernel α γ)
hκν : fst κ ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
A✝ : Set γ
hA✝ : MeasurableSet A✝
this : IsFiniteKernel κ
hA' : MeasurableSet A✝
A : Set γ
hA : MeasurableSet A
hA_eq : ∫ (x : γ) in A, density κ ν a x s ∂ν a = ((κ a) (A ×ˢ s)).toReal
h : ((κ a) (A ×ˢ s)).toReal + ∫ (x : γ) in Aᶜ, density κ ν a x s ∂ν a = ((κ a) (univ ×ˢ s)).toReal
⊢ (univ \ A) ×ˢ s = univ ×ˢ (s \ s) ∪ (univ \ A) ×ˢ s
state_after: no goals

tactic: intro x
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : ↥(kernel α (γ × β))
ν : ↥(kernel α γ)
hκν : fst κ ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
A✝ : Set γ
hA✝ : MeasurableSet A✝
this✝ : IsFiniteKernel κ
hA' : MeasurableSet A✝
A : Set γ
hA : MeasurableSet A
hA_eq : ∫ (x : γ) in A, density κ ν a x s ∂ν a = ((κ a) (A ×ˢ s)).toReal
h : ((κ a) (A ×ˢ s)).toReal + ∫ (x : γ) in Aᶜ, density κ ν a x s ∂ν a = ((κ a) (univ ×ˢ s)).toReal
this : Aᶜ ×ˢ s = univ ×ˢ s \ A ×ˢ s
⊢ A ×ˢ s ⊆ univ ×ˢ s
state_after: α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : ↥(kernel α (γ × β))
ν : ↥(kernel α γ)
hκν : fst κ ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
A✝ : Set γ
hA✝ : MeasurableSet A✝
this✝ : IsFiniteKernel κ
hA' : MeasurableSet A✝
A : Set γ
hA : MeasurableSet A
hA_eq : ∫ (x : γ) in A, density κ ν a x s ∂ν a = ((κ a) (A ×ˢ s)).toReal
h : ((κ a) (A ×ˢ s)).toReal + ∫ (x : γ) in Aᶜ, density κ ν a x s ∂ν a = ((κ a) (univ ×ˢ s)).toReal
this : Aᶜ ×ˢ s = univ ×ˢ s \ A ×ˢ s
x : γ × β
⊢ x ∈ A ×ˢ s → x ∈ univ ×ˢ s

tactic: simp
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : ↥(kernel α (γ × β))
ν : ↥(kernel α γ)
hκν : fst κ ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
A✝ : Set γ
hA✝ : MeasurableSet A✝
this✝ : IsFiniteKernel κ
hA' : MeasurableSet A✝
A : Set γ
hA : MeasurableSet A
hA_eq : ∫ (x : γ) in A, density κ ν a x s ∂ν a = ((κ a) (A ×ˢ s)).toReal
h : ((κ a) (A ×ˢ s)).toReal + ∫ (x : γ) in Aᶜ, density κ ν a x s ∂ν a = ((κ a) (univ ×ˢ s)).toReal
this : Aᶜ ×ˢ s = univ ×ˢ s \ A ×ˢ s
x : γ × β
⊢ x ∈ A ×ˢ s → x ∈ univ ×ˢ s
state_after: no goals

tactic: exact h
state_before: case refine_4
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : ↥(kernel α (γ × β))
ν : ↥(kernel α γ)
hκν : fst κ ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
A✝ : Set γ
hA✝ : MeasurableSet A✝
this✝ : IsFiniteKernel κ
hA' : MeasurableSet A✝
A : Set γ
hA : MeasurableSet A
hA_eq : ∫ (x : γ) in A, density κ ν a x s ∂ν a = ((κ a) (A ×ˢ s)).toReal
h : ((κ a) (A ×ˢ s)).toReal + ∫ (x : γ) in Aᶜ, density κ ν a x s ∂ν a = ((κ a) (univ ×ˢ s)).toReal
this : Aᶜ ×ˢ s = univ ×ˢ s \ A ×ˢ s
⊢ ((κ a) (A ×ˢ s)).toReal + ∫ (x : γ) in Aᶜ, density κ ν a x s ∂ν a = ((κ a) (univ ×ˢ s)).toReal
state_after: no goals

tactic: rw [ENNReal.toReal_le_toReal (measure_ne_top _ _) (measure_ne_top _ _)]
state_before: case refine_4
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : ↥(kernel α (γ × β))
ν : ↥(kernel α γ)
hκν : fst κ ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
A✝ : Set γ
hA✝ : MeasurableSet A✝
this✝ : IsFiniteKernel κ
hA' : MeasurableSet A✝
A : Set γ
hA : MeasurableSet A
hA_eq : ∫ (x : γ) in A, density κ ν a x s ∂ν a = ((κ a) (A ×ˢ s)).toReal
h : ((κ a) (A ×ˢ s)).toReal + ∫ (x : γ) in Aᶜ, density κ ν a x s ∂ν a = ((κ a) (univ ×ˢ s)).toReal
this : Aᶜ ×ˢ s = univ ×ˢ s \ A ×ˢ s
⊢ ((κ a) (A ×ˢ s)).toReal ≤ ((κ a) (univ ×ˢ s)).toReal
state_after: case refine_4
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : ↥(kernel α (γ × β))
ν : ↥(kernel α γ)
hκν : fst κ ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
A✝ : Set γ
hA✝ : MeasurableSet A✝
this✝ : IsFiniteKernel κ
hA' : MeasurableSet A✝
A : Set γ
hA : MeasurableSet A
hA_eq : ∫ (x : γ) in A, density κ ν a x s ∂ν a = ((κ a) (A ×ˢ s)).toReal
h : ((κ a) (A ×ˢ s)).toReal + ∫ (x : γ) in Aᶜ, density κ ν a x s ∂ν a = ((κ a) (univ ×ˢ s)).toReal
this : Aᶜ ×ˢ s = univ ×ˢ s \ A ×ˢ s
⊢ (κ a) (A ×ˢ s) ≤ (κ a) (univ ×ˢ s)

tactic: exact measure_mono (by intro x; simp)
state_before: case refine_4
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : ↥(kernel α (γ × β))
ν : ↥(kernel α γ)
hκν : fst κ ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
A✝ : Set γ
hA✝ : MeasurableSet A✝
this✝ : IsFiniteKernel κ
hA' : MeasurableSet A✝
A : Set γ
hA : MeasurableSet A
hA_eq : ∫ (x : γ) in A, density κ ν a x s ∂ν a = ((κ a) (A ×ˢ s)).toReal
h : ((κ a) (A ×ˢ s)).toReal + ∫ (x : γ) in Aᶜ, density κ ν a x s ∂ν a = ((κ a) (univ ×ˢ s)).toReal
this : Aᶜ ×ˢ s = univ ×ˢ s \ A ×ˢ s
⊢ (κ a) (A ×ˢ s) ≤ (κ a) (univ ×ˢ s)
state_after: no goals

tactic: intro f hf_disj hf h_eq
state_before: case refine_5
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : ↥(kernel α (γ × β))
ν : ↥(kernel α γ)
hκν : fst κ ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
A : Set γ
hA : MeasurableSet A
this : IsFiniteKernel κ
hA' : MeasurableSet A
⊢ ∀ (f : ℕ → Set γ),
    Pairwise (Disjoint on f) →
      (∀ (i : ℕ), MeasurableSet (f i)) →
        (∀ (i : ℕ), (fun A => ∫ (x : γ) in A, density κ ν a x s ∂ν a = ((κ a) (A ×ˢ s)).toReal) (f i)) →
          (fun A => ∫ (x : γ) in A, density κ ν a x s ∂ν a = ((κ a) (A ×ˢ s)).toReal) (⋃ i, f i)
state_after: case refine_5
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : ↥(kernel α (γ × β))
ν : ↥(kernel α γ)
hκν : fst κ ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
A : Set γ
hA : MeasurableSet A
this : IsFiniteKernel κ
hA' : MeasurableSet A
f : ℕ → Set γ
hf_disj : Pairwise (Disjoint on f)
hf : ∀ (i : ℕ), MeasurableSet (f i)
h_eq : ∀ (i : ℕ), (fun A => ∫ (x : γ) in A, density κ ν a x s ∂ν a = ((κ a) (A ×ˢ s)).toReal) (f i)
⊢ ∫ (x : γ) in ⋃ i, f i, density κ ν a x s ∂ν a = ((κ a) ((⋃ i, f i) ×ˢ s)).toReal

tactic: rw [integral_iUnion _ hf_disj (integrable_density hκν _ hs).integrableOn]
state_before: case refine_5
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : ↥(kernel α (γ × β))
ν : ↥(kernel α γ)
hκν : fst κ ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
A : Set γ
hA : MeasurableSet A
this : IsFiniteKernel κ
hA' : MeasurableSet A
f : ℕ → Set γ
hf_disj : Pairwise (Disjoint on f)
hf : ∀ (i : ℕ), MeasurableSet (f i)
h_eq : ∀ (i : ℕ), (fun A => ∫ (x : γ) in A, density κ ν a x s ∂ν a = ((κ a) (A ×ˢ s)).toReal) (f i)
⊢ ∫ (x : γ) in ⋃ i, f i, density κ ν a x s ∂ν a = ((κ a) ((⋃ i, f i) ×ˢ s)).toReal
state_after: case refine_5
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : ↥(kernel α (γ × β))
ν : ↥(kernel α γ)
hκν : fst κ ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
A : Set γ
hA : MeasurableSet A
this : IsFiniteKernel κ
hA' : MeasurableSet A
f : ℕ → Set γ
hf_disj : Pairwise (Disjoint on f)
hf : ∀ (i : ℕ), MeasurableSet (f i)
h_eq : ∀ (i : ℕ), (fun A => ∫ (x : γ) in A, density κ ν a x s ∂ν a = ((κ a) (A ×ˢ s)).toReal) (f i)
⊢ ∑' (n : ℕ), ∫ (x : γ) in f n, density κ ν a x s ∂ν a = ((κ a) ((⋃ i, f i) ×ˢ s)).toReal

α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : ↥(kernel α (γ × β))
ν : ↥(kernel α γ)
hκν : fst κ ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
A : Set γ
hA : MeasurableSet A
this : IsFiniteKernel κ
hA' : MeasurableSet A
f : ℕ → Set γ
hf_disj : Pairwise (Disjoint on f)
hf : ∀ (i : ℕ), MeasurableSet (f i)
h_eq : ∀ (i : ℕ), (fun A => ∫ (x : γ) in A, density κ ν a x s ∂ν a = ((κ a) (A ×ˢ s)).toReal) (f i)
⊢ ∀ (i : ℕ), MeasurableSet (f i)

tactic: simp_rw [h_eq]
state_before: case refine_5
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : ↥(kernel α (γ × β))
ν : ↥(kernel α γ)
hκν : fst κ ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
A : Set γ
hA : MeasurableSet A
this : IsFiniteKernel κ
hA' : MeasurableSet A
f : ℕ → Set γ
hf_disj : Pairwise (Disjoint on f)
hf : ∀ (i : ℕ), MeasurableSet (f i)
h_eq : ∀ (i : ℕ), (fun A => ∫ (x : γ) in A, density κ ν a x s ∂ν a = ((κ a) (A ×ˢ s)).toReal) (f i)
⊢ ∑' (n : ℕ), ∫ (x : γ) in f n, density κ ν a x s ∂ν a = ((κ a) ((⋃ i, f i) ×ˢ s)).toReal
state_after: case refine_5
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : ↥(kernel α (γ × β))
ν : ↥(kernel α γ)
hκν : fst κ ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
A : Set γ
hA : MeasurableSet A
this : IsFiniteKernel κ
hA' : MeasurableSet A
f : ℕ → Set γ
hf_disj : Pairwise (Disjoint on f)
hf : ∀ (i : ℕ), MeasurableSet (f i)
h_eq : ∀ (i : ℕ), (fun A => ∫ (x : γ) in A, density κ ν a x s ∂ν a = ((κ a) (A ×ˢ s)).toReal) (f i)
⊢ ∑' (n : ℕ), ((κ a) (f n ×ˢ s)).toReal = ((κ a) ((⋃ i, f i) ×ˢ s)).toReal

tactic: rw [← ENNReal.tsum_toReal_eq (fun _ ↦ measure_ne_top _ _)]
state_before: case refine_5
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : ↥(kernel α (γ × β))
ν : ↥(kernel α γ)
hκν : fst κ ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
A : Set γ
hA : MeasurableSet A
this : IsFiniteKernel κ
hA' : MeasurableSet A
f : ℕ → Set γ
hf_disj : Pairwise (Disjoint on f)
hf : ∀ (i : ℕ), MeasurableSet (f i)
h_eq : ∀ (i : ℕ), (fun A => ∫ (x : γ) in A, density κ ν a x s ∂ν a = ((κ a) (A ×ˢ s)).toReal) (f i)
⊢ ∑' (n : ℕ), ((κ a) (f n ×ˢ s)).toReal = ((κ a) ((⋃ i, f i) ×ˢ s)).toReal
state_after: case refine_5
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : ↥(kernel α (γ × β))
ν : ↥(kernel α γ)
hκν : fst κ ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
A : Set γ
hA : MeasurableSet A
this : IsFiniteKernel κ
hA' : MeasurableSet A
f : ℕ → Set γ
hf_disj : Pairwise (Disjoint on f)
hf : ∀ (i : ℕ), MeasurableSet (f i)
h_eq : ∀ (i : ℕ), (fun A => ∫ (x : γ) in A, density κ ν a x s ∂ν a = ((κ a) (A ×ˢ s)).toReal) (f i)
⊢ (∑' (a_1 : ℕ), (κ a) (f a_1 ×ˢ s)).toReal = ((κ a) ((⋃ i, f i) ×ˢ s)).toReal

tactic: congr
state_before: case refine_5
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : ↥(kernel α (γ × β))
ν : ↥(kernel α γ)
hκν : fst κ ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
A : Set γ
hA : MeasurableSet A
this : IsFiniteKernel κ
hA' : MeasurableSet A
f : ℕ → Set γ
hf_disj : Pairwise (Disjoint on f)
hf : ∀ (i : ℕ), MeasurableSet (f i)
h_eq : ∀ (i : ℕ), (fun A => ∫ (x : γ) in A, density κ ν a x s ∂ν a = ((κ a) (A ×ˢ s)).toReal) (f i)
⊢ (∑' (a_1 : ℕ), (κ a) (f a_1 ×ˢ s)).toReal = ((κ a) ((⋃ i, f i) ×ˢ s)).toReal
state_after: case refine_5.e_a
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : ↥(kernel α (γ × β))
ν : ↥(kernel α γ)
hκν : fst κ ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
A : Set γ
hA : MeasurableSet A
this : IsFiniteKernel κ
hA' : MeasurableSet A
f : ℕ → Set γ
hf_disj : Pairwise (Disjoint on f)
hf : ∀ (i : ℕ), MeasurableSet (f i)
h_eq : ∀ (i : ℕ), (fun A => ∫ (x : γ) in A, density κ ν a x s ∂ν a = ((κ a) (A ×ˢ s)).toReal) (f i)
⊢ ∑' (a_1 : ℕ), (κ a) (f a_1 ×ˢ s) = (κ a) ((⋃ i, f i) ×ˢ s)

tactic: rw [iUnion_prod_const, measure_iUnion]
state_before: case refine_5.e_a
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : ↥(kernel α (γ × β))
ν : ↥(kernel α γ)
hκν : fst κ ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
A : Set γ
hA : MeasurableSet A
this : IsFiniteKernel κ
hA' : MeasurableSet A
f : ℕ → Set γ
hf_disj : Pairwise (Disjoint on f)
hf : ∀ (i : ℕ), MeasurableSet (f i)
h_eq : ∀ (i : ℕ), (fun A => ∫ (x : γ) in A, density κ ν a x s ∂ν a = ((κ a) (A ×ˢ s)).toReal) (f i)
⊢ ∑' (a_1 : ℕ), (κ a) (f a_1 ×ˢ s) = (κ a) ((⋃ i, f i) ×ˢ s)
state_after: case refine_5.e_a.hn
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : ↥(kernel α (γ × β))
ν : ↥(kernel α γ)
hκν : fst κ ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
A : Set γ
hA : MeasurableSet A
this : IsFiniteKernel κ
hA' : MeasurableSet A
f : ℕ → Set γ
hf_disj : Pairwise (Disjoint on f)
hf : ∀ (i : ℕ), MeasurableSet (f i)
h_eq : ∀ (i : ℕ), (fun A => ∫ (x : γ) in A, density κ ν a x s ∂ν a = ((κ a) (A ×ˢ s)).toReal) (f i)
⊢ Pairwise (Disjoint on fun i => f i ×ˢ s)

case refine_5.e_a.h
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : ↥(kernel α (γ × β))
ν : ↥(kernel α γ)
hκν : fst κ ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
A : Set γ
hA : MeasurableSet A
this : IsFiniteKernel κ
hA' : MeasurableSet A
f : ℕ → Set γ
hf_disj : Pairwise (Disjoint on f)
hf : ∀ (i : ℕ), MeasurableSet (f i)
h_eq : ∀ (i : ℕ), (fun A => ∫ (x : γ) in A, density κ ν a x s ∂ν a = ((κ a) (A ×ˢ s)).toReal) (f i)
⊢ ∀ (i : ℕ), MeasurableSet (f i ×ˢ s)

tactic: intro i j hij
state_before: case refine_5.e_a.hn
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : ↥(kernel α (γ × β))
ν : ↥(kernel α γ)
hκν : fst κ ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
A : Set γ
hA : MeasurableSet A
this : IsFiniteKernel κ
hA' : MeasurableSet A
f : ℕ → Set γ
hf_disj : Pairwise (Disjoint on f)
hf : ∀ (i : ℕ), MeasurableSet (f i)
h_eq : ∀ (i : ℕ), (fun A => ∫ (x : γ) in A, density κ ν a x s ∂ν a = ((κ a) (A ×ˢ s)).toReal) (f i)
⊢ Pairwise (Disjoint on fun i => f i ×ˢ s)
state_after: case refine_5.e_a.hn
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : ↥(kernel α (γ × β))
ν : ↥(kernel α γ)
hκν : fst κ ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
A : Set γ
hA : MeasurableSet A
this : IsFiniteKernel κ
hA' : MeasurableSet A
f : ℕ → Set γ
hf_disj : Pairwise (Disjoint on f)
hf : ∀ (i : ℕ), MeasurableSet (f i)
h_eq : ∀ (i : ℕ), (fun A => ∫ (x : γ) in A, density κ ν a x s ∂ν a = ((κ a) (A ×ˢ s)).toReal) (f i)
i j : ℕ
hij : i ≠ j
⊢ (Disjoint on fun i => f i ×ˢ s) i j

tactic: rw [Function.onFun, Set.disjoint_prod]
state_before: case refine_5.e_a.hn
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : ↥(kernel α (γ × β))
ν : ↥(kernel α γ)
hκν : fst κ ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
A : Set γ
hA : MeasurableSet A
this : IsFiniteKernel κ
hA' : MeasurableSet A
f : ℕ → Set γ
hf_disj : Pairwise (Disjoint on f)
hf : ∀ (i : ℕ), MeasurableSet (f i)
h_eq : ∀ (i : ℕ), (fun A => ∫ (x : γ) in A, density κ ν a x s ∂ν a = ((κ a) (A ×ˢ s)).toReal) (f i)
i j : ℕ
hij : i ≠ j
⊢ (Disjoint on fun i => f i ×ˢ s) i j
state_after: case refine_5.e_a.hn
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : ↥(kernel α (γ × β))
ν : ↥(kernel α γ)
hκν : fst κ ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
A : Set γ
hA : MeasurableSet A
this : IsFiniteKernel κ
hA' : MeasurableSet A
f : ℕ → Set γ
hf_disj : Pairwise (Disjoint on f)
hf : ∀ (i : ℕ), MeasurableSet (f i)
h_eq : ∀ (i : ℕ), (fun A => ∫ (x : γ) in A, density κ ν a x s ∂ν a = ((κ a) (A ×ˢ s)).toReal) (f i)
i j : ℕ
hij : i ≠ j
⊢ Disjoint (f i) (f j) ∨ Disjoint s s

tactic: exact Or.inl (hf_disj hij)
state_before: case refine_5.e_a.hn
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : ↥(kernel α (γ × β))
ν : ↥(kernel α γ)
hκν : fst κ ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
A : Set γ
hA : MeasurableSet A
this : IsFiniteKernel κ
hA' : MeasurableSet A
f : ℕ → Set γ
hf_disj : Pairwise (Disjoint on f)
hf : ∀ (i : ℕ), MeasurableSet (f i)
h_eq : ∀ (i : ℕ), (fun A => ∫ (x : γ) in A, density κ ν a x s ∂ν a = ((κ a) (A ×ˢ s)).toReal) (f i)
i j : ℕ
hij : i ≠ j
⊢ Disjoint (f i) (f j) ∨ Disjoint s s
state_after: no goals

tactic: exact fun i ↦ (hf i).prod hs
state_before: case refine_5.e_a.h
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : ↥(kernel α (γ × β))
ν : ↥(kernel α γ)
hκν : fst κ ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
A : Set γ
hA : MeasurableSet A
this : IsFiniteKernel κ
hA' : MeasurableSet A
f : ℕ → Set γ
hf_disj : Pairwise (Disjoint on f)
hf : ∀ (i : ℕ), MeasurableSet (f i)
h_eq : ∀ (i : ℕ), (fun A => ∫ (x : γ) in A, density κ ν a x s ∂ν a = ((κ a) (A ×ˢ s)).toReal) (f i)
⊢ ∀ (i : ℕ), MeasurableSet (f i ×ˢ s)
state_after: no goals

tactic: rwa [iSup_countableFiltration] at hf
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : ↥(kernel α (γ × β))
ν : ↥(kernel α γ)
hκν : fst κ ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
A : Set γ
hA : MeasurableSet A
this : IsFiniteKernel κ
hA' : MeasurableSet A
f : ℕ → Set γ
hf_disj : Pairwise (Disjoint on f)
hf : ∀ (i : ℕ), MeasurableSet (f i)
h_eq : ∀ (i : ℕ), (fun A => ∫ (x : γ) in A, density κ ν a x s ∂ν a = ((κ a) (A ×ˢ s)).toReal) (f i)
⊢ ∀ (i : ℕ), MeasurableSet (f i)
state_after: no goals