Theorem: ContinuousOn.aestronglyMeasurable_of_isCompact
File: Mathlib/MeasureTheory/Integral/IntegrableOn.lean

Proof steps:
state_before: α : Type u_1
β : Type u_2
E : Type u_3
F : Type u_4
inst✝⁵ : MeasurableSpace α
inst✝⁴ : NormedAddCommGroup E
inst✝³ : TopologicalSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : TopologicalSpace β
inst✝ : PseudoMetrizableSpace β
f : α → β
s : Set α
μ : Measure α
hf : ContinuousOn f s
hs : IsCompact s
h's : MeasurableSet s
⊢ AEStronglyMeasurable f (μ.restrict s)
state_after: α : Type u_1
β : Type u_2
E : Type u_3
F : Type u_4
inst✝⁵ : MeasurableSpace α
inst✝⁴ : NormedAddCommGroup E
inst✝³ : TopologicalSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : TopologicalSpace β
inst✝ : PseudoMetrizableSpace β
f : α → β
s : Set α
μ : Measure α
hf : ContinuousOn f s
hs : IsCompact s
h's : MeasurableSet s
this : PseudoMetricSpace β := pseudoMetrizableSpacePseudoMetric β
⊢ AEStronglyMeasurable f (μ.restrict s)
tactic: letI := pseudoMetrizableSpacePseudoMetric β
state_before: α : Type u_1
β : Type u_2
E : Type u_3
F : Type u_4
inst✝⁵ : MeasurableSpace α
inst✝⁴ : NormedAddCommGroup E
inst✝³ : TopologicalSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : TopologicalSpace β
inst✝ : PseudoMetrizableSpace β
f : α → β
s : Set α
μ : Measure α
hf : ContinuousOn f s
hs : IsCompact s
h's : MeasurableSet s
this : PseudoMetricSpace β := pseudoMetrizableSpacePseudoMetric β
⊢ AEStronglyMeasurable f (μ.restrict s)
state_after: α : Type u_1
β : Type u_2
E : Type u_3
F : Type u_4
inst✝⁵ : MeasurableSpace α
inst✝⁴ : NormedAddCommGroup E
inst✝³ : TopologicalSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : TopologicalSpace β
inst✝ : PseudoMetrizableSpace β
f : α → β
s : Set α
μ : Measure α
hf : ContinuousOn f s
hs : IsCompact s
h's : MeasurableSet s
this : PseudoMetricSpace β := pseudoMetrizableSpacePseudoMetric β
this✝¹ : MeasurableSpace β := borel β
this✝ : BorelSpace β
⊢ AEStronglyMeasurable f (μ.restrict s)
tactic: borelize β
state_before: α : Type u_1
β : Type u_2
E : Type u_3
F : Type u_4
inst✝⁵ : MeasurableSpace α
inst✝⁴ : NormedAddCommGroup E
inst✝³ : TopologicalSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : TopologicalSpace β
inst✝ : PseudoMetrizableSpace β
f : α → β
s : Set α
μ : Measure α
hf : ContinuousOn f s
hs : IsCompact s
h's : MeasurableSet s
this : PseudoMetricSpace β := pseudoMetrizableSpacePseudoMetric β
this✝¹ : MeasurableSpace β := borel β
this✝ : BorelSpace β
⊢ AEStronglyMeasurable f (μ.restrict s)
state_after: α : Type u_1
β : Type u_2
E : Type u_3
F : Type u_4
inst✝⁵ : MeasurableSpace α
inst✝⁴ : NormedAddCommGroup E
inst✝³ : TopologicalSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : TopologicalSpace β
inst✝ : PseudoMetrizableSpace β
f : α → β
s : Set α
μ : Measure α
hf : ContinuousOn f s
hs : IsCompact s
h's : MeasurableSet s
this : PseudoMetricSpace β := pseudoMetrizableSpacePseudoMetric β
this✝¹ : MeasurableSpace β := borel β
this✝ : BorelSpace β
⊢ AEMeasurable f (μ.restrict s) ∧ ∃ t, IsSeparable t ∧ ∀ᵐ (x : α) ∂μ.restrict s, f x ∈ t
tactic: rw [aestronglyMeasurable_iff_aemeasurable_separable]
state_before: α : Type u_1
β : Type u_2
E : Type u_3
F : Type u_4
inst✝⁵ : MeasurableSpace α
inst✝⁴ : NormedAddCommGroup E
inst✝³ : TopologicalSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : TopologicalSpace β
inst✝ : PseudoMetrizableSpace β
f : α → β
s : Set α
μ : Measure α
hf : ContinuousOn f s
hs : IsCompact s
h's : MeasurableSet s
this : PseudoMetricSpace β := pseudoMetrizableSpacePseudoMetric β
this✝¹ : MeasurableSpace β := borel β
this✝ : BorelSpace β
⊢ AEMeasurable f (μ.restrict s) ∧ ∃ t, IsSeparable t ∧ ∀ᵐ (x : α) ∂μ.restrict s, f x ∈ t
state_after: case refine_1
α : Type u_1
β : Type u_2
E : Type u_3
F : Type u_4
inst✝⁵ : MeasurableSpace α
inst✝⁴ : NormedAddCommGroup E
inst✝³ : TopologicalSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : TopologicalSpace β
inst✝ : PseudoMetrizableSpace β
f : α → β
s : Set α
μ : Measure α
hf : ContinuousOn f s
hs : IsCompact s
h's : MeasurableSet s
this : PseudoMetricSpace β := pseudoMetrizableSpacePseudoMetric β
this✝¹ : MeasurableSpace β := borel β
this✝ : BorelSpace β
⊢ IsSeparable (f '' s)

case refine_2
α : Type u_1
β : Type u_2
E : Type u_3
F : Type u_4
inst✝⁵ : MeasurableSpace α
inst✝⁴ : NormedAddCommGroup E
inst✝³ : TopologicalSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : TopologicalSpace β
inst✝ : PseudoMetrizableSpace β
f : α → β
s : Set α
μ : Measure α
hf : ContinuousOn f s
hs : IsCompact s
h's : MeasurableSet s
this : PseudoMetricSpace β := pseudoMetrizableSpacePseudoMetric β
this✝¹ : MeasurableSpace β := borel β
this✝ : BorelSpace β
⊢ ∀ᵐ (x : α) ∂μ.restrict s, f x ∈ f '' s
tactic: refine ⟨hf.aemeasurable h's, f '' s, ?_, ?_⟩