-- Mathlib/MeasureTheory/Measure/AEMeasurable.lean
theorem aemeasurable_Ioi_of_forall_Ioc {β} {mβ : MeasurableSpace β} [LinearOrder α]
    [(atTop : Filter α).IsCountablyGenerated] {x : α} {g : α → β}
    (g_meas : ∀ t > x, AEMeasurable g (μ.restrict (Ioc x t))) :
    AEMeasurable g (μ.restrict (Ioi x))

Proof state:
case intro
α : Type u_2
m0 : MeasurableSpace α
μ : 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

Result:
case intro
α : Type u_2
m0 : MeasurableSpace α
μ : 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)))

Proof state:
case intro
α : Type u_2
m0 : MeasurableSpace α
μ : 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

Result:
case intro
α : Type u_2
m0 : MeasurableSpace α
μ : 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))

Proof state:
α : Type u_2
m0 : MeasurableSpace α
μ : 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)

Result:
case intro
α : Type u_2
m0 : MeasurableSpace α
μ : 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))

Proof state:
case intro.inl
α : Type u_2
m0 : MeasurableSpace α
μ : 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)))

Tactic:
exact g_meas (u n) h

Result:
no goals

Proof state:
case intro
α : Type u_2
m0 : MeasurableSpace α
μ : 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:
rcases lt_or_le x (u n) with h | h

Result:
case intro.inl
α : Type u_2
m0 : MeasurableSpace α
μ : 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
α : Type u_2
m0 : MeasurableSpace α
μ : 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)))

Proof state:
α : Type u_2
m0 : MeasurableSpace α
μ : 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)

Tactic:
rw [iUnion_Ioc_eq_Ioi_self_iff.mpr _]

Result:
α : Type u_2
m0 : MeasurableSpace α
μ : 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

Proof state:
α : Type u_2
m0 : MeasurableSpace α
μ : 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))

Tactic:
haveI : Nonempty α := ⟨x⟩

Result:
α : Type u_2
m0 : MeasurableSpace α
μ : 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))

Proof state:
α : Type u_2
m0 : MeasurableSpace α
μ : 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

Result:
no goals

Proof state:
case intro
α : Type u_2
m0 : MeasurableSpace α
μ : 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]

Result:
case intro
α : Type u_2
m0 : MeasurableSpace α
μ : 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)))

Proof state:
case intro.inr
α : Type u_2
m0 : MeasurableSpace α
μ : 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 [Ioc_eq_empty (not_lt.mpr h), Measure.restrict_empty]

Result:
case intro.inr
α : Type u_2
m0 : MeasurableSpace α
μ : 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