Theorem: MeasureTheory.NullMeasurable.aemeasurable
File: Mathlib/MeasureTheory/Measure/AEMeasurable.lean

tactic: nontriviality β
state_before: ι : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
δ : Type u_5
R : Type u_6
m0 : MeasurableSpace α
inst✝² : MeasurableSpace β
inst✝¹ : MeasurableSpace γ
inst✝ : MeasurableSpace δ
f✝ g : α → β
μ ν : Measure α
f : α → β
hc : MeasurableSpace.CountablyGenerated β
h : NullMeasurable f μ
⊢ AEMeasurable f μ
state_after: ι : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
δ : Type u_5
R : Type u_6
m0 : MeasurableSpace α
inst✝² : MeasurableSpace β
inst✝¹ : MeasurableSpace γ
inst✝ : MeasurableSpace δ
f✝ g : α → β
μ ν : Measure α
f : α → β
hc : MeasurableSpace.CountablyGenerated β
h : NullMeasurable f μ
a✝ : Nontrivial β
⊢ AEMeasurable f μ

tactic: inhabit β
state_before: ι : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
δ : Type u_5
R : Type u_6
m0 : MeasurableSpace α
inst✝² : MeasurableSpace β
inst✝¹ : MeasurableSpace γ
inst✝ : MeasurableSpace δ
f✝ g : α → β
μ ν : Measure α
f : α → β
hc : MeasurableSpace.CountablyGenerated β
h : NullMeasurable f μ
a✝ : Nontrivial β
⊢ AEMeasurable f μ
state_after: ι : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
δ : Type u_5
R : Type u_6
m0 : MeasurableSpace α
inst✝² : MeasurableSpace β
inst✝¹ : MeasurableSpace γ
inst✝ : MeasurableSpace δ
f✝ g : α → β
μ ν : Measure α
f : α → β
hc : MeasurableSpace.CountablyGenerated β
h : NullMeasurable f μ
a✝ : Nontrivial β
inhabited_h : Inhabited β
⊢ AEMeasurable f μ

tactic: rcases hc.1 with ⟨S, hSc, rfl⟩
state_before: ι : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
δ : Type u_5
R : Type u_6
m0 : MeasurableSpace α
inst✝² : MeasurableSpace β
inst✝¹ : MeasurableSpace γ
inst✝ : MeasurableSpace δ
f✝ g : α → β
μ ν : Measure α
f : α → β
hc : MeasurableSpace.CountablyGenerated β
h : NullMeasurable f μ
a✝ : Nontrivial β
inhabited_h : Inhabited β
⊢ AEMeasurable f μ
state_after: case intro.intro
ι : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
δ : Type u_5
R : Type u_6
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace γ
inst✝ : MeasurableSpace δ
f✝ g : α → β
μ ν : 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
⊢ AEMeasurable f μ

tactic: choose! T hTf hTm hTeq using fun s hs ↦ (h <| .basic s hs).exists_measurable_subset_ae_eq
state_before: case intro.intro
ι : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
δ : Type u_5
R : Type u_6
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace γ
inst✝ : MeasurableSpace δ
f✝ g : α → β
μ ν : 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
⊢ AEMeasurable f μ
state_after: case intro.intro
ι : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
δ : Type u_5
R : Type u_6
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace γ
inst✝ : MeasurableSpace δ
f✝ g : α → β
μ ν : 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
⊢ AEMeasurable f μ

tactic: choose! U hUf hUm hUeq using fun s hs ↦ (h <| .basic s hs).exists_measurable_superset_ae_eq
state_before: case intro.intro
ι : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
δ : Type u_5
R : Type u_6
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace γ
inst✝ : MeasurableSpace δ
f✝ g : α → β
μ ν : 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
⊢ AEMeasurable f μ
state_after: case intro.intro
ι : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
δ : Type u_5
R : Type u_6
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace γ
inst✝ : MeasurableSpace δ
f✝ g : α → β
μ ν : 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
state_before: case intro.intro
ι : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
δ : Type u_5
R : Type u_6
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace γ
inst✝ : MeasurableSpace δ
f✝ g : α → β
μ ν : 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
state_after: case intro.intro
ι : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
δ : Type u_5
R : Type u_6
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace γ
inst✝ : MeasurableSpace δ
f✝ g : α → β
μ ν : 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)
state_before: case intro.intro
ι : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
δ : Type u_5
R : Type u_6
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace γ
inst✝ : MeasurableSpace δ
f✝ g : α → β
μ ν : 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
state_after: case intro.intro
ι : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
δ : Type u_5
R : Type u_6
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace γ
inst✝ : MeasurableSpace δ
f✝ g : α → β
μ ν : 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

tactic: have hvμ : μ v = 0 := (measure_biUnion_null_iff hSc).2 fun s hs ↦ ae_le_set.1 <|
  ((hUeq s hs).trans (hTeq s hs).symm).le
state_before: case intro.intro
ι : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
δ : Type u_5
R : Type u_6
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace γ
inst✝ : MeasurableSpace δ
f✝ g : α → β
μ ν : 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
state_after: case intro.intro
ι : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
δ : Type u_5
R : Type u_6
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace γ
inst✝ : MeasurableSpace δ
f✝ g : α → β
μ ν : 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
⊢ AEMeasurable f μ

tactic: refine ⟨v.piecewise (fun _ ↦ default) f, ?_, measure_mono_null (fun x ↦
  not_imp_comm.2 fun hxv ↦ (piecewise_eq_of_not_mem _ _ _ hxv).symm) hvμ⟩
state_before: case intro.intro
ι : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
δ : Type u_5
R : Type u_6
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace γ
inst✝ : MeasurableSpace δ
f✝ g : α → β
μ ν : 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
⊢ AEMeasurable f μ
state_after: case intro.intro
ι : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
δ : Type u_5
R : Type u_6
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace γ
inst✝ : MeasurableSpace δ
f✝ g : α → β
μ ν : 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.piecewise (fun x => default) f)

tactic: refine measurable_of_restrict_of_restrict_compl hvm ?_ ?_
state_before: case intro.intro
ι : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
δ : Type u_5
R : Type u_6
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace γ
inst✝ : MeasurableSpace δ
f✝ g : α → β
μ ν : 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.piecewise (fun x => default) f)
state_after: case intro.intro.refine_1
ι : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
δ : Type u_5
R : Type u_6
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace γ
inst✝ : MeasurableSpace δ
f✝ g : α → β
μ ν : 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))

case intro.intro.refine_2
ι : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
δ : Type u_5
R : Type u_6
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace γ
inst✝ : MeasurableSpace δ
f✝ g : α → β
μ ν : 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 (vᶜ.restrict (v.piecewise (fun x => default) f))

tactic: rw [restrict_piecewise]
state_before: case intro.intro.refine_1
ι : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
δ : Type u_5
R : Type u_6
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace γ
inst✝ : MeasurableSpace δ
f✝ g : α → β
μ ν : 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))
state_after: case intro.intro.refine_1
ι : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
δ : Type u_5
R : Type u_6
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace γ
inst✝ : MeasurableSpace δ
f✝ g : α → β
μ ν : 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)

tactic: apply measurable_const
state_before: case intro.intro.refine_1
ι : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
δ : Type u_5
R : Type u_6
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace γ
inst✝ : MeasurableSpace δ
f✝ g : α → β
μ ν : 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)
state_after: no goals

tactic: rw [restrict_piecewise_compl, restrict_eq]
state_before: case intro.intro.refine_2
ι : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
δ : Type u_5
R : Type u_6
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace γ
inst✝ : MeasurableSpace δ
f✝ g : α → β
μ ν : 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 (vᶜ.restrict (v.piecewise (fun x => default) f))
state_after: case intro.intro.refine_2
ι : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
δ : Type u_5
R : Type u_6
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace γ
inst✝ : MeasurableSpace δ
f✝ g : α → β
μ ν : 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 (f ∘ Subtype.val)

tactic: refine measurable_generateFrom fun s hs ↦ .of_subtype_image ?_
state_before: case intro.intro.refine_2
ι : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
δ : Type u_5
R : Type u_6
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace γ
inst✝ : MeasurableSpace δ
f✝ g : α → β
μ ν : 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 (f ∘ Subtype.val)
state_after: case intro.intro.refine_2
ι : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
δ : Type u_5
R : Type u_6
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace γ
inst✝ : MeasurableSpace δ
f✝ g : α → β
μ ν : 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 (Subtype.val '' (f ∘ Subtype.val ⁻¹' s))

tactic: rw [preimage_comp, Subtype.image_preimage_coe]
state_before: case intro.intro.refine_2
ι : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
δ : Type u_5
R : Type u_6
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace γ
inst✝ : MeasurableSpace δ
f✝ g : α → β
μ ν : 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 (Subtype.val '' (f ∘ Subtype.val ⁻¹' s))
state_after: case intro.intro.refine_2
ι : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
δ : Type u_5
R : Type u_6
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace γ
inst✝ : MeasurableSpace δ
f✝ g : α → β
μ ν : 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
state_before: case intro.intro.refine_2
ι : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
δ : Type u_5
R : Type u_6
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace γ
inst✝ : MeasurableSpace δ
f✝ g : α → β
μ ν : 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)
state_after: case h.e'_3
ι : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
δ : Type u_5
R : Type u_6
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace γ
inst✝ : MeasurableSpace δ
f✝ g : α → β
μ ν : 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]
state_before: case h.e'_3
ι : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
δ : Type u_5
R : Type u_6
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace γ
inst✝ : MeasurableSpace δ
f✝ g : α → β
μ ν : 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
state_after: case h.e'_3
ι : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
δ : Type u_5
R : Type u_6
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace γ
inst✝ : MeasurableSpace δ
f✝ g : α → β
μ ν : 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

tactic: refine Set.ext fun x ↦ and_congr_left fun hxv ↦ ⟨fun hx ↦ ?_, fun hx ↦ hTf s hs hx⟩
state_before: case h.e'_3
ι : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
δ : Type u_5
R : Type u_6
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace γ
inst✝ : MeasurableSpace δ
f✝ g : α → β
μ ν : 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
state_after: case h.e'_3
ι : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
δ : Type u_5
R : Type u_6
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace γ
inst✝ : MeasurableSpace δ
f✝ g : α → β
μ ν : 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'⟩
state_before: case h.e'_3
ι : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
δ : Type u_5
R : Type u_6
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace γ
inst✝ : MeasurableSpace δ
f✝ g : α → β
μ ν : 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
state_after: no goals