Theorem: Measurable.stronglyMeasurable
File: Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean

tactic: letI := pseudoMetrizableSpacePseudoMetric β
state_before: α : Type u_2
β : Type u_1
γ : Type ?u.134098
ι : Type ?u.134101
inst✝⁵ : Countable ι
f g : α → β
mα : MeasurableSpace α
inst✝⁴ : MeasurableSpace β
inst✝³ : TopologicalSpace β
inst✝² : PseudoMetrizableSpace β
inst✝¹ : SecondCountableTopology β
inst✝ : OpensMeasurableSpace β
hf : Measurable f
⊢ StronglyMeasurable f

tactic: rcases isEmpty_or_nonempty β with ⟨⟩ <;> skip
state_before: α : Type u_2
β : Type u_1
γ : Type ?u.134098
ι : Type ?u.134101
inst✝⁵ : Countable ι
f g : α → β
mα : MeasurableSpace α
inst✝⁴ : MeasurableSpace β
inst✝³ : TopologicalSpace β
inst✝² : PseudoMetrizableSpace β
inst✝¹ : SecondCountableTopology β
inst✝ : OpensMeasurableSpace β
hf : Measurable f
this : PseudoMetricSpace β := pseudoMetrizableSpacePseudoMetric β
⊢ StronglyMeasurable f

tactic: exact Subsingleton.stronglyMeasurable f
state_before: case inl
α : Type u_2
β : Type u_1
γ : Type ?u.134098
ι : Type ?u.134101
inst✝⁵ : Countable ι
f g : α → β
mα : MeasurableSpace α
inst✝⁴ : MeasurableSpace β
inst✝³ : TopologicalSpace β
inst✝² : PseudoMetrizableSpace β
inst✝¹ : SecondCountableTopology β
inst✝ : OpensMeasurableSpace β
hf : Measurable f
this : PseudoMetricSpace β := pseudoMetrizableSpacePseudoMetric β
h✝ : IsEmpty β
⊢ StronglyMeasurable f

tactic: inhabit β
state_before: case inr
α : Type u_2
β : Type u_1
γ : Type ?u.134098
ι : Type ?u.134101
inst✝⁵ : Countable ι
f g : α → β
mα : MeasurableSpace α
inst✝⁴ : MeasurableSpace β
inst✝³ : TopologicalSpace β
inst✝² : PseudoMetrizableSpace β
inst✝¹ : SecondCountableTopology β
inst✝ : OpensMeasurableSpace β
hf : Measurable f
this : PseudoMetricSpace β := pseudoMetrizableSpacePseudoMetric β
h✝ : Nonempty β
⊢ StronglyMeasurable f

tactic: exact
  ⟨SimpleFunc.approxOn f hf Set.univ default (Set.mem_univ _), fun x =>
    SimpleFunc.tendsto_approxOn hf (Set.mem_univ _) (by rw [closure_univ]; simp)⟩
state_before: case inr
α : Type u_2
β : Type u_1
γ : Type ?u.134098
ι : Type ?u.134101
inst✝⁵ : Countable ι
f g : α → β
mα : MeasurableSpace α
inst✝⁴ : MeasurableSpace β
inst✝³ : TopologicalSpace β
inst✝² : PseudoMetrizableSpace β
inst✝¹ : SecondCountableTopology β
inst✝ : OpensMeasurableSpace β
hf : Measurable f
this : PseudoMetricSpace β := pseudoMetrizableSpacePseudoMetric β
h✝ : Nonempty β
inhabited_h : Inhabited β
⊢ StronglyMeasurable f

tactic: rw [closure_univ]
state_before: α : Type u_2
β : Type u_1
γ : Type ?u.134098
ι : Type ?u.134101
inst✝⁵ : Countable ι
f g : α → β
mα : MeasurableSpace α
inst✝⁴ : MeasurableSpace β
inst✝³ : TopologicalSpace β
inst✝² : PseudoMetrizableSpace β
inst✝¹ : SecondCountableTopology β
inst✝ : OpensMeasurableSpace β
hf : Measurable f
this : PseudoMetricSpace β := pseudoMetrizableSpacePseudoMetric β
h✝ : Nonempty β
inhabited_h : Inhabited β
x : α
⊢ f x ∈ closure univ

tactic: simp
state_before: α : Type u_2
β : Type u_1
γ : Type ?u.134098
ι : Type ?u.134101
inst✝⁵ : Countable ι
f g : α → β
mα : MeasurableSpace α
inst✝⁴ : MeasurableSpace β
inst✝³ : TopologicalSpace β
inst✝² : PseudoMetrizableSpace β
inst✝¹ : SecondCountableTopology β
inst✝ : OpensMeasurableSpace β
hf : Measurable f
this : PseudoMetricSpace β := pseudoMetrizableSpacePseudoMetric β
h✝ : Nonempty β
inhabited_h : Inhabited β
x : α
⊢ f x ∈ univ