Theorem: MeasureTheory.StronglyMeasurable.stronglyMeasurable_of_measurableSpace_le_on
File: Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean

tactic: have hs_m₂ : MeasurableSet[m₂] s := by
  rw [← Set.inter_univ s]
  refine hs Set.univ ?_
  rwa [Set.inter_univ]
state_before: α✝ : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝² : Countable ι
f✝ g : α✝ → β
α : Type u_5
E : Type u_6
m m₂ : MeasurableSpace α
inst✝¹ : TopologicalSpace E
inst✝ : Zero E
s : Set α
f : α → E
hs_m : MeasurableSet s
hs : ∀ (t : Set α), MeasurableSet (s ∩ t) → MeasurableSet (s ∩ t)
hf : StronglyMeasurable f
hf_zero : ∀ x ∉ s, f x = 0
⊢ StronglyMeasurable f
state_after: α✝ : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝² : Countable ι
f✝ g : α✝ → β
α : Type u_5
E : Type u_6
m m₂ : MeasurableSpace α
inst✝¹ : TopologicalSpace E
inst✝ : Zero E
s : Set α
f : α → E
hs_m : MeasurableSet s
hs : ∀ (t : Set α), MeasurableSet (s ∩ t) → MeasurableSet (s ∩ t)
hf : StronglyMeasurable f
hf_zero : ∀ x ∉ s, f x = 0
hs_m₂ : MeasurableSet s
⊢ StronglyMeasurable f

tactic: obtain ⟨g_seq_s, hg_seq_tendsto, hg_seq_zero⟩ := stronglyMeasurable_in_set hs_m hf hf_zero
state_before: α✝ : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝² : Countable ι
f✝ g : α✝ → β
α : Type u_5
E : Type u_6
m m₂ : MeasurableSpace α
inst✝¹ : TopologicalSpace E
inst✝ : Zero E
s : Set α
f : α → E
hs_m : MeasurableSet s
hs : ∀ (t : Set α), MeasurableSet (s ∩ t) → MeasurableSet (s ∩ t)
hf : StronglyMeasurable f
hf_zero : ∀ x ∉ s, f x = 0
hs_m₂ : MeasurableSet s
⊢ StronglyMeasurable f
state_after: case intro.intro
α✝ : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝² : Countable ι
f✝ g : α✝ → β
α : Type u_5
E : Type u_6
m m₂ : MeasurableSpace α
inst✝¹ : TopologicalSpace E
inst✝ : Zero E
s : Set α
f : α → E
hs_m : MeasurableSet s
hs : ∀ (t : Set α), MeasurableSet (s ∩ t) → MeasurableSet (s ∩ t)
hf : StronglyMeasurable f
hf_zero : ∀ x ∉ s, f x = 0
hs_m₂ : MeasurableSet s
g_seq_s : ℕ → α →ₛ E
hg_seq_tendsto : ∀ (x : α), Tendsto (fun n => ↑(g_seq_s n) x) atTop (𝓝 (f x))
hg_seq_zero : ∀ x ∉ s, ∀ (n : ℕ), ↑(g_seq_s n) x = 0
⊢ StronglyMeasurable f

tactic: exact ⟨g_seq_s₂, hg_seq_tendsto⟩
state_before: case intro.intro
α✝ : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝² : Countable ι
f✝ g : α✝ → β
α : Type u_5
E : Type u_6
m m₂ : MeasurableSpace α
inst✝¹ : TopologicalSpace E
inst✝ : Zero E
s : Set α
f : α → E
hs_m : MeasurableSet s
hs : ∀ (t : Set α), MeasurableSet (s ∩ t) → MeasurableSet (s ∩ t)
hf : StronglyMeasurable f
hf_zero : ∀ x ∉ s, f x = 0
hs_m₂ : MeasurableSet s
g_seq_s : ℕ → α →ₛ E
hg_seq_tendsto : ∀ (x : α), Tendsto (fun n => ↑(g_seq_s n) x) atTop (𝓝 (f x))
hg_seq_zero : ∀ x ∉ s, ∀ (n : ℕ), ↑(g_seq_s n) x = 0
g_seq_s₂ : ℕ → α →ₛ E := fun n => { toFun := ↑(g_seq_s n), measurableSet_fiber' := ⋯, finite_range' := ⋯ }
⊢ StronglyMeasurable f
state_after: no goals

tactic: rw [← Set.inter_univ s]
state_before: α✝ : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝² : Countable ι
f✝ g : α✝ → β
α : Type u_5
E : Type u_6
m m₂ : MeasurableSpace α
inst✝¹ : TopologicalSpace E
inst✝ : Zero E
s : Set α
f : α → E
hs_m : MeasurableSet s
hs : ∀ (t : Set α), MeasurableSet (s ∩ t) → MeasurableSet (s ∩ t)
hf : StronglyMeasurable f
hf_zero : ∀ x ∉ s, f x = 0
⊢ MeasurableSet s
state_after: α✝ : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝² : Countable ι
f✝ g : α✝ → β
α : Type u_5
E : Type u_6
m m₂ : MeasurableSpace α
inst✝¹ : TopologicalSpace E
inst✝ : Zero E
s : Set α
f : α → E
hs_m : MeasurableSet s
hs : ∀ (t : Set α), MeasurableSet (s ∩ t) → MeasurableSet (s ∩ t)
hf : StronglyMeasurable f
hf_zero : ∀ x ∉ s, f x = 0
⊢ MeasurableSet (s ∩ univ)

tactic: rwa [Set.inter_univ]
state_before: α✝ : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝² : Countable ι
f✝ g : α✝ → β
α : Type u_5
E : Type u_6
m m₂ : MeasurableSpace α
inst✝¹ : TopologicalSpace E
inst✝ : Zero E
s : Set α
f : α → E
hs_m : MeasurableSet s
hs : ∀ (t : Set α), MeasurableSet (s ∩ t) → MeasurableSet (s ∩ t)
hf : StronglyMeasurable f
hf_zero : ∀ x ∉ s, f x = 0
⊢ MeasurableSet (s ∩ univ)
state_after: no goals

tactic: rw [← Set.inter_univ (g_seq_s n ⁻¹' {x}), ← Set.union_compl_self s,
  Set.inter_union_distrib_left, Set.inter_comm (g_seq_s n ⁻¹' {x})]
state_before: α✝ : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝² : Countable ι
f✝ g : α✝ → β
α : Type u_5
E : Type u_6
m m₂ : MeasurableSpace α
inst✝¹ : TopologicalSpace E
inst✝ : Zero E
s : Set α
f : α → E
hs_m : MeasurableSet s
hs : ∀ (t : Set α), MeasurableSet (s ∩ t) → MeasurableSet (s ∩ t)
hf : StronglyMeasurable f
hf_zero : ∀ x ∉ s, f x = 0
hs_m₂ : MeasurableSet s
g_seq_s : ℕ → α →ₛ E
hg_seq_tendsto : ∀ (x : α), Tendsto (fun n => ↑(g_seq_s n) x) atTop (𝓝 (f x))
hg_seq_zero : ∀ x ∉ s, ∀ (n : ℕ), ↑(g_seq_s n) x = 0
n : ℕ
x : E
⊢ MeasurableSet (↑(g_seq_s n) ⁻¹' {x})
state_after: α✝ : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝² : Countable ι
f✝ g : α✝ → β
α : Type u_5
E : Type u_6
m m₂ : MeasurableSpace α
inst✝¹ : TopologicalSpace E
inst✝ : Zero E
s : Set α
f : α → E
hs_m : MeasurableSet s
hs : ∀ (t : Set α), MeasurableSet (s ∩ t) → MeasurableSet (s ∩ t)
hf : StronglyMeasurable f
hf_zero : ∀ x ∉ s, f x = 0
hs_m₂ : MeasurableSet s
g_seq_s : ℕ → α →ₛ E
hg_seq_tendsto : ∀ (x : α), Tendsto (fun n => ↑(g_seq_s n) x) atTop (𝓝 (f x))
hg_seq_zero : ∀ x ∉ s, ∀ (n : ℕ), ↑(g_seq_s n) x = 0
n : ℕ
x : E
⊢ MeasurableSet (s ∩ ↑(g_seq_s n) ⁻¹' {x} ∪ ↑(g_seq_s n) ⁻¹' {x} ∩ sᶜ)

tactic: refine MeasurableSet.union (hs _ (hs_m.inter ?_)) ?_
state_before: α✝ : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝² : Countable ι
f✝ g : α✝ → β
α : Type u_5
E : Type u_6
m m₂ : MeasurableSpace α
inst✝¹ : TopologicalSpace E
inst✝ : Zero E
s : Set α
f : α → E
hs_m : MeasurableSet s
hs : ∀ (t : Set α), MeasurableSet (s ∩ t) → MeasurableSet (s ∩ t)
hf : StronglyMeasurable f
hf_zero : ∀ x ∉ s, f x = 0
hs_m₂ : MeasurableSet s
g_seq_s : ℕ → α →ₛ E
hg_seq_tendsto : ∀ (x : α), Tendsto (fun n => ↑(g_seq_s n) x) atTop (𝓝 (f x))
hg_seq_zero : ∀ x ∉ s, ∀ (n : ℕ), ↑(g_seq_s n) x = 0
n : ℕ
x : E
⊢ MeasurableSet (s ∩ ↑(g_seq_s n) ⁻¹' {x} ∪ ↑(g_seq_s n) ⁻¹' {x} ∩ sᶜ)
state_after: case refine_1
α✝ : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝² : Countable ι
f✝ g : α✝ → β
α : Type u_5
E : Type u_6
m m₂ : MeasurableSpace α
inst✝¹ : TopologicalSpace E
inst✝ : Zero E
s : Set α
f : α → E
hs_m : MeasurableSet s
hs : ∀ (t : Set α), MeasurableSet (s ∩ t) → MeasurableSet (s ∩ t)
hf : StronglyMeasurable f
hf_zero : ∀ x ∉ s, f x = 0
hs_m₂ : MeasurableSet s
g_seq_s : ℕ → α →ₛ E
hg_seq_tendsto : ∀ (x : α), Tendsto (fun n => ↑(g_seq_s n) x) atTop (𝓝 (f x))
hg_seq_zero : ∀ x ∉ s, ∀ (n : ℕ), ↑(g_seq_s n) x = 0
n : ℕ
x : E
⊢ MeasurableSet (↑(g_seq_s n) ⁻¹' {x})

case refine_2
α✝ : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝² : Countable ι
f✝ g : α✝ → β
α : Type u_5
E : Type u_6
m m₂ : MeasurableSpace α
inst✝¹ : TopologicalSpace E
inst✝ : Zero E
s : Set α
f : α → E
hs_m : MeasurableSet s
hs : ∀ (t : Set α), MeasurableSet (s ∩ t) → MeasurableSet (s ∩ t)
hf : StronglyMeasurable f
hf_zero : ∀ x ∉ s, f x = 0
hs_m₂ : MeasurableSet s
g_seq_s : ℕ → α →ₛ E
hg_seq_tendsto : ∀ (x : α), Tendsto (fun n => ↑(g_seq_s n) x) atTop (𝓝 (f x))
hg_seq_zero : ∀ x ∉ s, ∀ (n : ℕ), ↑(g_seq_s n) x = 0
n : ℕ
x : E
⊢ MeasurableSet (↑(g_seq_s n) ⁻¹' {x} ∩ sᶜ)

tactic: by_cases hx : x = 0
state_before: case refine_2
α✝ : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝² : Countable ι
f✝ g : α✝ → β
α : Type u_5
E : Type u_6
m m₂ : MeasurableSpace α
inst✝¹ : TopologicalSpace E
inst✝ : Zero E
s : Set α
f : α → E
hs_m : MeasurableSet s
hs : ∀ (t : Set α), MeasurableSet (s ∩ t) → MeasurableSet (s ∩ t)
hf : StronglyMeasurable f
hf_zero : ∀ x ∉ s, f x = 0
hs_m₂ : MeasurableSet s
g_seq_s : ℕ → α →ₛ E
hg_seq_tendsto : ∀ (x : α), Tendsto (fun n => ↑(g_seq_s n) x) atTop (𝓝 (f x))
hg_seq_zero : ∀ x ∉ s, ∀ (n : ℕ), ↑(g_seq_s n) x = 0
n : ℕ
x : E
⊢ MeasurableSet (↑(g_seq_s n) ⁻¹' {x} ∩ sᶜ)
state_after: case pos
α✝ : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝² : Countable ι
f✝ g : α✝ → β
α : Type u_5
E : Type u_6
m m₂ : MeasurableSpace α
inst✝¹ : TopologicalSpace E
inst✝ : Zero E
s : Set α
f : α → E
hs_m : MeasurableSet s
hs : ∀ (t : Set α), MeasurableSet (s ∩ t) → MeasurableSet (s ∩ t)
hf : StronglyMeasurable f
hf_zero : ∀ x ∉ s, f x = 0
hs_m₂ : MeasurableSet s
g_seq_s : ℕ → α →ₛ E
hg_seq_tendsto : ∀ (x : α), Tendsto (fun n => ↑(g_seq_s n) x) atTop (𝓝 (f x))
hg_seq_zero : ∀ x ∉ s, ∀ (n : ℕ), ↑(g_seq_s n) x = 0
n : ℕ
x : E
hx : x = 0
⊢ MeasurableSet (↑(g_seq_s n) ⁻¹' {x} ∩ sᶜ)

case neg
α✝ : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝² : Countable ι
f✝ g : α✝ → β
α : Type u_5
E : Type u_6
m m₂ : MeasurableSpace α
inst✝¹ : TopologicalSpace E
inst✝ : Zero E
s : Set α
f : α → E
hs_m : MeasurableSet s
hs : ∀ (t : Set α), MeasurableSet (s ∩ t) → MeasurableSet (s ∩ t)
hf : StronglyMeasurable f
hf_zero : ∀ x ∉ s, f x = 0
hs_m₂ : MeasurableSet s
g_seq_s : ℕ → α →ₛ E
hg_seq_tendsto : ∀ (x : α), Tendsto (fun n => ↑(g_seq_s n) x) atTop (𝓝 (f x))
hg_seq_zero : ∀ x ∉ s, ∀ (n : ℕ), ↑(g_seq_s n) x = 0
n : ℕ
x : E
hx : ¬x = 0
⊢ MeasurableSet (↑(g_seq_s n) ⁻¹' {x} ∩ sᶜ)

tactic: exact @SimpleFunc.measurableSet_fiber _ _ m _ _
state_before: case refine_1
α✝ : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝² : Countable ι
f✝ g : α✝ → β
α : Type u_5
E : Type u_6
m m₂ : MeasurableSpace α
inst✝¹ : TopologicalSpace E
inst✝ : Zero E
s : Set α
f : α → E
hs_m : MeasurableSet s
hs : ∀ (t : Set α), MeasurableSet (s ∩ t) → MeasurableSet (s ∩ t)
hf : StronglyMeasurable f
hf_zero : ∀ x ∉ s, f x = 0
hs_m₂ : MeasurableSet s
g_seq_s : ℕ → α →ₛ E
hg_seq_tendsto : ∀ (x : α), Tendsto (fun n => ↑(g_seq_s n) x) atTop (𝓝 (f x))
hg_seq_zero : ∀ x ∉ s, ∀ (n : ℕ), ↑(g_seq_s n) x = 0
n : ℕ
x : E
⊢ MeasurableSet (↑(g_seq_s n) ⁻¹' {x})
state_after: no goals

tactic: suffices g_seq_s n ⁻¹' {x} ∩ sᶜ = sᶜ by
  rw [this]
  exact hs_m₂.compl
state_before: case pos
α✝ : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝² : Countable ι
f✝ g : α✝ → β
α : Type u_5
E : Type u_6
m m₂ : MeasurableSpace α
inst✝¹ : TopologicalSpace E
inst✝ : Zero E
s : Set α
f : α → E
hs_m : MeasurableSet s
hs : ∀ (t : Set α), MeasurableSet (s ∩ t) → MeasurableSet (s ∩ t)
hf : StronglyMeasurable f
hf_zero : ∀ x ∉ s, f x = 0
hs_m₂ : MeasurableSet s
g_seq_s : ℕ → α →ₛ E
hg_seq_tendsto : ∀ (x : α), Tendsto (fun n => ↑(g_seq_s n) x) atTop (𝓝 (f x))
hg_seq_zero : ∀ x ∉ s, ∀ (n : ℕ), ↑(g_seq_s n) x = 0
n : ℕ
x : E
hx : x = 0
⊢ MeasurableSet (↑(g_seq_s n) ⁻¹' {x} ∩ sᶜ)
state_after: case pos
α✝ : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝² : Countable ι
f✝ g : α✝ → β
α : Type u_5
E : Type u_6
m m₂ : MeasurableSpace α
inst✝¹ : TopologicalSpace E
inst✝ : Zero E
s : Set α
f : α → E
hs_m : MeasurableSet s
hs : ∀ (t : Set α), MeasurableSet (s ∩ t) → MeasurableSet (s ∩ t)
hf : StronglyMeasurable f
hf_zero : ∀ x ∉ s, f x = 0
hs_m₂ : MeasurableSet s
g_seq_s : ℕ → α →ₛ E
hg_seq_tendsto : ∀ (x : α), Tendsto (fun n => ↑(g_seq_s n) x) atTop (𝓝 (f x))
hg_seq_zero : ∀ x ∉ s, ∀ (n : ℕ), ↑(g_seq_s n) x = 0
n : ℕ
x : E
hx : x = 0
⊢ ↑(g_seq_s n) ⁻¹' {x} ∩ sᶜ = sᶜ

tactic: ext1 y
state_before: case pos
α✝ : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝² : Countable ι
f✝ g : α✝ → β
α : Type u_5
E : Type u_6
m m₂ : MeasurableSpace α
inst✝¹ : TopologicalSpace E
inst✝ : Zero E
s : Set α
f : α → E
hs_m : MeasurableSet s
hs : ∀ (t : Set α), MeasurableSet (s ∩ t) → MeasurableSet (s ∩ t)
hf : StronglyMeasurable f
hf_zero : ∀ x ∉ s, f x = 0
hs_m₂ : MeasurableSet s
g_seq_s : ℕ → α →ₛ E
hg_seq_tendsto : ∀ (x : α), Tendsto (fun n => ↑(g_seq_s n) x) atTop (𝓝 (f x))
hg_seq_zero : ∀ x ∉ s, ∀ (n : ℕ), ↑(g_seq_s n) x = 0
n : ℕ
x : E
hx : x = 0
⊢ ↑(g_seq_s n) ⁻¹' {x} ∩ sᶜ = sᶜ
state_after: case pos.h
α✝ : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝² : Countable ι
f✝ g : α✝ → β
α : Type u_5
E : Type u_6
m m₂ : MeasurableSpace α
inst✝¹ : TopologicalSpace E
inst✝ : Zero E
s : Set α
f : α → E
hs_m : MeasurableSet s
hs : ∀ (t : Set α), MeasurableSet (s ∩ t) → MeasurableSet (s ∩ t)
hf : StronglyMeasurable f
hf_zero : ∀ x ∉ s, f x = 0
hs_m₂ : MeasurableSet s
g_seq_s : ℕ → α →ₛ E
hg_seq_tendsto : ∀ (x : α), Tendsto (fun n => ↑(g_seq_s n) x) atTop (𝓝 (f x))
hg_seq_zero : ∀ x ∉ s, ∀ (n : ℕ), ↑(g_seq_s n) x = 0
n : ℕ
x : E
hx : x = 0
y : α
⊢ y ∈ ↑(g_seq_s n) ⁻¹' {x} ∩ sᶜ ↔ y ∈ sᶜ

tactic: rw [hx, Set.mem_inter_iff, Set.mem_preimage, Set.mem_singleton_iff]
state_before: case pos.h
α✝ : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝² : Countable ι
f✝ g : α✝ → β
α : Type u_5
E : Type u_6
m m₂ : MeasurableSpace α
inst✝¹ : TopologicalSpace E
inst✝ : Zero E
s : Set α
f : α → E
hs_m : MeasurableSet s
hs : ∀ (t : Set α), MeasurableSet (s ∩ t) → MeasurableSet (s ∩ t)
hf : StronglyMeasurable f
hf_zero : ∀ x ∉ s, f x = 0
hs_m₂ : MeasurableSet s
g_seq_s : ℕ → α →ₛ E
hg_seq_tendsto : ∀ (x : α), Tendsto (fun n => ↑(g_seq_s n) x) atTop (𝓝 (f x))
hg_seq_zero : ∀ x ∉ s, ∀ (n : ℕ), ↑(g_seq_s n) x = 0
n : ℕ
x : E
hx : x = 0
y : α
⊢ y ∈ ↑(g_seq_s n) ⁻¹' {x} ∩ sᶜ ↔ y ∈ sᶜ
state_after: case pos.h
α✝ : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝² : Countable ι
f✝ g : α✝ → β
α : Type u_5
E : Type u_6
m m₂ : MeasurableSpace α
inst✝¹ : TopologicalSpace E
inst✝ : Zero E
s : Set α
f : α → E
hs_m : MeasurableSet s
hs : ∀ (t : Set α), MeasurableSet (s ∩ t) → MeasurableSet (s ∩ t)
hf : StronglyMeasurable f
hf_zero : ∀ x ∉ s, f x = 0
hs_m₂ : MeasurableSet s
g_seq_s : ℕ → α →ₛ E
hg_seq_tendsto : ∀ (x : α), Tendsto (fun n => ↑(g_seq_s n) x) atTop (𝓝 (f x))
hg_seq_zero : ∀ x ∉ s, ∀ (n : ℕ), ↑(g_seq_s n) x = 0
n : ℕ
x : E
hx : x = 0
y : α
⊢ ↑(g_seq_s n) y = 0 ∧ y ∈ sᶜ ↔ y ∈ sᶜ

tactic: exact ⟨fun h => h.2, fun h => ⟨hg_seq_zero y h n, h⟩⟩
state_before: case pos.h
α✝ : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝² : Countable ι
f✝ g : α✝ → β
α : Type u_5
E : Type u_6
m m₂ : MeasurableSpace α
inst✝¹ : TopologicalSpace E
inst✝ : Zero E
s : Set α
f : α → E
hs_m : MeasurableSet s
hs : ∀ (t : Set α), MeasurableSet (s ∩ t) → MeasurableSet (s ∩ t)
hf : StronglyMeasurable f
hf_zero : ∀ x ∉ s, f x = 0
hs_m₂ : MeasurableSet s
g_seq_s : ℕ → α →ₛ E
hg_seq_tendsto : ∀ (x : α), Tendsto (fun n => ↑(g_seq_s n) x) atTop (𝓝 (f x))
hg_seq_zero : ∀ x ∉ s, ∀ (n : ℕ), ↑(g_seq_s n) x = 0
n : ℕ
x : E
hx : x = 0
y : α
⊢ ↑(g_seq_s n) y = 0 ∧ y ∈ sᶜ ↔ y ∈ sᶜ
state_after: no goals

tactic: rw [this]
state_before: α✝ : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝² : Countable ι
f✝ g : α✝ → β
α : Type u_5
E : Type u_6
m m₂ : MeasurableSpace α
inst✝¹ : TopologicalSpace E
inst✝ : Zero E
s : Set α
f : α → E
hs_m : MeasurableSet s
hs : ∀ (t : Set α), MeasurableSet (s ∩ t) → MeasurableSet (s ∩ t)
hf : StronglyMeasurable f
hf_zero : ∀ x ∉ s, f x = 0
hs_m₂ : MeasurableSet s
g_seq_s : ℕ → α →ₛ E
hg_seq_tendsto : ∀ (x : α), Tendsto (fun n => ↑(g_seq_s n) x) atTop (𝓝 (f x))
hg_seq_zero : ∀ x ∉ s, ∀ (n : ℕ), ↑(g_seq_s n) x = 0
n : ℕ
x : E
hx : x = 0
this : ↑(g_seq_s n) ⁻¹' {x} ∩ sᶜ = sᶜ
⊢ MeasurableSet (↑(g_seq_s n) ⁻¹' {x} ∩ sᶜ)
state_after: α✝ : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝² : Countable ι
f✝ g : α✝ → β
α : Type u_5
E : Type u_6
m m₂ : MeasurableSpace α
inst✝¹ : TopologicalSpace E
inst✝ : Zero E
s : Set α
f : α → E
hs_m : MeasurableSet s
hs : ∀ (t : Set α), MeasurableSet (s ∩ t) → MeasurableSet (s ∩ t)
hf : StronglyMeasurable f
hf_zero : ∀ x ∉ s, f x = 0
hs_m₂ : MeasurableSet s
g_seq_s : ℕ → α →ₛ E
hg_seq_tendsto : ∀ (x : α), Tendsto (fun n => ↑(g_seq_s n) x) atTop (𝓝 (f x))
hg_seq_zero : ∀ x ∉ s, ∀ (n : ℕ), ↑(g_seq_s n) x = 0
n : ℕ
x : E
hx : x = 0
this : ↑(g_seq_s n) ⁻¹' {x} ∩ sᶜ = sᶜ
⊢ MeasurableSet sᶜ

tactic: exact hs_m₂.compl
state_before: α✝ : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝² : Countable ι
f✝ g : α✝ → β
α : Type u_5
E : Type u_6
m m₂ : MeasurableSpace α
inst✝¹ : TopologicalSpace E
inst✝ : Zero E
s : Set α
f : α → E
hs_m : MeasurableSet s
hs : ∀ (t : Set α), MeasurableSet (s ∩ t) → MeasurableSet (s ∩ t)
hf : StronglyMeasurable f
hf_zero : ∀ x ∉ s, f x = 0
hs_m₂ : MeasurableSet s
g_seq_s : ℕ → α →ₛ E
hg_seq_tendsto : ∀ (x : α), Tendsto (fun n => ↑(g_seq_s n) x) atTop (𝓝 (f x))
hg_seq_zero : ∀ x ∉ s, ∀ (n : ℕ), ↑(g_seq_s n) x = 0
n : ℕ
x : E
hx : x = 0
this : ↑(g_seq_s n) ⁻¹' {x} ∩ sᶜ = sᶜ
⊢ MeasurableSet sᶜ
state_after: no goals

tactic: suffices g_seq_s n ⁻¹' {x} ∩ sᶜ = ∅ by
  rw [this]
  exact MeasurableSet.empty
state_before: case neg
α✝ : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝² : Countable ι
f✝ g : α✝ → β
α : Type u_5
E : Type u_6
m m₂ : MeasurableSpace α
inst✝¹ : TopologicalSpace E
inst✝ : Zero E
s : Set α
f : α → E
hs_m : MeasurableSet s
hs : ∀ (t : Set α), MeasurableSet (s ∩ t) → MeasurableSet (s ∩ t)
hf : StronglyMeasurable f
hf_zero : ∀ x ∉ s, f x = 0
hs_m₂ : MeasurableSet s
g_seq_s : ℕ → α →ₛ E
hg_seq_tendsto : ∀ (x : α), Tendsto (fun n => ↑(g_seq_s n) x) atTop (𝓝 (f x))
hg_seq_zero : ∀ x ∉ s, ∀ (n : ℕ), ↑(g_seq_s n) x = 0
n : ℕ
x : E
hx : ¬x = 0
⊢ MeasurableSet (↑(g_seq_s n) ⁻¹' {x} ∩ sᶜ)
state_after: case neg
α✝ : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝² : Countable ι
f✝ g : α✝ → β
α : Type u_5
E : Type u_6
m m₂ : MeasurableSpace α
inst✝¹ : TopologicalSpace E
inst✝ : Zero E
s : Set α
f : α → E
hs_m : MeasurableSet s
hs : ∀ (t : Set α), MeasurableSet (s ∩ t) → MeasurableSet (s ∩ t)
hf : StronglyMeasurable f
hf_zero : ∀ x ∉ s, f x = 0
hs_m₂ : MeasurableSet s
g_seq_s : ℕ → α →ₛ E
hg_seq_tendsto : ∀ (x : α), Tendsto (fun n => ↑(g_seq_s n) x) atTop (𝓝 (f x))
hg_seq_zero : ∀ x ∉ s, ∀ (n : ℕ), ↑(g_seq_s n) x = 0
n : ℕ
x : E
hx : ¬x = 0
⊢ ↑(g_seq_s n) ⁻¹' {x} ∩ sᶜ = ∅

tactic: ext1 y
state_before: case neg
α✝ : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝² : Countable ι
f✝ g : α✝ → β
α : Type u_5
E : Type u_6
m m₂ : MeasurableSpace α
inst✝¹ : TopologicalSpace E
inst✝ : Zero E
s : Set α
f : α → E
hs_m : MeasurableSet s
hs : ∀ (t : Set α), MeasurableSet (s ∩ t) → MeasurableSet (s ∩ t)
hf : StronglyMeasurable f
hf_zero : ∀ x ∉ s, f x = 0
hs_m₂ : MeasurableSet s
g_seq_s : ℕ → α →ₛ E
hg_seq_tendsto : ∀ (x : α), Tendsto (fun n => ↑(g_seq_s n) x) atTop (𝓝 (f x))
hg_seq_zero : ∀ x ∉ s, ∀ (n : ℕ), ↑(g_seq_s n) x = 0
n : ℕ
x : E
hx : ¬x = 0
⊢ ↑(g_seq_s n) ⁻¹' {x} ∩ sᶜ = ∅
state_after: case neg.h
α✝ : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝² : Countable ι
f✝ g : α✝ → β
α : Type u_5
E : Type u_6
m m₂ : MeasurableSpace α
inst✝¹ : TopologicalSpace E
inst✝ : Zero E
s : Set α
f : α → E
hs_m : MeasurableSet s
hs : ∀ (t : Set α), MeasurableSet (s ∩ t) → MeasurableSet (s ∩ t)
hf : StronglyMeasurable f
hf_zero : ∀ x ∉ s, f x = 0
hs_m₂ : MeasurableSet s
g_seq_s : ℕ → α →ₛ E
hg_seq_tendsto : ∀ (x : α), Tendsto (fun n => ↑(g_seq_s n) x) atTop (𝓝 (f x))
hg_seq_zero : ∀ x ∉ s, ∀ (n : ℕ), ↑(g_seq_s n) x = 0
n : ℕ
x : E
hx : ¬x = 0
y : α
⊢ y ∈ ↑(g_seq_s n) ⁻¹' {x} ∩ sᶜ ↔ y ∈ ∅

tactic: simp only [mem_inter_iff, mem_preimage, mem_singleton_iff, mem_compl_iff,
  mem_empty_iff_false, iff_false_iff, not_and, not_not_mem]
state_before: case neg.h
α✝ : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝² : Countable ι
f✝ g : α✝ → β
α : Type u_5
E : Type u_6
m m₂ : MeasurableSpace α
inst✝¹ : TopologicalSpace E
inst✝ : Zero E
s : Set α
f : α → E
hs_m : MeasurableSet s
hs : ∀ (t : Set α), MeasurableSet (s ∩ t) → MeasurableSet (s ∩ t)
hf : StronglyMeasurable f
hf_zero : ∀ x ∉ s, f x = 0
hs_m₂ : MeasurableSet s
g_seq_s : ℕ → α →ₛ E
hg_seq_tendsto : ∀ (x : α), Tendsto (fun n => ↑(g_seq_s n) x) atTop (𝓝 (f x))
hg_seq_zero : ∀ x ∉ s, ∀ (n : ℕ), ↑(g_seq_s n) x = 0
n : ℕ
x : E
hx : ¬x = 0
y : α
⊢ y ∈ ↑(g_seq_s n) ⁻¹' {x} ∩ sᶜ ↔ y ∈ ∅
state_after: case neg.h
α✝ : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝² : Countable ι
f✝ g : α✝ → β
α : Type u_5
E : Type u_6
m m₂ : MeasurableSpace α
inst✝¹ : TopologicalSpace E
inst✝ : Zero E
s : Set α
f : α → E
hs_m : MeasurableSet s
hs : ∀ (t : Set α), MeasurableSet (s ∩ t) → MeasurableSet (s ∩ t)
hf : StronglyMeasurable f
hf_zero : ∀ x ∉ s, f x = 0
hs_m₂ : MeasurableSet s
g_seq_s : ℕ → α →ₛ E
hg_seq_tendsto : ∀ (x : α), Tendsto (fun n => ↑(g_seq_s n) x) atTop (𝓝 (f x))
hg_seq_zero : ∀ x ∉ s, ∀ (n : ℕ), ↑(g_seq_s n) x = 0
n : ℕ
x : E
hx : ¬x = 0
y : α
⊢ ↑(g_seq_s n) y = x → y ∈ s

tactic: refine Function.mtr fun hys => ?_
state_before: case neg.h
α✝ : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝² : Countable ι
f✝ g : α✝ → β
α : Type u_5
E : Type u_6
m m₂ : MeasurableSpace α
inst✝¹ : TopologicalSpace E
inst✝ : Zero E
s : Set α
f : α → E
hs_m : MeasurableSet s
hs : ∀ (t : Set α), MeasurableSet (s ∩ t) → MeasurableSet (s ∩ t)
hf : StronglyMeasurable f
hf_zero : ∀ x ∉ s, f x = 0
hs_m₂ : MeasurableSet s
g_seq_s : ℕ → α →ₛ E
hg_seq_tendsto : ∀ (x : α), Tendsto (fun n => ↑(g_seq_s n) x) atTop (𝓝 (f x))
hg_seq_zero : ∀ x ∉ s, ∀ (n : ℕ), ↑(g_seq_s n) x = 0
n : ℕ
x : E
hx : ¬x = 0
y : α
⊢ ↑(g_seq_s n) y = x → y ∈ s
state_after: case neg.h
α✝ : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝² : Countable ι
f✝ g : α✝ → β
α : Type u_5
E : Type u_6
m m₂ : MeasurableSpace α
inst✝¹ : TopologicalSpace E
inst✝ : Zero E
s : Set α
f : α → E
hs_m : MeasurableSet s
hs : ∀ (t : Set α), MeasurableSet (s ∩ t) → MeasurableSet (s ∩ t)
hf : StronglyMeasurable f
hf_zero : ∀ x ∉ s, f x = 0
hs_m₂ : MeasurableSet s
g_seq_s : ℕ → α →ₛ E
hg_seq_tendsto : ∀ (x : α), Tendsto (fun n => ↑(g_seq_s n) x) atTop (𝓝 (f x))
hg_seq_zero : ∀ x ∉ s, ∀ (n : ℕ), ↑(g_seq_s n) x = 0
n : ℕ
x : E
hx : ¬x = 0
y : α
hys : y ∉ s
⊢ ¬↑(g_seq_s n) y = x

tactic: rw [hg_seq_zero y hys n]
state_before: case neg.h
α✝ : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝² : Countable ι
f✝ g : α✝ → β
α : Type u_5
E : Type u_6
m m₂ : MeasurableSpace α
inst✝¹ : TopologicalSpace E
inst✝ : Zero E
s : Set α
f : α → E
hs_m : MeasurableSet s
hs : ∀ (t : Set α), MeasurableSet (s ∩ t) → MeasurableSet (s ∩ t)
hf : StronglyMeasurable f
hf_zero : ∀ x ∉ s, f x = 0
hs_m₂ : MeasurableSet s
g_seq_s : ℕ → α →ₛ E
hg_seq_tendsto : ∀ (x : α), Tendsto (fun n => ↑(g_seq_s n) x) atTop (𝓝 (f x))
hg_seq_zero : ∀ x ∉ s, ∀ (n : ℕ), ↑(g_seq_s n) x = 0
n : ℕ
x : E
hx : ¬x = 0
y : α
hys : y ∉ s
⊢ ¬↑(g_seq_s n) y = x
state_after: case neg.h
α✝ : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝² : Countable ι
f✝ g : α✝ → β
α : Type u_5
E : Type u_6
m m₂ : MeasurableSpace α
inst✝¹ : TopologicalSpace E
inst✝ : Zero E
s : Set α
f : α → E
hs_m : MeasurableSet s
hs : ∀ (t : Set α), MeasurableSet (s ∩ t) → MeasurableSet (s ∩ t)
hf : StronglyMeasurable f
hf_zero : ∀ x ∉ s, f x = 0
hs_m₂ : MeasurableSet s
g_seq_s : ℕ → α →ₛ E
hg_seq_tendsto : ∀ (x : α), Tendsto (fun n => ↑(g_seq_s n) x) atTop (𝓝 (f x))
hg_seq_zero : ∀ x ∉ s, ∀ (n : ℕ), ↑(g_seq_s n) x = 0
n : ℕ
x : E
hx : ¬x = 0
y : α
hys : y ∉ s
⊢ ¬0 = x

tactic: exact Ne.symm hx
state_before: case neg.h
α✝ : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝² : Countable ι
f✝ g : α✝ → β
α : Type u_5
E : Type u_6
m m₂ : MeasurableSpace α
inst✝¹ : TopologicalSpace E
inst✝ : Zero E
s : Set α
f : α → E
hs_m : MeasurableSet s
hs : ∀ (t : Set α), MeasurableSet (s ∩ t) → MeasurableSet (s ∩ t)
hf : StronglyMeasurable f
hf_zero : ∀ x ∉ s, f x = 0
hs_m₂ : MeasurableSet s
g_seq_s : ℕ → α →ₛ E
hg_seq_tendsto : ∀ (x : α), Tendsto (fun n => ↑(g_seq_s n) x) atTop (𝓝 (f x))
hg_seq_zero : ∀ x ∉ s, ∀ (n : ℕ), ↑(g_seq_s n) x = 0
n : ℕ
x : E
hx : ¬x = 0
y : α
hys : y ∉ s
⊢ ¬0 = x
state_after: no goals

tactic: rw [this]
state_before: α✝ : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝² : Countable ι
f✝ g : α✝ → β
α : Type u_5
E : Type u_6
m m₂ : MeasurableSpace α
inst✝¹ : TopologicalSpace E
inst✝ : Zero E
s : Set α
f : α → E
hs_m : MeasurableSet s
hs : ∀ (t : Set α), MeasurableSet (s ∩ t) → MeasurableSet (s ∩ t)
hf : StronglyMeasurable f
hf_zero : ∀ x ∉ s, f x = 0
hs_m₂ : MeasurableSet s
g_seq_s : ℕ → α →ₛ E
hg_seq_tendsto : ∀ (x : α), Tendsto (fun n => ↑(g_seq_s n) x) atTop (𝓝 (f x))
hg_seq_zero : ∀ x ∉ s, ∀ (n : ℕ), ↑(g_seq_s n) x = 0
n : ℕ
x : E
hx : ¬x = 0
this : ↑(g_seq_s n) ⁻¹' {x} ∩ sᶜ = ∅
⊢ MeasurableSet (↑(g_seq_s n) ⁻¹' {x} ∩ sᶜ)
state_after: α✝ : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝² : Countable ι
f✝ g : α✝ → β
α : Type u_5
E : Type u_6
m m₂ : MeasurableSpace α
inst✝¹ : TopologicalSpace E
inst✝ : Zero E
s : Set α
f : α → E
hs_m : MeasurableSet s
hs : ∀ (t : Set α), MeasurableSet (s ∩ t) → MeasurableSet (s ∩ t)
hf : StronglyMeasurable f
hf_zero : ∀ x ∉ s, f x = 0
hs_m₂ : MeasurableSet s
g_seq_s : ℕ → α →ₛ E
hg_seq_tendsto : ∀ (x : α), Tendsto (fun n => ↑(g_seq_s n) x) atTop (𝓝 (f x))
hg_seq_zero : ∀ x ∉ s, ∀ (n : ℕ), ↑(g_seq_s n) x = 0
n : ℕ
x : E
hx : ¬x = 0
this : ↑(g_seq_s n) ⁻¹' {x} ∩ sᶜ = ∅
⊢ MeasurableSet ∅

tactic: exact MeasurableSet.empty
state_before: α✝ : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝² : Countable ι
f✝ g : α✝ → β
α : Type u_5
E : Type u_6
m m₂ : MeasurableSpace α
inst✝¹ : TopologicalSpace E
inst✝ : Zero E
s : Set α
f : α → E
hs_m : MeasurableSet s
hs : ∀ (t : Set α), MeasurableSet (s ∩ t) → MeasurableSet (s ∩ t)
hf : StronglyMeasurable f
hf_zero : ∀ x ∉ s, f x = 0
hs_m₂ : MeasurableSet s
g_seq_s : ℕ → α →ₛ E
hg_seq_tendsto : ∀ (x : α), Tendsto (fun n => ↑(g_seq_s n) x) atTop (𝓝 (f x))
hg_seq_zero : ∀ x ∉ s, ∀ (n : ℕ), ↑(g_seq_s n) x = 0
n : ℕ
x : E
hx : ¬x = 0
this : ↑(g_seq_s n) ⁻¹' {x} ∩ sᶜ = ∅
⊢ MeasurableSet ∅
state_after: no goals