Theorem: MeasurableSpace.induction_on_inter
File: Mathlib/MeasureTheory/PiSystem.lean

Proof steps:
state_before: α : Type u_1
C : Set α → Prop
s : Set (Set α)
m : MeasurableSpace α
h_eq : m = generateFrom s
h_inter : IsPiSystem s
h_empty : C ∅
h_basic : ∀ t ∈ s, C t
h_compl : ∀ (t : Set α), MeasurableSet t → C t → C tᶜ
h_union :
  ∀ (f : ℕ → Set α), Pairwise (Disjoint on f) → (∀ (i : ℕ), MeasurableSet (f i)) → (∀ (i : ℕ), C (f i)) → C (⋃ i, f i)
⊢ MeasurableSet = DynkinSystem.GenerateHas s
state_after: no goals
tactic: rfl
state_before: α : Type u_1
C : Set α → Prop
s : Set (Set α)
m : MeasurableSpace α
h_eq : m = generateFrom s
h_inter : IsPiSystem s
h_empty : C ∅
h_basic : ∀ t ∈ s, C t
h_compl : ∀ (t : Set α), MeasurableSet t → C t → C tᶜ
h_union :
  ∀ (f : ℕ → Set α), Pairwise (Disjoint on f) → (∀ (i : ℕ), MeasurableSet (f i)) → (∀ (i : ℕ), C (f i)) → C (⋃ i, f i)
eq : MeasurableSet = DynkinSystem.GenerateHas s
t : Set α
ht : MeasurableSet t
⊢ DynkinSystem.GenerateHas s t
state_after: no goals
tactic: rwa [eq] at ht
state_before: α : Type u_1
C : Set α → Prop
s : Set (Set α)
m : MeasurableSpace α
h_eq : m = generateFrom s
h_inter : IsPiSystem s
h_empty : C ∅
h_basic : ∀ t ∈ s, C t
h_compl : ∀ (t : Set α), MeasurableSet t → C t → C tᶜ
h_union :
  ∀ (f : ℕ → Set α), Pairwise (Disjoint on f) → (∀ (i : ℕ), MeasurableSet (f i)) → (∀ (i : ℕ), C (f i)) → C (⋃ i, f i)
eq : MeasurableSet = DynkinSystem.GenerateHas s
t✝ : Set α
ht✝ : MeasurableSet t✝
this : DynkinSystem.GenerateHas s t✝
t : Set α
ht : DynkinSystem.GenerateHas s t
⊢ MeasurableSet t
state_after: α : Type u_1
C : Set α → Prop
s : Set (Set α)
m : MeasurableSpace α
h_eq : m = generateFrom s
h_inter : IsPiSystem s
h_empty : C ∅
h_basic : ∀ t ∈ s, C t
h_compl : ∀ (t : Set α), MeasurableSet t → C t → C tᶜ
h_union :
  ∀ (f : ℕ → Set α), Pairwise (Disjoint on f) → (∀ (i : ℕ), MeasurableSet (f i)) → (∀ (i : ℕ), C (f i)) → C (⋃ i, f i)
eq : MeasurableSet = DynkinSystem.GenerateHas s
t✝ : Set α
ht✝ : MeasurableSet t✝
this : DynkinSystem.GenerateHas s t✝
t : Set α
ht : DynkinSystem.GenerateHas s t
⊢ DynkinSystem.GenerateHas s t
tactic: rw [eq]
state_before: α : Type u_1
C : Set α → Prop
s : Set (Set α)
m : MeasurableSpace α
h_eq : m = generateFrom s
h_inter : IsPiSystem s
h_empty : C ∅
h_basic : ∀ t ∈ s, C t
h_compl : ∀ (t : Set α), MeasurableSet t → C t → C tᶜ
h_union :
  ∀ (f : ℕ → Set α), Pairwise (Disjoint on f) → (∀ (i : ℕ), MeasurableSet (f i)) → (∀ (i : ℕ), C (f i)) → C (⋃ i, f i)
eq : MeasurableSet = DynkinSystem.GenerateHas s
t✝ : Set α
ht✝ : MeasurableSet t✝
this : DynkinSystem.GenerateHas s t✝
t : Set α
ht : DynkinSystem.GenerateHas s t
⊢ DynkinSystem.GenerateHas s t
state_after: no goals
tactic: exact ht
state_before: α : Type u_1
C : Set α → Prop
s : Set (Set α)
m : MeasurableSpace α
h_eq : m = generateFrom s
h_inter : IsPiSystem s
h_empty : C ∅
h_basic : ∀ t ∈ s, C t
h_compl : ∀ (t : Set α), MeasurableSet t → C t → C tᶜ
h_union :
  ∀ (f : ℕ → Set α), Pairwise (Disjoint on f) → (∀ (i : ℕ), MeasurableSet (f i)) → (∀ (i : ℕ), C (f i)) → C (⋃ i, f i)
eq : MeasurableSet = DynkinSystem.GenerateHas s
t : Set α
ht✝ : MeasurableSet t
this : DynkinSystem.GenerateHas s t
f : ℕ → Set α
hf : Pairwise (Disjoint on f)
ht : ∀ (i : ℕ), DynkinSystem.GenerateHas s (f i)
i : ℕ
⊢ MeasurableSet (f i)
state_after: α : Type u_1
C : Set α → Prop
s : Set (Set α)
m : MeasurableSpace α
h_eq : m = generateFrom s
h_inter : IsPiSystem s
h_empty : C ∅
h_basic : ∀ t ∈ s, C t
h_compl : ∀ (t : Set α), MeasurableSet t → C t → C tᶜ
h_union :
  ∀ (f : ℕ → Set α), Pairwise (Disjoint on f) → (∀ (i : ℕ), MeasurableSet (f i)) → (∀ (i : ℕ), C (f i)) → C (⋃ i, f i)
eq : MeasurableSet = DynkinSystem.GenerateHas s
t : Set α
ht✝ : MeasurableSet t
this : DynkinSystem.GenerateHas s t
f : ℕ → Set α
hf : Pairwise (Disjoint on f)
ht : ∀ (i : ℕ), DynkinSystem.GenerateHas s (f i)
i : ℕ
⊢ DynkinSystem.GenerateHas s (f i)
tactic: rw [eq]
state_before: α : Type u_1
C : Set α → Prop
s : Set (Set α)
m : MeasurableSpace α
h_eq : m = generateFrom s
h_inter : IsPiSystem s
h_empty : C ∅
h_basic : ∀ t ∈ s, C t
h_compl : ∀ (t : Set α), MeasurableSet t → C t → C tᶜ
h_union :
  ∀ (f : ℕ → Set α), Pairwise (Disjoint on f) → (∀ (i : ℕ), MeasurableSet (f i)) → (∀ (i : ℕ), C (f i)) → C (⋃ i, f i)
eq : MeasurableSet = DynkinSystem.GenerateHas s
t : Set α
ht✝ : MeasurableSet t
this : DynkinSystem.GenerateHas s t
f : ℕ → Set α
hf : Pairwise (Disjoint on f)
ht : ∀ (i : ℕ), DynkinSystem.GenerateHas s (f i)
i : ℕ
⊢ DynkinSystem.GenerateHas s (f i)
state_after: no goals
tactic: exact ht _