α : Type u_2
β : Type u_1
γ : Type ?u.398194
ι : Type u_3
inst✝³ : Countable ι
m✝ : MeasurableSpace α
μ✝ : Measure α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
f g : α → β
inst✝ : PseudoMetrizableSpace β
m : MeasurableSpace α
μ : ι → Measure α
h : ∀ (i : ι), AEStronglyMeasurable f (μ i)
this✝¹ : MeasurableSpace β := borel β
this✝ : BorelSpace β
⊢ AEStronglyMeasurable f (sum μ)
α : Type u_2
β : Type u_1
γ : Type ?u.398194
ι : Type u_3
inst✝³ : Countable ι
m✝ : MeasurableSpace α
μ✝ : Measure α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
f g : α → β
inst✝ : PseudoMetrizableSpace β
m : MeasurableSpace α
μ : ι → Measure α
h : ∀ (i : ι), AEStronglyMeasurable f (μ i)
⊢ AEStronglyMeasurable f (sum μ)
borelize β
α : Type u_2
β : Type u_1
γ : Type ?u.398194
ι : Type u_3
inst✝³ : Countable ι
m✝ : MeasurableSpace α
μ✝ : Measure α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
f g : α → β
inst✝ : PseudoMetrizableSpace β
m : MeasurableSpace α
μ : ι → Measure α
h : ∀ (i : ι), AEStronglyMeasurable f (μ i)
this✝¹ : MeasurableSpace β := borel β
this✝ : BorelSpace β
⊢ ∃ t, IsSeparable t ∧ ∀ᵐ (x : α) ∂sum μ, f x ∈ t
α : Type u_2
β : Type u_1
γ : Type ?u.398194
ι : Type u_3
inst✝³ : Countable ι
m✝ : MeasurableSpace α
μ✝ : Measure α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
f g : α → β
inst✝ : PseudoMetrizableSpace β
m : MeasurableSpace α
μ : ι → Measure α
h : ∀ (i : ι), AEStronglyMeasurable f (μ i)
this✝¹ : MeasurableSpace β := borel β
this✝ : BorelSpace β
⊢ AEStronglyMeasurable f (sum μ)
refine'
  aestronglyMeasurable_iff_aemeasurable_separable.2
    ⟨AEMeasurable.sum_measure fun i => (h i).aemeasurable, _⟩
α : Type u_2
β : Type u_1
γ : Type ?u.398194
ι : Type u_3
inst✝³ : Countable ι
m✝ : MeasurableSpace α
μ✝ : Measure α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
f g : α → β
inst✝ : PseudoMetrizableSpace β
m : MeasurableSpace α
μ : ι → Measure α
h : ∀ (i : ι), AEStronglyMeasurable f (μ i)
this✝¹ : MeasurableSpace β := borel β
this✝ : BorelSpace β
A : ∀ (i : ι), ∃ t, IsSeparable t ∧ f ⁻¹' t ∈ ae (μ i)
⊢ ∃ t, IsSeparable t ∧ ∀ᵐ (x : α) ∂sum μ, f x ∈ t
α : Type u_2
β : Type u_1
γ : Type ?u.398194
ι : Type u_3
inst✝³ : Countable ι
m✝ : MeasurableSpace α
μ✝ : Measure α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
f g : α → β
inst✝ : PseudoMetrizableSpace β
m : MeasurableSpace α
μ : ι → Measure α
h : ∀ (i : ι), AEStronglyMeasurable f (μ i)
this✝¹ : MeasurableSpace β := borel β
this✝ : BorelSpace β
⊢ ∃ t, IsSeparable t ∧ ∀ᵐ (x : α) ∂sum μ, f x ∈ t
have A : ∀ i : ι, ∃ t : Set β, IsSeparable t ∧ f ⁻¹' t ∈ (μ i).ae := fun i =>
  (aestronglyMeasurable_iff_aemeasurable_separable.1 (h i)).2
α : Type u_2
β : Type u_1
γ : Type ?u.398194
ι : Type u_3
inst✝³ : Countable ι
m✝ : MeasurableSpace α
μ✝ : Measure α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
f g : α → β
inst✝ : PseudoMetrizableSpace β
m : MeasurableSpace α
μ : ι → Measure α
h : ∀ (i : ι), AEStronglyMeasurable f (μ i)
this✝¹ : MeasurableSpace β := borel β
this✝ : BorelSpace β
t : ι → Set β
t_sep : ∀ (i : ι), IsSeparable (t i)
ht : ∀ (i : ι), f ⁻¹' t i ∈ ae (μ i)
⊢ ∃ t, IsSeparable t ∧ ∀ᵐ (x : α) ∂sum μ, f x ∈ t
α : Type u_2
β : Type u_1
γ : Type ?u.398194
ι : Type u_3
inst✝³ : Countable ι
m✝ : MeasurableSpace α
μ✝ : Measure α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
f g : α → β
inst✝ : PseudoMetrizableSpace β
m : MeasurableSpace α
μ : ι → Measure α
h : ∀ (i : ι), AEStronglyMeasurable f (μ i)
this✝¹ : MeasurableSpace β := borel β
this✝ : BorelSpace β
A : ∀ (i : ι), ∃ t, IsSeparable t ∧ f ⁻¹' t ∈ ae (μ i)
⊢ ∃ t, IsSeparable t ∧ ∀ᵐ (x : α) ∂sum μ, f x ∈ t
choose t t_sep ht using A
α : Type u_2
β : Type u_1
γ : Type ?u.398194
ι : Type u_3
inst✝³ : Countable ι
m✝ : MeasurableSpace α
μ✝ : Measure α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
f g : α → β
inst✝ : PseudoMetrizableSpace β
m : MeasurableSpace α
μ : ι → Measure α
h : ∀ (i : ι), AEStronglyMeasurable f (μ i)
this✝¹ : MeasurableSpace β := borel β
this✝ : BorelSpace β
t : ι → Set β
t_sep : ∀ (i : ι), IsSeparable (t i)
ht : ∀ (i : ι), f ⁻¹' t i ∈ ae (μ i)
⊢ ∀ᵐ (x : α) ∂sum μ, f x ∈ ⋃ (i : ι), t i
α : Type u_2
β : Type u_1
γ : Type ?u.398194
ι : Type u_3
inst✝³ : Countable ι
m✝ : MeasurableSpace α
μ✝ : Measure α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
f g : α → β
inst✝ : PseudoMetrizableSpace β
m : MeasurableSpace α
μ : ι → Measure α
h : ∀ (i : ι), AEStronglyMeasurable f (μ i)
this✝¹ : MeasurableSpace β := borel β
this✝ : BorelSpace β
t : ι → Set β
t_sep : ∀ (i : ι), IsSeparable (t i)
ht : ∀ (i : ι), f ⁻¹' t i ∈ ae (μ i)
⊢ ∃ t, IsSeparable t ∧ ∀ᵐ (x : α) ∂sum μ, f x ∈ t
refine' ⟨⋃ i, t i, isSeparable_iUnion t_sep, _⟩
α : Type u_2
β : Type u_1
γ : Type ?u.398194
ι : Type u_3
inst✝³ : Countable ι
m✝ : MeasurableSpace α
μ✝ : Measure α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
f g : α → β
inst✝ : PseudoMetrizableSpace β
m : MeasurableSpace α
μ : ι → Measure α
h : ∀ (i : ι), AEStronglyMeasurable f (μ i)
this✝¹ : MeasurableSpace β := borel β
this✝ : BorelSpace β
t : ι → Set β
t_sep : ∀ (i : ι), IsSeparable (t i)
ht : ∀ (i : ι), f ⁻¹' t i ∈ ae (μ i)
⊢ ∀ (b : ι), ∀ᵐ (x : α) ∂μ b, ∃ i, f x ∈ t i
α : Type u_2
β : Type u_1
γ : Type ?u.398194
ι : Type u_3
inst✝³ : Countable ι
m✝ : MeasurableSpace α
μ✝ : Measure α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
f g : α → β
inst✝ : PseudoMetrizableSpace β
m : MeasurableSpace α
μ : ι → Measure α
h : ∀ (i : ι), AEStronglyMeasurable f (μ i)
this✝¹ : MeasurableSpace β := borel β
this✝ : BorelSpace β
t : ι → Set β
t_sep : ∀ (i : ι), IsSeparable (t i)
ht : ∀ (i : ι), f ⁻¹' t i ∈ ae (μ i)
⊢ ∀ᵐ (x : α) ∂sum μ, f x ∈ ⋃ (i : ι), t i
simp only [Measure.ae_sum_eq, mem_iUnion, eventually_iSup]
α : Type u_2
β : Type u_1
γ : Type ?u.398194
ι : Type u_3
inst✝³ : Countable ι
m✝ : MeasurableSpace α
μ✝ : Measure α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
f g : α → β
inst✝ : PseudoMetrizableSpace β
m : MeasurableSpace α
μ : ι → Measure α
h : ∀ (i : ι), AEStronglyMeasurable f (μ i)
this✝¹ : MeasurableSpace β := borel β
this✝ : BorelSpace β
t : ι → Set β
t_sep : ∀ (i : ι), IsSeparable (t i)
ht : ∀ (i : ι), f ⁻¹' t i ∈ ae (μ i)
i : ι
⊢ ∀ᵐ (x : α) ∂μ i, ∃ i, f x ∈ t i
α : Type u_2
β : Type u_1
γ : Type ?u.398194
ι : Type u_3
inst✝³ : Countable ι
m✝ : MeasurableSpace α
μ✝ : Measure α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
f g : α → β
inst✝ : PseudoMetrizableSpace β
m : MeasurableSpace α
μ : ι → Measure α
h : ∀ (i : ι), AEStronglyMeasurable f (μ i)
this✝¹ : MeasurableSpace β := borel β
this✝ : BorelSpace β
t : ι → Set β
t_sep : ∀ (i : ι), IsSeparable (t i)
ht : ∀ (i : ι), f ⁻¹' t i ∈ ae (μ i)
⊢ ∀ (b : ι), ∀ᵐ (x : α) ∂μ b, ∃ i, f x ∈ t i
intro i
case h
α : Type u_2
β : Type u_1
γ : Type ?u.398194
ι : Type u_3
inst✝³ : Countable ι
m✝ : MeasurableSpace α
μ✝ : Measure α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
f g : α → β
inst✝ : PseudoMetrizableSpace β
m : MeasurableSpace α
μ : ι → Measure α
h : ∀ (i : ι), AEStronglyMeasurable f (μ i)
this✝¹ : MeasurableSpace β := borel β
this✝ : BorelSpace β
t : ι → Set β
t_sep : ∀ (i : ι), IsSeparable (t i)
ht : ∀ (i : ι), f ⁻¹' t i ∈ ae (μ i)
i : ι
x : α
hx : x ∈ f ⁻¹' t i
⊢ ∃ i, f x ∈ t i
α : Type u_2
β : Type u_1
γ : Type ?u.398194
ι : Type u_3
inst✝³ : Countable ι
m✝ : MeasurableSpace α
μ✝ : Measure α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
f g : α → β
inst✝ : PseudoMetrizableSpace β
m : MeasurableSpace α
μ : ι → Measure α
h : ∀ (i : ι), AEStronglyMeasurable f (μ i)
this✝¹ : MeasurableSpace β := borel β
this✝ : BorelSpace β
t : ι → Set β
t_sep : ∀ (i : ι), IsSeparable (t i)
ht : ∀ (i : ι), f ⁻¹' t i ∈ ae (μ i)
i : ι
⊢ ∀ᵐ (x : α) ∂μ i, ∃ i, f x ∈ t i
filter_upwards [ht i]with x hx
no goals
case h
α : Type u_2
β : Type u_1
γ : Type ?u.398194
ι : Type u_3
inst✝³ : Countable ι
m✝ : MeasurableSpace α
μ✝ : Measure α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
f g : α → β
inst✝ : PseudoMetrizableSpace β
m : MeasurableSpace α
μ : ι → Measure α
h : ∀ (i : ι), AEStronglyMeasurable f (μ i)
this✝¹ : MeasurableSpace β := borel β
this✝ : BorelSpace β
t : ι → Set β
t_sep : ∀ (i : ι), IsSeparable (t i)
ht : ∀ (i : ι), f ⁻¹' t i ∈ ae (μ i)
i : ι
x : α
hx : x ∈ f ⁻¹' t i
⊢ ∃ i, f x ∈ t i
exact ⟨i, hx⟩