Theorem: aemeasurable_indicator_const_iff
File: Mathlib/MeasureTheory/Measure/AEMeasurable.lean

tactic: constructor <;> intro h
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✝⁵ : MeasurableSpace β
inst✝⁴ : MeasurableSpace γ
inst✝³ : MeasurableSpace δ
f g : α → β
μ ν : Measure α
inst✝² : Zero β
s : Set α
inst✝¹ : MeasurableSingletonClass β
b : β
inst✝ : NeZero b
⊢ AEMeasurable (s.indicator fun x => b) μ ↔ NullMeasurableSet s μ
state_after: case mp
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✝⁵ : MeasurableSpace β
inst✝⁴ : MeasurableSpace γ
inst✝³ : MeasurableSpace δ
f g : α → β
μ ν : Measure α
inst✝² : Zero β
s : Set α
inst✝¹ : MeasurableSingletonClass β
b : β
inst✝ : NeZero b
h : AEMeasurable (s.indicator fun x => b) μ
⊢ NullMeasurableSet s μ

case mpr
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✝⁵ : MeasurableSpace β
inst✝⁴ : MeasurableSpace γ
inst✝³ : MeasurableSpace δ
f g : α → β
μ ν : Measure α
inst✝² : Zero β
s : Set α
inst✝¹ : MeasurableSingletonClass β
b : β
inst✝ : NeZero b
h : NullMeasurableSet s μ
⊢ AEMeasurable (s.indicator fun x => b) μ

tactic: convert h.nullMeasurable (MeasurableSet.singleton (0 : β)).compl
state_before: case mp
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✝⁵ : MeasurableSpace β
inst✝⁴ : MeasurableSpace γ
inst✝³ : MeasurableSpace δ
f g : α → β
μ ν : Measure α
inst✝² : Zero β
s : Set α
inst✝¹ : MeasurableSingletonClass β
b : β
inst✝ : NeZero b
h : AEMeasurable (s.indicator fun x => b) μ
⊢ NullMeasurableSet s μ
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✝⁵ : MeasurableSpace β
inst✝⁴ : MeasurableSpace γ
inst✝³ : MeasurableSpace δ
f g : α → β
μ ν : Measure α
inst✝² : Zero β
s : Set α
inst✝¹ : MeasurableSingletonClass β
b : β
inst✝ : NeZero b
h : AEMeasurable (s.indicator fun x => b) μ
⊢ s = (s.indicator fun x => b) ⁻¹' {0}ᶜ

tactic: rw [indicator_const_preimage_eq_union s {0}ᶜ b]
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✝⁵ : MeasurableSpace β
inst✝⁴ : MeasurableSpace γ
inst✝³ : MeasurableSpace δ
f g : α → β
μ ν : Measure α
inst✝² : Zero β
s : Set α
inst✝¹ : MeasurableSingletonClass β
b : β
inst✝ : NeZero b
h : AEMeasurable (s.indicator fun x => b) μ
⊢ s = (s.indicator fun x => b) ⁻¹' {0}ᶜ
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✝⁵ : MeasurableSpace β
inst✝⁴ : MeasurableSpace γ
inst✝³ : MeasurableSpace δ
f g : α → β
μ ν : Measure α
inst✝² : Zero β
s : Set α
inst✝¹ : MeasurableSingletonClass β
b : β
inst✝ : NeZero b
h : AEMeasurable (s.indicator fun x => b) μ
⊢ s = (if b ∈ {0}ᶜ then s else ∅) ∪ if 0 ∈ {0}ᶜ then sᶜ else ∅

tactic: simp [NeZero.ne b]
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✝⁵ : MeasurableSpace β
inst✝⁴ : MeasurableSpace γ
inst✝³ : MeasurableSpace δ
f g : α → β
μ ν : Measure α
inst✝² : Zero β
s : Set α
inst✝¹ : MeasurableSingletonClass β
b : β
inst✝ : NeZero b
h : AEMeasurable (s.indicator fun x => b) μ
⊢ s = (if b ∈ {0}ᶜ then s else ∅) ∪ if 0 ∈ {0}ᶜ then sᶜ else ∅
state_after: no goals

tactic: exact (aemeasurable_indicator_iff₀ h).mpr aemeasurable_const
state_before: case mpr
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✝⁵ : MeasurableSpace β
inst✝⁴ : MeasurableSpace γ
inst✝³ : MeasurableSpace δ
f g : α → β
μ ν : Measure α
inst✝² : Zero β
s : Set α
inst✝¹ : MeasurableSingletonClass β
b : β
inst✝ : NeZero b
h : NullMeasurableSet s μ
⊢ AEMeasurable (s.indicator fun x => b) μ
state_after: no goals