Theorem: ProbabilityTheory.eq_condKernel_of_measure_eq_compProd_real
File: Mathlib/Probability/Kernel/Disintegration/Unique.lean

tactic: have huniv : ∀ᵐ x ∂ρ.fst, κ x Set.univ = ρ.condKernel x Set.univ :=
  eq_condKernel_of_measure_eq_compProd' κ hκ MeasurableSet.univ
state_before: α : Type u_1
β : Type u_2
Ω : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
inst✝⁵ : MeasurableSpace Ω
inst✝⁴ : StandardBorelSpace Ω
inst✝³ : Nonempty Ω
ρ✝ : Measure (α × Ω)
inst✝² : IsFiniteMeasure ρ✝
ρ : Measure (α × ℝ)
inst✝¹ : IsFiniteMeasure ρ
κ : ↥(kernel α ℝ)
inst✝ : IsFiniteKernel κ
hκ : ρ = ρ.fst ⊗ₘ κ
⊢ ∀ᵐ (x : α) ∂ρ.fst, κ x = ρ.condKernel x
state_after: α : Type u_1
β : Type u_2
Ω : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
inst✝⁵ : MeasurableSpace Ω
inst✝⁴ : StandardBorelSpace Ω
inst✝³ : Nonempty Ω
ρ✝ : Measure (α × Ω)
inst✝² : IsFiniteMeasure ρ✝
ρ : Measure (α × ℝ)
inst✝¹ : IsFiniteMeasure ρ
κ : ↥(kernel α ℝ)
inst✝ : IsFiniteKernel κ
hκ : ρ = ρ.fst ⊗ₘ κ
huniv : ∀ᵐ (x : α) ∂ρ.fst, (κ x) univ = (ρ.condKernel x) univ
⊢ ∀ᵐ (x : α) ∂ρ.fst, κ x = ρ.condKernel x

tactic: suffices ∀ᵐ x ∂ρ.fst, ∀ ⦃t⦄, MeasurableSet t → κ x t = ρ.condKernel x t by
  filter_upwards [this] with x hx
  ext t ht; exact hx ht
state_before: α : Type u_1
β : Type u_2
Ω : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
inst✝⁵ : MeasurableSpace Ω
inst✝⁴ : StandardBorelSpace Ω
inst✝³ : Nonempty Ω
ρ✝ : Measure (α × Ω)
inst✝² : IsFiniteMeasure ρ✝
ρ : Measure (α × ℝ)
inst✝¹ : IsFiniteMeasure ρ
κ : ↥(kernel α ℝ)
inst✝ : IsFiniteKernel κ
hκ : ρ = ρ.fst ⊗ₘ κ
huniv : ∀ᵐ (x : α) ∂ρ.fst, (κ x) univ = (ρ.condKernel x) univ
⊢ ∀ᵐ (x : α) ∂ρ.fst, κ x = ρ.condKernel x
state_after: α : Type u_1
β : Type u_2
Ω : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
inst✝⁵ : MeasurableSpace Ω
inst✝⁴ : StandardBorelSpace Ω
inst✝³ : Nonempty Ω
ρ✝ : Measure (α × Ω)
inst✝² : IsFiniteMeasure ρ✝
ρ : Measure (α × ℝ)
inst✝¹ : IsFiniteMeasure ρ
κ : ↥(kernel α ℝ)
inst✝ : IsFiniteKernel κ
hκ : ρ = ρ.fst ⊗ₘ κ
huniv : ∀ᵐ (x : α) ∂ρ.fst, (κ x) univ = (ρ.condKernel x) univ
⊢ ∀ᵐ (x : α) ∂ρ.fst, ∀ ⦃t : Set ℝ⦄, MeasurableSet t → (κ x) t = (ρ.condKernel x) t

tactic: apply MeasurableSpace.ae_induction_on_inter Real.borel_eq_generateFrom_Iic_rat
  Real.isPiSystem_Iic_rat
state_before: α : Type u_1
β : Type u_2
Ω : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
inst✝⁵ : MeasurableSpace Ω
inst✝⁴ : StandardBorelSpace Ω
inst✝³ : Nonempty Ω
ρ✝ : Measure (α × Ω)
inst✝² : IsFiniteMeasure ρ✝
ρ : Measure (α × ℝ)
inst✝¹ : IsFiniteMeasure ρ
κ : ↥(kernel α ℝ)
inst✝ : IsFiniteKernel κ
hκ : ρ = ρ.fst ⊗ₘ κ
huniv : ∀ᵐ (x : α) ∂ρ.fst, (κ x) univ = (ρ.condKernel x) univ
⊢ ∀ᵐ (x : α) ∂ρ.fst, ∀ ⦃t : Set ℝ⦄, MeasurableSet t → (κ x) t = (ρ.condKernel x) t
state_after: case h_empty
α : Type u_1
β : Type u_2
Ω : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
inst✝⁵ : MeasurableSpace Ω
inst✝⁴ : StandardBorelSpace Ω
inst✝³ : Nonempty Ω
ρ✝ : Measure (α × Ω)
inst✝² : IsFiniteMeasure ρ✝
ρ : Measure (α × ℝ)
inst✝¹ : IsFiniteMeasure ρ
κ : ↥(kernel α ℝ)
inst✝ : IsFiniteKernel κ
hκ : ρ = ρ.fst ⊗ₘ κ
huniv : ∀ᵐ (x : α) ∂ρ.fst, (κ x) univ = (ρ.condKernel x) univ
⊢ ∀ᵐ (x : α) ∂ρ.fst, (κ x) ∅ = (ρ.condKernel x) ∅

case h_basic
α : Type u_1
β : Type u_2
Ω : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
inst✝⁵ : MeasurableSpace Ω
inst✝⁴ : StandardBorelSpace Ω
inst✝³ : Nonempty Ω
ρ✝ : Measure (α × Ω)
inst✝² : IsFiniteMeasure ρ✝
ρ : Measure (α × ℝ)
inst✝¹ : IsFiniteMeasure ρ
κ : ↥(kernel α ℝ)
inst✝ : IsFiniteKernel κ
hκ : ρ = ρ.fst ⊗ₘ κ
huniv : ∀ᵐ (x : α) ∂ρ.fst, (κ x) univ = (ρ.condKernel x) univ
⊢ ∀ᵐ (x : α) ∂ρ.fst, ∀ t ∈ ⋃ a, {Iic ↑a}, (κ x) t = (ρ.condKernel x) t

case h_compl
α : Type u_1
β : Type u_2
Ω : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
inst✝⁵ : MeasurableSpace Ω
inst✝⁴ : StandardBorelSpace Ω
inst✝³ : Nonempty Ω
ρ✝ : Measure (α × Ω)
inst✝² : IsFiniteMeasure ρ✝
ρ : Measure (α × ℝ)
inst✝¹ : IsFiniteMeasure ρ
κ : ↥(kernel α ℝ)
inst✝ : IsFiniteKernel κ
hκ : ρ = ρ.fst ⊗ₘ κ
huniv : ∀ᵐ (x : α) ∂ρ.fst, (κ x) univ = (ρ.condKernel x) univ
⊢ ∀ᵐ (x : α) ∂ρ.fst, ∀ (t : Set ℝ), MeasurableSet t → (κ x) t = (ρ.condKernel x) t → (κ x) tᶜ = (ρ.condKernel x) tᶜ

case h_union
α : Type u_1
β : Type u_2
Ω : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
inst✝⁵ : MeasurableSpace Ω
inst✝⁴ : StandardBorelSpace Ω
inst✝³ : Nonempty Ω
ρ✝ : Measure (α × Ω)
inst✝² : IsFiniteMeasure ρ✝
ρ : Measure (α × ℝ)
inst✝¹ : IsFiniteMeasure ρ
κ : ↥(kernel α ℝ)
inst✝ : IsFiniteKernel κ
hκ : ρ = ρ.fst ⊗ₘ κ
huniv : ∀ᵐ (x : α) ∂ρ.fst, (κ x) univ = (ρ.condKernel x) univ
⊢ ∀ᵐ (x : α) ∂ρ.fst,
    ∀ (f : ℕ → Set ℝ),
      Pairwise (Disjoint on f) →
        (∀ (i : ℕ), MeasurableSet (f i)) →
          (∀ (i : ℕ), (κ x) (f i) = (ρ.condKernel x) (f i)) → (κ x) (⋃ i, f i) = (ρ.condKernel x) (⋃ i, f i)

tactic: filter_upwards [this] with x hx
state_before: α : Type u_1
β : Type u_2
Ω : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
inst✝⁵ : MeasurableSpace Ω
inst✝⁴ : StandardBorelSpace Ω
inst✝³ : Nonempty Ω
ρ✝ : Measure (α × Ω)
inst✝² : IsFiniteMeasure ρ✝
ρ : Measure (α × ℝ)
inst✝¹ : IsFiniteMeasure ρ
κ : ↥(kernel α ℝ)
inst✝ : IsFiniteKernel κ
hκ : ρ = ρ.fst ⊗ₘ κ
huniv : ∀ᵐ (x : α) ∂ρ.fst, (κ x) univ = (ρ.condKernel x) univ
this : ∀ᵐ (x : α) ∂ρ.fst, ∀ ⦃t : Set ℝ⦄, MeasurableSet t → (κ x) t = (ρ.condKernel x) t
⊢ ∀ᵐ (x : α) ∂ρ.fst, κ x = ρ.condKernel x
state_after: case h
α : Type u_1
β : Type u_2
Ω : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
inst✝⁵ : MeasurableSpace Ω
inst✝⁴ : StandardBorelSpace Ω
inst✝³ : Nonempty Ω
ρ✝ : Measure (α × Ω)
inst✝² : IsFiniteMeasure ρ✝
ρ : Measure (α × ℝ)
inst✝¹ : IsFiniteMeasure ρ
κ : ↥(kernel α ℝ)
inst✝ : IsFiniteKernel κ
hκ : ρ = ρ.fst ⊗ₘ κ
huniv : ∀ᵐ (x : α) ∂ρ.fst, (κ x) univ = (ρ.condKernel x) univ
this : ∀ᵐ (x : α) ∂ρ.fst, ∀ ⦃t : Set ℝ⦄, MeasurableSet t → (κ x) t = (ρ.condKernel x) t
x : α
hx : ∀ ⦃t : Set ℝ⦄, MeasurableSet t → (κ x) t = (ρ.condKernel x) t
⊢ κ x = ρ.condKernel x

tactic: ext t ht
state_before: case h
α : Type u_1
β : Type u_2
Ω : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
inst✝⁵ : MeasurableSpace Ω
inst✝⁴ : StandardBorelSpace Ω
inst✝³ : Nonempty Ω
ρ✝ : Measure (α × Ω)
inst✝² : IsFiniteMeasure ρ✝
ρ : Measure (α × ℝ)
inst✝¹ : IsFiniteMeasure ρ
κ : ↥(kernel α ℝ)
inst✝ : IsFiniteKernel κ
hκ : ρ = ρ.fst ⊗ₘ κ
huniv : ∀ᵐ (x : α) ∂ρ.fst, (κ x) univ = (ρ.condKernel x) univ
this : ∀ᵐ (x : α) ∂ρ.fst, ∀ ⦃t : Set ℝ⦄, MeasurableSet t → (κ x) t = (ρ.condKernel x) t
x : α
hx : ∀ ⦃t : Set ℝ⦄, MeasurableSet t → (κ x) t = (ρ.condKernel x) t
⊢ κ x = ρ.condKernel x
state_after: case h.h
α : Type u_1
β : Type u_2
Ω : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
inst✝⁵ : MeasurableSpace Ω
inst✝⁴ : StandardBorelSpace Ω
inst✝³ : Nonempty Ω
ρ✝ : Measure (α × Ω)
inst✝² : IsFiniteMeasure ρ✝
ρ : Measure (α × ℝ)
inst✝¹ : IsFiniteMeasure ρ
κ : ↥(kernel α ℝ)
inst✝ : IsFiniteKernel κ
hκ : ρ = ρ.fst ⊗ₘ κ
huniv : ∀ᵐ (x : α) ∂ρ.fst, (κ x) univ = (ρ.condKernel x) univ
this : ∀ᵐ (x : α) ∂ρ.fst, ∀ ⦃t : Set ℝ⦄, MeasurableSet t → (κ x) t = (ρ.condKernel x) t
x : α
hx : ∀ ⦃t : Set ℝ⦄, MeasurableSet t → (κ x) t = (ρ.condKernel x) t
t : Set ℝ
ht : MeasurableSet t
⊢ (κ x) t = (ρ.condKernel x) t

tactic: exact hx ht
state_before: case h.h
α : Type u_1
β : Type u_2
Ω : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
inst✝⁵ : MeasurableSpace Ω
inst✝⁴ : StandardBorelSpace Ω
inst✝³ : Nonempty Ω
ρ✝ : Measure (α × Ω)
inst✝² : IsFiniteMeasure ρ✝
ρ : Measure (α × ℝ)
inst✝¹ : IsFiniteMeasure ρ
κ : ↥(kernel α ℝ)
inst✝ : IsFiniteKernel κ
hκ : ρ = ρ.fst ⊗ₘ κ
huniv : ∀ᵐ (x : α) ∂ρ.fst, (κ x) univ = (ρ.condKernel x) univ
this : ∀ᵐ (x : α) ∂ρ.fst, ∀ ⦃t : Set ℝ⦄, MeasurableSet t → (κ x) t = (ρ.condKernel x) t
x : α
hx : ∀ ⦃t : Set ℝ⦄, MeasurableSet t → (κ x) t = (ρ.condKernel x) t
t : Set ℝ
ht : MeasurableSet t
⊢ (κ x) t = (ρ.condKernel x) t
state_after: no goals

tactic: simp
state_before: case h_empty
α : Type u_1
β : Type u_2
Ω : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
inst✝⁵ : MeasurableSpace Ω
inst✝⁴ : StandardBorelSpace Ω
inst✝³ : Nonempty Ω
ρ✝ : Measure (α × Ω)
inst✝² : IsFiniteMeasure ρ✝
ρ : Measure (α × ℝ)
inst✝¹ : IsFiniteMeasure ρ
κ : ↥(kernel α ℝ)
inst✝ : IsFiniteKernel κ
hκ : ρ = ρ.fst ⊗ₘ κ
huniv : ∀ᵐ (x : α) ∂ρ.fst, (κ x) univ = (ρ.condKernel x) univ
⊢ ∀ᵐ (x : α) ∂ρ.fst, (κ x) ∅ = (ρ.condKernel x) ∅
state_after: no goals

tactic: simp only [iUnion_singleton_eq_range, mem_range, forall_exists_index, forall_apply_eq_imp_iff]
state_before: case h_basic
α : Type u_1
β : Type u_2
Ω : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
inst✝⁵ : MeasurableSpace Ω
inst✝⁴ : StandardBorelSpace Ω
inst✝³ : Nonempty Ω
ρ✝ : Measure (α × Ω)
inst✝² : IsFiniteMeasure ρ✝
ρ : Measure (α × ℝ)
inst✝¹ : IsFiniteMeasure ρ
κ : ↥(kernel α ℝ)
inst✝ : IsFiniteKernel κ
hκ : ρ = ρ.fst ⊗ₘ κ
huniv : ∀ᵐ (x : α) ∂ρ.fst, (κ x) univ = (ρ.condKernel x) univ
⊢ ∀ᵐ (x : α) ∂ρ.fst, ∀ t ∈ ⋃ a, {Iic ↑a}, (κ x) t = (ρ.condKernel x) t
state_after: case h_basic
α : Type u_1
β : Type u_2
Ω : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
inst✝⁵ : MeasurableSpace Ω
inst✝⁴ : StandardBorelSpace Ω
inst✝³ : Nonempty Ω
ρ✝ : Measure (α × Ω)
inst✝² : IsFiniteMeasure ρ✝
ρ : Measure (α × ℝ)
inst✝¹ : IsFiniteMeasure ρ
κ : ↥(kernel α ℝ)
inst✝ : IsFiniteKernel κ
hκ : ρ = ρ.fst ⊗ₘ κ
huniv : ∀ᵐ (x : α) ∂ρ.fst, (κ x) univ = (ρ.condKernel x) univ
⊢ ∀ᵐ (x : α) ∂ρ.fst, ∀ (a : ℚ), (κ x) (Iic ↑a) = (ρ.condKernel x) (Iic ↑a)

tactic: exact ae_all_iff.2 fun q ↦ eq_condKernel_of_measure_eq_compProd' κ hκ measurableSet_Iic
state_before: case h_basic
α : Type u_1
β : Type u_2
Ω : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
inst✝⁵ : MeasurableSpace Ω
inst✝⁴ : StandardBorelSpace Ω
inst✝³ : Nonempty Ω
ρ✝ : Measure (α × Ω)
inst✝² : IsFiniteMeasure ρ✝
ρ : Measure (α × ℝ)
inst✝¹ : IsFiniteMeasure ρ
κ : ↥(kernel α ℝ)
inst✝ : IsFiniteKernel κ
hκ : ρ = ρ.fst ⊗ₘ κ
huniv : ∀ᵐ (x : α) ∂ρ.fst, (κ x) univ = (ρ.condKernel x) univ
⊢ ∀ᵐ (x : α) ∂ρ.fst, ∀ (a : ℚ), (κ x) (Iic ↑a) = (ρ.condKernel x) (Iic ↑a)
state_after: no goals

tactic: filter_upwards [huniv] with x hxuniv t ht heq
state_before: case h_compl
α : Type u_1
β : Type u_2
Ω : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
inst✝⁵ : MeasurableSpace Ω
inst✝⁴ : StandardBorelSpace Ω
inst✝³ : Nonempty Ω
ρ✝ : Measure (α × Ω)
inst✝² : IsFiniteMeasure ρ✝
ρ : Measure (α × ℝ)
inst✝¹ : IsFiniteMeasure ρ
κ : ↥(kernel α ℝ)
inst✝ : IsFiniteKernel κ
hκ : ρ = ρ.fst ⊗ₘ κ
huniv : ∀ᵐ (x : α) ∂ρ.fst, (κ x) univ = (ρ.condKernel x) univ
⊢ ∀ᵐ (x : α) ∂ρ.fst, ∀ (t : Set ℝ), MeasurableSet t → (κ x) t = (ρ.condKernel x) t → (κ x) tᶜ = (ρ.condKernel x) tᶜ
state_after: case h
α : Type u_1
β : Type u_2
Ω : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
inst✝⁵ : MeasurableSpace Ω
inst✝⁴ : StandardBorelSpace Ω
inst✝³ : Nonempty Ω
ρ✝ : Measure (α × Ω)
inst✝² : IsFiniteMeasure ρ✝
ρ : Measure (α × ℝ)
inst✝¹ : IsFiniteMeasure ρ
κ : ↥(kernel α ℝ)
inst✝ : IsFiniteKernel κ
hκ : ρ = ρ.fst ⊗ₘ κ
huniv : ∀ᵐ (x : α) ∂ρ.fst, (κ x) univ = (ρ.condKernel x) univ
x : α
hxuniv : (κ x) univ = (ρ.condKernel x) univ
t : Set ℝ
ht : MeasurableSet t
heq : (κ x) t = (ρ.condKernel x) t
⊢ (κ x) tᶜ = (ρ.condKernel x) tᶜ

tactic: rw [measure_compl ht <| measure_ne_top _ _, heq, hxuniv, measure_compl ht <| measure_ne_top _ _]
state_before: case h
α : Type u_1
β : Type u_2
Ω : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
inst✝⁵ : MeasurableSpace Ω
inst✝⁴ : StandardBorelSpace Ω
inst✝³ : Nonempty Ω
ρ✝ : Measure (α × Ω)
inst✝² : IsFiniteMeasure ρ✝
ρ : Measure (α × ℝ)
inst✝¹ : IsFiniteMeasure ρ
κ : ↥(kernel α ℝ)
inst✝ : IsFiniteKernel κ
hκ : ρ = ρ.fst ⊗ₘ κ
huniv : ∀ᵐ (x : α) ∂ρ.fst, (κ x) univ = (ρ.condKernel x) univ
x : α
hxuniv : (κ x) univ = (ρ.condKernel x) univ
t : Set ℝ
ht : MeasurableSet t
heq : (κ x) t = (ρ.condKernel x) t
⊢ (κ x) tᶜ = (ρ.condKernel x) tᶜ
state_after: no goals

tactic: refine ae_of_all _ (fun x f hdisj hf heq ↦ ?_)
state_before: case h_union
α : Type u_1
β : Type u_2
Ω : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
inst✝⁵ : MeasurableSpace Ω
inst✝⁴ : StandardBorelSpace Ω
inst✝³ : Nonempty Ω
ρ✝ : Measure (α × Ω)
inst✝² : IsFiniteMeasure ρ✝
ρ : Measure (α × ℝ)
inst✝¹ : IsFiniteMeasure ρ
κ : ↥(kernel α ℝ)
inst✝ : IsFiniteKernel κ
hκ : ρ = ρ.fst ⊗ₘ κ
huniv : ∀ᵐ (x : α) ∂ρ.fst, (κ x) univ = (ρ.condKernel x) univ
⊢ ∀ᵐ (x : α) ∂ρ.fst,
    ∀ (f : ℕ → Set ℝ),
      Pairwise (Disjoint on f) →
        (∀ (i : ℕ), MeasurableSet (f i)) →
          (∀ (i : ℕ), (κ x) (f i) = (ρ.condKernel x) (f i)) → (κ x) (⋃ i, f i) = (ρ.condKernel x) (⋃ i, f i)
state_after: case h_union
α : Type u_1
β : Type u_2
Ω : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
inst✝⁵ : MeasurableSpace Ω
inst✝⁴ : StandardBorelSpace Ω
inst✝³ : Nonempty Ω
ρ✝ : Measure (α × Ω)
inst✝² : IsFiniteMeasure ρ✝
ρ : Measure (α × ℝ)
inst✝¹ : IsFiniteMeasure ρ
κ : ↥(kernel α ℝ)
inst✝ : IsFiniteKernel κ
hκ : ρ = ρ.fst ⊗ₘ κ
huniv : ∀ᵐ (x : α) ∂ρ.fst, (κ x) univ = (ρ.condKernel x) univ
x : α
f : ℕ → Set ℝ
hdisj : Pairwise (Disjoint on f)
hf : ∀ (i : ℕ), MeasurableSet (f i)
heq : ∀ (i : ℕ), (κ x) (f i) = (ρ.condKernel x) (f i)
⊢ (κ x) (⋃ i, f i) = (ρ.condKernel x) (⋃ i, f i)

tactic: rw [measure_iUnion hdisj hf, measure_iUnion hdisj hf]
state_before: case h_union
α : Type u_1
β : Type u_2
Ω : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
inst✝⁵ : MeasurableSpace Ω
inst✝⁴ : StandardBorelSpace Ω
inst✝³ : Nonempty Ω
ρ✝ : Measure (α × Ω)
inst✝² : IsFiniteMeasure ρ✝
ρ : Measure (α × ℝ)
inst✝¹ : IsFiniteMeasure ρ
κ : ↥(kernel α ℝ)
inst✝ : IsFiniteKernel κ
hκ : ρ = ρ.fst ⊗ₘ κ
huniv : ∀ᵐ (x : α) ∂ρ.fst, (κ x) univ = (ρ.condKernel x) univ
x : α
f : ℕ → Set ℝ
hdisj : Pairwise (Disjoint on f)
hf : ∀ (i : ℕ), MeasurableSet (f i)
heq : ∀ (i : ℕ), (κ x) (f i) = (ρ.condKernel x) (f i)
⊢ (κ x) (⋃ i, f i) = (ρ.condKernel x) (⋃ i, f i)
state_after: case h_union
α : Type u_1
β : Type u_2
Ω : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
inst✝⁵ : MeasurableSpace Ω
inst✝⁴ : StandardBorelSpace Ω
inst✝³ : Nonempty Ω
ρ✝ : Measure (α × Ω)
inst✝² : IsFiniteMeasure ρ✝
ρ : Measure (α × ℝ)
inst✝¹ : IsFiniteMeasure ρ
κ : ↥(kernel α ℝ)
inst✝ : IsFiniteKernel κ
hκ : ρ = ρ.fst ⊗ₘ κ
huniv : ∀ᵐ (x : α) ∂ρ.fst, (κ x) univ = (ρ.condKernel x) univ
x : α
f : ℕ → Set ℝ
hdisj : Pairwise (Disjoint on f)
hf : ∀ (i : ℕ), MeasurableSet (f i)
heq : ∀ (i : ℕ), (κ x) (f i) = (ρ.condKernel x) (f i)
⊢ ∑' (i : ℕ), (κ x) (f i) = ∑' (i : ℕ), (ρ.condKernel x) (f i)

tactic: exact tsum_congr heq
state_before: case h_union
α : Type u_1
β : Type u_2
Ω : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
inst✝⁵ : MeasurableSpace Ω
inst✝⁴ : StandardBorelSpace Ω
inst✝³ : Nonempty Ω
ρ✝ : Measure (α × Ω)
inst✝² : IsFiniteMeasure ρ✝
ρ : Measure (α × ℝ)
inst✝¹ : IsFiniteMeasure ρ
κ : ↥(kernel α ℝ)
inst✝ : IsFiniteKernel κ
hκ : ρ = ρ.fst ⊗ₘ κ
huniv : ∀ᵐ (x : α) ∂ρ.fst, (κ x) univ = (ρ.condKernel x) univ
x : α
f : ℕ → Set ℝ
hdisj : Pairwise (Disjoint on f)
hf : ∀ (i : ℕ), MeasurableSet (f i)
heq : ∀ (i : ℕ), (κ x) (f i) = (ρ.condKernel x) (f i)
⊢ ∑' (i : ℕ), (κ x) (f i) = ∑' (i : ℕ), (ρ.condKernel x) (f i)
state_after: no goals