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

tactic: haveI : Nonempty α := ⟨x⟩
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✝⁴ : MeasurableSpace β✝
inst✝³ : MeasurableSpace γ
inst✝² : MeasurableSpace δ
f g✝ : α → β✝
μ ν : Measure α
β : Type u_7
mβ : MeasurableSpace β
inst✝¹ : LinearOrder α
inst✝ : atTop.IsCountablyGenerated
x : α
g : α → β
g_meas : ∀ t > x, AEMeasurable g (μ.restrict (Ioc x t))
⊢ AEMeasurable g (μ.restrict (Ioi x))
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✝⁴ : MeasurableSpace β✝
inst✝³ : MeasurableSpace γ
inst✝² : MeasurableSpace δ
f g✝ : α → β✝
μ ν : Measure α
β : Type u_7
mβ : MeasurableSpace β
inst✝¹ : LinearOrder α
inst✝ : atTop.IsCountablyGenerated
x : α
g : α → β
g_meas : ∀ t > x, AEMeasurable g (μ.restrict (Ioc x t))
this : Nonempty α
⊢ AEMeasurable g (μ.restrict (Ioi x))

tactic: obtain ⟨u, hu_tendsto⟩ := exists_seq_tendsto (atTop : Filter α)
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✝⁴ : MeasurableSpace β✝
inst✝³ : MeasurableSpace γ
inst✝² : MeasurableSpace δ
f g✝ : α → β✝
μ ν : Measure α
β : Type u_7
mβ : MeasurableSpace β
inst✝¹ : LinearOrder α
inst✝ : atTop.IsCountablyGenerated
x : α
g : α → β
g_meas : ∀ t > x, AEMeasurable g (μ.restrict (Ioc x t))
this : Nonempty α
⊢ AEMeasurable g (μ.restrict (Ioi x))
state_after: case 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✝⁴ : MeasurableSpace β✝
inst✝³ : MeasurableSpace γ
inst✝² : MeasurableSpace δ
f g✝ : α → β✝
μ ν : Measure α
β : Type u_7
mβ : MeasurableSpace β
inst✝¹ : LinearOrder α
inst✝ : atTop.IsCountablyGenerated
x : α
g : α → β
g_meas : ∀ t > x, AEMeasurable g (μ.restrict (Ioc x t))
this : Nonempty α
u : ℕ → α
hu_tendsto : Tendsto u atTop atTop
⊢ AEMeasurable g (μ.restrict (Ioi x))

tactic: have Ioi_eq_iUnion : Ioi x = ⋃ n : ℕ, Ioc x (u n) := by
  rw [iUnion_Ioc_eq_Ioi_self_iff.mpr _]
  exact fun y _ => (hu_tendsto.eventually (eventually_ge_atTop y)).exists
state_before: case 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✝⁴ : MeasurableSpace β✝
inst✝³ : MeasurableSpace γ
inst✝² : MeasurableSpace δ
f g✝ : α → β✝
μ ν : Measure α
β : Type u_7
mβ : MeasurableSpace β
inst✝¹ : LinearOrder α
inst✝ : atTop.IsCountablyGenerated
x : α
g : α → β
g_meas : ∀ t > x, AEMeasurable g (μ.restrict (Ioc x t))
this : Nonempty α
u : ℕ → α
hu_tendsto : Tendsto u atTop atTop
⊢ AEMeasurable g (μ.restrict (Ioi x))
state_after: case 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✝⁴ : MeasurableSpace β✝
inst✝³ : MeasurableSpace γ
inst✝² : MeasurableSpace δ
f g✝ : α → β✝
μ ν : Measure α
β : Type u_7
mβ : MeasurableSpace β
inst✝¹ : LinearOrder α
inst✝ : atTop.IsCountablyGenerated
x : α
g : α → β
g_meas : ∀ t > x, AEMeasurable g (μ.restrict (Ioc x t))
this : Nonempty α
u : ℕ → α
hu_tendsto : Tendsto u atTop atTop
Ioi_eq_iUnion : Ioi x = ⋃ n, Ioc x (u n)
⊢ AEMeasurable g (μ.restrict (Ioi x))

tactic: rw [Ioi_eq_iUnion, aemeasurable_iUnion_iff]
state_before: case 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✝⁴ : MeasurableSpace β✝
inst✝³ : MeasurableSpace γ
inst✝² : MeasurableSpace δ
f g✝ : α → β✝
μ ν : Measure α
β : Type u_7
mβ : MeasurableSpace β
inst✝¹ : LinearOrder α
inst✝ : atTop.IsCountablyGenerated
x : α
g : α → β
g_meas : ∀ t > x, AEMeasurable g (μ.restrict (Ioc x t))
this : Nonempty α
u : ℕ → α
hu_tendsto : Tendsto u atTop atTop
Ioi_eq_iUnion : Ioi x = ⋃ n, Ioc x (u n)
⊢ AEMeasurable g (μ.restrict (Ioi x))
state_after: case 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✝⁴ : MeasurableSpace β✝
inst✝³ : MeasurableSpace γ
inst✝² : MeasurableSpace δ
f g✝ : α → β✝
μ ν : Measure α
β : Type u_7
mβ : MeasurableSpace β
inst✝¹ : LinearOrder α
inst✝ : atTop.IsCountablyGenerated
x : α
g : α → β
g_meas : ∀ t > x, AEMeasurable g (μ.restrict (Ioc x t))
this : Nonempty α
u : ℕ → α
hu_tendsto : Tendsto u atTop atTop
Ioi_eq_iUnion : Ioi x = ⋃ n, Ioc x (u n)
⊢ ∀ (i : ℕ), AEMeasurable g (μ.restrict (Ioc x (u i)))

tactic: intro n
state_before: case 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✝⁴ : MeasurableSpace β✝
inst✝³ : MeasurableSpace γ
inst✝² : MeasurableSpace δ
f g✝ : α → β✝
μ ν : Measure α
β : Type u_7
mβ : MeasurableSpace β
inst✝¹ : LinearOrder α
inst✝ : atTop.IsCountablyGenerated
x : α
g : α → β
g_meas : ∀ t > x, AEMeasurable g (μ.restrict (Ioc x t))
this : Nonempty α
u : ℕ → α
hu_tendsto : Tendsto u atTop atTop
Ioi_eq_iUnion : Ioi x = ⋃ n, Ioc x (u n)
⊢ ∀ (i : ℕ), AEMeasurable g (μ.restrict (Ioc x (u i)))
state_after: case 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✝⁴ : MeasurableSpace β✝
inst✝³ : MeasurableSpace γ
inst✝² : MeasurableSpace δ
f g✝ : α → β✝
μ ν : Measure α
β : Type u_7
mβ : MeasurableSpace β
inst✝¹ : LinearOrder α
inst✝ : atTop.IsCountablyGenerated
x : α
g : α → β
g_meas : ∀ t > x, AEMeasurable g (μ.restrict (Ioc x t))
this : Nonempty α
u : ℕ → α
hu_tendsto : Tendsto u atTop atTop
Ioi_eq_iUnion : Ioi x = ⋃ n, Ioc x (u n)
n : ℕ
⊢ AEMeasurable g (μ.restrict (Ioc x (u n)))

tactic: cases' lt_or_le x (u n) with h h
state_before: case 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✝⁴ : MeasurableSpace β✝
inst✝³ : MeasurableSpace γ
inst✝² : MeasurableSpace δ
f g✝ : α → β✝
μ ν : Measure α
β : Type u_7
mβ : MeasurableSpace β
inst✝¹ : LinearOrder α
inst✝ : atTop.IsCountablyGenerated
x : α
g : α → β
g_meas : ∀ t > x, AEMeasurable g (μ.restrict (Ioc x t))
this : Nonempty α
u : ℕ → α
hu_tendsto : Tendsto u atTop atTop
Ioi_eq_iUnion : Ioi x = ⋃ n, Ioc x (u n)
n : ℕ
⊢ AEMeasurable g (μ.restrict (Ioc x (u n)))
state_after: case intro.inl
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✝⁴ : MeasurableSpace β✝
inst✝³ : MeasurableSpace γ
inst✝² : MeasurableSpace δ
f g✝ : α → β✝
μ ν : Measure α
β : Type u_7
mβ : MeasurableSpace β
inst✝¹ : LinearOrder α
inst✝ : atTop.IsCountablyGenerated
x : α
g : α → β
g_meas : ∀ t > x, AEMeasurable g (μ.restrict (Ioc x t))
this : Nonempty α
u : ℕ → α
hu_tendsto : Tendsto u atTop atTop
Ioi_eq_iUnion : Ioi x = ⋃ n, Ioc x (u n)
n : ℕ
h : x < u n
⊢ AEMeasurable g (μ.restrict (Ioc x (u n)))

case intro.inr
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✝⁴ : MeasurableSpace β✝
inst✝³ : MeasurableSpace γ
inst✝² : MeasurableSpace δ
f g✝ : α → β✝
μ ν : Measure α
β : Type u_7
mβ : MeasurableSpace β
inst✝¹ : LinearOrder α
inst✝ : atTop.IsCountablyGenerated
x : α
g : α → β
g_meas : ∀ t > x, AEMeasurable g (μ.restrict (Ioc x t))
this : Nonempty α
u : ℕ → α
hu_tendsto : Tendsto u atTop atTop
Ioi_eq_iUnion : Ioi x = ⋃ n, Ioc x (u n)
n : ℕ
h : u n ≤ x
⊢ AEMeasurable g (μ.restrict (Ioc x (u n)))

tactic: rw [iUnion_Ioc_eq_Ioi_self_iff.mpr _]
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✝⁴ : MeasurableSpace β✝
inst✝³ : MeasurableSpace γ
inst✝² : MeasurableSpace δ
f g✝ : α → β✝
μ ν : Measure α
β : Type u_7
mβ : MeasurableSpace β
inst✝¹ : LinearOrder α
inst✝ : atTop.IsCountablyGenerated
x : α
g : α → β
g_meas : ∀ t > x, AEMeasurable g (μ.restrict (Ioc x t))
this : Nonempty α
u : ℕ → α
hu_tendsto : Tendsto u atTop atTop
⊢ Ioi x = ⋃ n, Ioc x (u n)
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✝⁴ : MeasurableSpace β✝
inst✝³ : MeasurableSpace γ
inst✝² : MeasurableSpace δ
f g✝ : α → β✝
μ ν : Measure α
β : Type u_7
mβ : MeasurableSpace β
inst✝¹ : LinearOrder α
inst✝ : atTop.IsCountablyGenerated
x : α
g : α → β
g_meas : ∀ t > x, AEMeasurable g (μ.restrict (Ioc x t))
this : Nonempty α
u : ℕ → α
hu_tendsto : Tendsto u atTop atTop
⊢ ∀ (x_1 : α), x < x_1 → ∃ i, x_1 ≤ u i

tactic: exact fun y _ => (hu_tendsto.eventually (eventually_ge_atTop y)).exists
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✝⁴ : MeasurableSpace β✝
inst✝³ : MeasurableSpace γ
inst✝² : MeasurableSpace δ
f g✝ : α → β✝
μ ν : Measure α
β : Type u_7
mβ : MeasurableSpace β
inst✝¹ : LinearOrder α
inst✝ : atTop.IsCountablyGenerated
x : α
g : α → β
g_meas : ∀ t > x, AEMeasurable g (μ.restrict (Ioc x t))
this : Nonempty α
u : ℕ → α
hu_tendsto : Tendsto u atTop atTop
⊢ ∀ (x_1 : α), x < x_1 → ∃ i, x_1 ≤ u i
state_after: no goals

tactic: exact g_meas (u n) h
state_before: case intro.inl
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✝⁴ : MeasurableSpace β✝
inst✝³ : MeasurableSpace γ
inst✝² : MeasurableSpace δ
f g✝ : α → β✝
μ ν : Measure α
β : Type u_7
mβ : MeasurableSpace β
inst✝¹ : LinearOrder α
inst✝ : atTop.IsCountablyGenerated
x : α
g : α → β
g_meas : ∀ t > x, AEMeasurable g (μ.restrict (Ioc x t))
this : Nonempty α
u : ℕ → α
hu_tendsto : Tendsto u atTop atTop
Ioi_eq_iUnion : Ioi x = ⋃ n, Ioc x (u n)
n : ℕ
h : x < u n
⊢ AEMeasurable g (μ.restrict (Ioc x (u n)))
state_after: no goals

tactic: rw [Ioc_eq_empty (not_lt.mpr h), Measure.restrict_empty]
state_before: case intro.inr
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✝⁴ : MeasurableSpace β✝
inst✝³ : MeasurableSpace γ
inst✝² : MeasurableSpace δ
f g✝ : α → β✝
μ ν : Measure α
β : Type u_7
mβ : MeasurableSpace β
inst✝¹ : LinearOrder α
inst✝ : atTop.IsCountablyGenerated
x : α
g : α → β
g_meas : ∀ t > x, AEMeasurable g (μ.restrict (Ioc x t))
this : Nonempty α
u : ℕ → α
hu_tendsto : Tendsto u atTop atTop
Ioi_eq_iUnion : Ioi x = ⋃ n, Ioc x (u n)
n : ℕ
h : u n ≤ x
⊢ AEMeasurable g (μ.restrict (Ioc x (u n)))
state_after: case intro.inr
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✝⁴ : MeasurableSpace β✝
inst✝³ : MeasurableSpace γ
inst✝² : MeasurableSpace δ
f g✝ : α → β✝
μ ν : Measure α
β : Type u_7
mβ : MeasurableSpace β
inst✝¹ : LinearOrder α
inst✝ : atTop.IsCountablyGenerated
x : α
g : α → β
g_meas : ∀ t > x, AEMeasurable g (μ.restrict (Ioc x t))
this : Nonempty α
u : ℕ → α
hu_tendsto : Tendsto u atTop atTop
Ioi_eq_iUnion : Ioi x = ⋃ n, Ioc x (u n)
n : ℕ
h : u n ≤ x
⊢ AEMeasurable g 0

tactic: exact aemeasurable_zero_measure
state_before: case intro.inr
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✝⁴ : MeasurableSpace β✝
inst✝³ : MeasurableSpace γ
inst✝² : MeasurableSpace δ
f g✝ : α → β✝
μ ν : Measure α
β : Type u_7
mβ : MeasurableSpace β
inst✝¹ : LinearOrder α
inst✝ : atTop.IsCountablyGenerated
x : α
g : α → β
g_meas : ∀ t > x, AEMeasurable g (μ.restrict (Ioc x t))
this : Nonempty α
u : ℕ → α
hu_tendsto : Tendsto u atTop atTop
Ioi_eq_iUnion : Ioi x = ⋃ n, Ioc x (u n)
n : ℕ
h : u n ≤ x
⊢ AEMeasurable g 0
state_after: no goals