Theorem: HasCompactSupport.stronglyMeasurable_of_prod
File: Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean

Proof steps:
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝⁹ : Countable ι
f✝ g : α → β
X : Type u_5
Y : Type u_6
inst✝⁸ : Zero α
inst✝⁷ : TopologicalSpace X
inst✝⁶ : TopologicalSpace Y
inst✝⁵ : MeasurableSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : OpensMeasurableSpace X
inst✝² : OpensMeasurableSpace Y
inst✝¹ : TopologicalSpace α
inst✝ : PseudoMetrizableSpace α
f : X × Y → α
hf : Continuous f
h'f : HasCompactSupport f
⊢ StronglyMeasurable f
state_after: α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝⁹ : Countable ι
f✝ g : α → β
X : Type u_5
Y : Type u_6
inst✝⁸ : Zero α
inst✝⁷ : TopologicalSpace X
inst✝⁶ : TopologicalSpace Y
inst✝⁵ : MeasurableSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : OpensMeasurableSpace X
inst✝² : OpensMeasurableSpace Y
inst✝¹ : TopologicalSpace α
inst✝ : PseudoMetrizableSpace α
f : X × Y → α
hf : Continuous f
h'f : HasCompactSupport f
this✝¹ : MeasurableSpace α := borel α
this✝ : BorelSpace α
⊢ StronglyMeasurable f
tactic: borelize α
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝⁹ : Countable ι
f✝ g : α → β
X : Type u_5
Y : Type u_6
inst✝⁸ : Zero α
inst✝⁷ : TopologicalSpace X
inst✝⁶ : TopologicalSpace Y
inst✝⁵ : MeasurableSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : OpensMeasurableSpace X
inst✝² : OpensMeasurableSpace Y
inst✝¹ : TopologicalSpace α
inst✝ : PseudoMetrizableSpace α
f : X × Y → α
hf : Continuous f
h'f : HasCompactSupport f
this✝¹ : MeasurableSpace α := borel α
this✝ : BorelSpace α
⊢ StronglyMeasurable f
state_after: α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝⁹ : Countable ι
f✝ g : α → β
X : Type u_5
Y : Type u_6
inst✝⁸ : Zero α
inst✝⁷ : TopologicalSpace X
inst✝⁶ : TopologicalSpace Y
inst✝⁵ : MeasurableSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : OpensMeasurableSpace X
inst✝² : OpensMeasurableSpace Y
inst✝¹ : TopologicalSpace α
inst✝ : PseudoMetrizableSpace α
f : X × Y → α
hf : Continuous f
h'f : HasCompactSupport f
this✝¹ : MeasurableSpace α := borel α
this✝ : BorelSpace α
⊢ IsSeparable (range f)
tactic: apply stronglyMeasurable_iff_measurable_separable.2 ⟨h'f.measurable_of_prod hf, ?_⟩
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝⁹ : Countable ι
f✝ g : α → β
X : Type u_5
Y : Type u_6
inst✝⁸ : Zero α
inst✝⁷ : TopologicalSpace X
inst✝⁶ : TopologicalSpace Y
inst✝⁵ : MeasurableSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : OpensMeasurableSpace X
inst✝² : OpensMeasurableSpace Y
inst✝¹ : TopologicalSpace α
inst✝ : PseudoMetrizableSpace α
f : X × Y → α
hf : Continuous f
h'f : HasCompactSupport f
this✝¹ : MeasurableSpace α := borel α
this✝ : BorelSpace α
⊢ IsSeparable (range f)
state_after: α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝⁹ : Countable ι
f✝ g : α → β
X : Type u_5
Y : Type u_6
inst✝⁸ : Zero α
inst✝⁷ : TopologicalSpace X
inst✝⁶ : TopologicalSpace Y
inst✝⁵ : MeasurableSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : OpensMeasurableSpace X
inst✝² : OpensMeasurableSpace Y
inst✝¹ : TopologicalSpace α
inst✝ : PseudoMetrizableSpace α
f : X × Y → α
hf : Continuous f
h'f : HasCompactSupport f
this✝¹ : MeasurableSpace α := borel α
this✝ : BorelSpace α
this : PseudoMetricSpace α := pseudoMetrizableSpacePseudoMetric α
⊢ IsSeparable (range f)
tactic: letI : PseudoMetricSpace α := pseudoMetrizableSpacePseudoMetric α
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝⁹ : Countable ι
f✝ g : α → β
X : Type u_5
Y : Type u_6
inst✝⁸ : Zero α
inst✝⁷ : TopologicalSpace X
inst✝⁶ : TopologicalSpace Y
inst✝⁵ : MeasurableSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : OpensMeasurableSpace X
inst✝² : OpensMeasurableSpace Y
inst✝¹ : TopologicalSpace α
inst✝ : PseudoMetrizableSpace α
f : X × Y → α
hf : Continuous f
h'f : HasCompactSupport f
this✝¹ : MeasurableSpace α := borel α
this✝ : BorelSpace α
this : PseudoMetricSpace α := pseudoMetrizableSpacePseudoMetric α
⊢ IsSeparable (range f)
state_after: no goals
tactic: exact IsCompact.isSeparable (s := range f) (h'f.isCompact_range hf)