-- Mathlib/MeasureTheory/Measure/AEMeasurable.lean
/-- If the `σ`-algebra of the codomain of a null measurable function is countably generated,
then the function is a.e.-measurable. -/
lemma MeasureTheory.NullMeasurable.aemeasurable {f : α → β}
    [hc : MeasurableSpace.CountablyGenerated β] (h : NullMeasurable f μ) : AEMeasurable f μ

Proof state:
case intro.intro.refine_2
α : Type u_2
β : Type u_3
m0 : MeasurableSpace α
μ : Measure α
f : α → β
a✝ : Nontrivial β
inhabited_h : Inhabited β
S : Set (Set β)
hSc : S.Countable
hc : MeasurableSpace.CountablyGenerated β
h : NullMeasurable f μ
T : Set β → Set α
hTf : ∀ s ∈ S, T s ⊆ f ⁻¹' s
hTm : ∀ s ∈ S, MeasurableSet (T s)
hTeq : ∀ s ∈ S, T s =ᶠ[ae μ] f ⁻¹' s
U : Set β → Set α
hUf : ∀ s ∈ S, U s ⊇ f ⁻¹' s
hUm : ∀ s ∈ S, MeasurableSet (U s)
hUeq : ∀ s ∈ S, U s =ᶠ[ae μ] f ⁻¹' s
v : Set α := ⋃ s ∈ S, U s \ T s
hvm : MeasurableSet v
hvμ : μ v = 0
s : Set β
hs : s ∈ S
⊢ MeasurableSet (vᶜ ∩ f ⁻¹' s)

Tactic:
convert (hTm s hs).diff hvm using 1

Result:
case h.e'_3
α : Type u_2
β : Type u_3
m0 : MeasurableSpace α
μ : Measure α
f : α → β
a✝ : Nontrivial β
inhabited_h : Inhabited β
S : Set (Set β)
hSc : S.Countable
hc : MeasurableSpace.CountablyGenerated β
h : NullMeasurable f μ
T : Set β → Set α
hTf : ∀ s ∈ S, T s ⊆ f ⁻¹' s
hTm : ∀ s ∈ S, MeasurableSet (T s)
hTeq : ∀ s ∈ S, T s =ᶠ[ae μ] f ⁻¹' s
U : Set β → Set α
hUf : ∀ s ∈ S, U s ⊇ f ⁻¹' s
hUm : ∀ s ∈ S, MeasurableSet (U s)
hUeq : ∀ s ∈ S, U s =ᶠ[ae μ] f ⁻¹' s
v : Set α := ⋃ s ∈ S, U s \ T s
hvm : MeasurableSet v
hvμ : μ v = 0
s : Set β
hs : s ∈ S
⊢ vᶜ ∩ f ⁻¹' s = T s \ v

Proof state:
case h.e'_3
α : Type u_2
β : Type u_3
m0 : MeasurableSpace α
μ : Measure α
f : α → β
a✝ : Nontrivial β
inhabited_h : Inhabited β
S : Set (Set β)
hSc : S.Countable
hc : MeasurableSpace.CountablyGenerated β
h : NullMeasurable f μ
T : Set β → Set α
hTf : ∀ s ∈ S, T s ⊆ f ⁻¹' s
hTm : ∀ s ∈ S, MeasurableSet (T s)
hTeq : ∀ s ∈ S, T s =ᶠ[ae μ] f ⁻¹' s
U : Set β → Set α
hUf : ∀ s ∈ S, U s ⊇ f ⁻¹' s
hUm : ∀ s ∈ S, MeasurableSet (U s)
hUeq : ∀ s ∈ S, U s =ᶠ[ae μ] f ⁻¹' s
v : Set α := ⋃ s ∈ S, U s \ T s
hvm : MeasurableSet v
hvμ : μ v = 0
s : Set β
hs : s ∈ S
x : α
hxv : x ∈ vᶜ
hx : x ∈ f ⁻¹' s
⊢ x ∈ T s

Tactic:
exact by_contra fun hx' ↦ hxv <| mem_biUnion hs ⟨hUf s hs hx, hx'⟩

Result:
no goals

Proof state:
α : Type u_2
β : Type u_3
m0 : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
f : α → β
hc : MeasurableSpace.CountablyGenerated β
h : NullMeasurable f μ
⊢ AEMeasurable f μ

Tactic:
nontriviality β

Result:
α : Type u_2
β : Type u_3
m0 : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
f : α → β
hc : MeasurableSpace.CountablyGenerated β
h : NullMeasurable f μ
a✝ : Nontrivial β
⊢ AEMeasurable f μ

Proof state:
α : Type u_2
β : Type u_3
m0 : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
f : α → β
hc : MeasurableSpace.CountablyGenerated β
h : NullMeasurable f μ
a✝ : Nontrivial β
⊢ AEMeasurable f μ

Tactic:
inhabit β

Result:
α : Type u_2
β : Type u_3
m0 : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
f : α → β
hc : MeasurableSpace.CountablyGenerated β
h : NullMeasurable f μ
a✝ : Nontrivial β
inhabited_h : Inhabited β
⊢ AEMeasurable f μ

Proof state:
case intro.intro
α : Type u_2
β : Type u_3
m0 : MeasurableSpace α
μ : Measure α
f : α → β
a✝ : Nontrivial β
inhabited_h : Inhabited β
S : Set (Set β)
hSc : S.Countable
hc : MeasurableSpace.CountablyGenerated β
h : NullMeasurable f μ
T : Set β → Set α
hTf : ∀ s ∈ S, T s ⊆ f ⁻¹' s
hTm : ∀ s ∈ S, MeasurableSet (T s)
hTeq : ∀ s ∈ S, T s =ᶠ[ae μ] f ⁻¹' s
U : Set β → Set α
hUf : ∀ s ∈ S, U s ⊇ f ⁻¹' s
hUm : ∀ s ∈ S, MeasurableSet (U s)
hUeq : ∀ s ∈ S, U s =ᶠ[ae μ] f ⁻¹' s
⊢ AEMeasurable f μ

Tactic:
set v := ⋃ s ∈ S, U s \ T s

Result:
case intro.intro
α : Type u_2
β : Type u_3
m0 : MeasurableSpace α
μ : Measure α
f : α → β
a✝ : Nontrivial β
inhabited_h : Inhabited β
S : Set (Set β)
hSc : S.Countable
hc : MeasurableSpace.CountablyGenerated β
h : NullMeasurable f μ
T : Set β → Set α
hTf : ∀ s ∈ S, T s ⊆ f ⁻¹' s
hTm : ∀ s ∈ S, MeasurableSet (T s)
hTeq : ∀ s ∈ S, T s =ᶠ[ae μ] f ⁻¹' s
U : Set β → Set α
hUf : ∀ s ∈ S, U s ⊇ f ⁻¹' s
hUm : ∀ s ∈ S, MeasurableSet (U s)
hUeq : ∀ s ∈ S, U s =ᶠ[ae μ] f ⁻¹' s
v : Set α := ⋃ s ∈ S, U s \ T s
⊢ AEMeasurable f μ

Proof state:
case intro.intro.refine_1
α : Type u_2
β : Type u_3
m0 : MeasurableSpace α
μ : Measure α
f : α → β
a✝ : Nontrivial β
inhabited_h : Inhabited β
S : Set (Set β)
hSc : S.Countable
hc : MeasurableSpace.CountablyGenerated β
h : NullMeasurable f μ
T : Set β → Set α
hTf : ∀ s ∈ S, T s ⊆ f ⁻¹' s
hTm : ∀ s ∈ S, MeasurableSet (T s)
hTeq : ∀ s ∈ S, T s =ᶠ[ae μ] f ⁻¹' s
U : Set β → Set α
hUf : ∀ s ∈ S, U s ⊇ f ⁻¹' s
hUm : ∀ s ∈ S, MeasurableSet (U s)
hUeq : ∀ s ∈ S, U s =ᶠ[ae μ] f ⁻¹' s
v : Set α := ⋃ s ∈ S, U s \ T s
hvm : MeasurableSet v
hvμ : μ v = 0
⊢ Measurable (v.restrict (v.piecewise (fun x => default) f))

Tactic:
rw [restrict_piecewise]

Result:
case intro.intro.refine_1
α : Type u_2
β : Type u_3
m0 : MeasurableSpace α
μ : Measure α
f : α → β
a✝ : Nontrivial β
inhabited_h : Inhabited β
S : Set (Set β)
hSc : S.Countable
hc : MeasurableSpace.CountablyGenerated β
h : NullMeasurable f μ
T : Set β → Set α
hTf : ∀ s ∈ S, T s ⊆ f ⁻¹' s
hTm : ∀ s ∈ S, MeasurableSet (T s)
hTeq : ∀ s ∈ S, T s =ᶠ[ae μ] f ⁻¹' s
U : Set β → Set α
hUf : ∀ s ∈ S, U s ⊇ f ⁻¹' s
hUm : ∀ s ∈ S, MeasurableSet (U s)
hUeq : ∀ s ∈ S, U s =ᶠ[ae μ] f ⁻¹' s
v : Set α := ⋃ s ∈ S, U s \ T s
hvm : MeasurableSet v
hvμ : μ v = 0
⊢ Measurable (v.restrict fun x => default)

Proof state:
case intro.intro
α : Type u_2
β : Type u_3
m0 : MeasurableSpace α
μ : Measure α
f : α → β
a✝ : Nontrivial β
inhabited_h : Inhabited β
S : Set (Set β)
hSc : S.Countable
hc : MeasurableSpace.CountablyGenerated β
h : NullMeasurable f μ
T : Set β → Set α
hTf : ∀ s ∈ S, T s ⊆ f ⁻¹' s
hTm : ∀ s ∈ S, MeasurableSet (T s)
hTeq : ∀ s ∈ S, T s =ᶠ[ae μ] f ⁻¹' s
U : Set β → Set α
hUf : ∀ s ∈ S, U s ⊇ f ⁻¹' s
hUm : ∀ s ∈ S, MeasurableSet (U s)
hUeq : ∀ s ∈ S, U s =ᶠ[ae μ] f ⁻¹' s
v : Set α := ⋃ s ∈ S, U s \ T s
⊢ AEMeasurable f μ

Tactic:
have hvm : MeasurableSet v := .biUnion hSc fun s hs ↦ (hUm s hs).diff (hTm s hs)

Result:
case intro.intro
α : Type u_2
β : Type u_3
m0 : MeasurableSpace α
μ : Measure α
f : α → β
a✝ : Nontrivial β
inhabited_h : Inhabited β
S : Set (Set β)
hSc : S.Countable
hc : MeasurableSpace.CountablyGenerated β
h : NullMeasurable f μ
T : Set β → Set α
hTf : ∀ s ∈ S, T s ⊆ f ⁻¹' s
hTm : ∀ s ∈ S, MeasurableSet (T s)
hTeq : ∀ s ∈ S, T s =ᶠ[ae μ] f ⁻¹' s
U : Set β → Set α
hUf : ∀ s ∈ S, U s ⊇ f ⁻¹' s
hUm : ∀ s ∈ S, MeasurableSet (U s)
hUeq : ∀ s ∈ S, U s =ᶠ[ae μ] f ⁻¹' s
v : Set α := ⋃ s ∈ S, U s \ T s
hvm : MeasurableSet v
⊢ AEMeasurable f μ

Proof state:
case h.e'_3
α : Type u_2
β : Type u_3
m0 : MeasurableSpace α
μ : Measure α
f : α → β
a✝ : Nontrivial β
inhabited_h : Inhabited β
S : Set (Set β)
hSc : S.Countable
hc : MeasurableSpace.CountablyGenerated β
h : NullMeasurable f μ
T : Set β → Set α
hTf : ∀ s ∈ S, T s ⊆ f ⁻¹' s
hTm : ∀ s ∈ S, MeasurableSet (T s)
hTeq : ∀ s ∈ S, T s =ᶠ[ae μ] f ⁻¹' s
U : Set β → Set α
hUf : ∀ s ∈ S, U s ⊇ f ⁻¹' s
hUm : ∀ s ∈ S, MeasurableSet (U s)
hUeq : ∀ s ∈ S, U s =ᶠ[ae μ] f ⁻¹' s
v : Set α := ⋃ s ∈ S, U s \ T s
hvm : MeasurableSet v
hvμ : μ v = 0
s : Set β
hs : s ∈ S
⊢ vᶜ ∩ f ⁻¹' s = T s \ v

Tactic:
rw [inter_comm]

Result:
case h.e'_3
α : Type u_2
β : Type u_3
m0 : MeasurableSpace α
μ : Measure α
f : α → β
a✝ : Nontrivial β
inhabited_h : Inhabited β
S : Set (Set β)
hSc : S.Countable
hc : MeasurableSpace.CountablyGenerated β
h : NullMeasurable f μ
T : Set β → Set α
hTf : ∀ s ∈ S, T s ⊆ f ⁻¹' s
hTm : ∀ s ∈ S, MeasurableSet (T s)
hTeq : ∀ s ∈ S, T s =ᶠ[ae μ] f ⁻¹' s
U : Set β → Set α
hUf : ∀ s ∈ S, U s ⊇ f ⁻¹' s
hUm : ∀ s ∈ S, MeasurableSet (U s)
hUeq : ∀ s ∈ S, U s =ᶠ[ae μ] f ⁻¹' s
v : Set α := ⋃ s ∈ S, U s \ T s
hvm : MeasurableSet v
hvμ : μ v = 0
s : Set β
hs : s ∈ S
⊢ f ⁻¹' s ∩ vᶜ = T s \ v