mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Measure.MeasureSpace

Goal state before:
pp: case h
α : Type u_1
ι : Type u_5
m : MeasurableSpace α
μ : Measure α
inst✝ : Countable ι
s : ι → Set α
h : ∀ (i : ι), NullMeasurableSet (s i) μ
hd : Directed (fun x1 x2 ↦ x1 ⊇ x2) s
k : ι
hk : μ (s k) ≠ ∞
this : ∀ t ⊆ s k, μ t ≠ ∞
i j : ι
hji : s i ⊇ s j
hjk : s k ⊇ s j
⊢ μ (s k \ s i) ≤ μ (s k \ s j)

Tactic:
]

Goal state after:
case h
α : Type u_1
ι : Type u_5
m : MeasurableSpace α
μ : Measure α
inst✝ : Countable ι
s : ι → Set α
h : ∀ (i : ι), NullMeasurableSet (s i) μ
hd : Directed (fun x1 x2 ↦ x1 ⊇ x2) s
k : ι
hk : μ (s k) ≠ ∞
this : ∀ t ⊆ s k, μ t ≠ ∞
i j : ι
hji : s i ⊇ s j
hjk : s k ⊇ s j
⊢ μ (s k \ s i) ≤ μ (s k \ s j)
Module: Mathlib.MeasureTheory.Measure.Typeclasses.SFinite

Goal state before:
pp: α : Type u_1
ι : Type u_4
x✝ : MeasurableSpace α
μ : Measure α
inst✝ : SFinite μ
As : ι → Set α
As_mble : ∀ (i : ι), NullMeasurableSet (As i) (sum (sfiniteSeq μ))
As_disj : Pairwise (AEDisjoint (sum (sfiniteSeq μ)) on As)
⊢ {i | 0 < (sum (sfiniteSeq μ)) (As i)}.Countable

Tactic:
]

Goal state after:
α : Type u_1
ι : Type u_4
x✝ : MeasurableSpace α
μ : Measure α
inst✝ : SFinite μ
As : ι → Set α
As_mble : ∀ (i : ι), NullMeasurableSet (As i) (sum (sfiniteSeq μ))
As_disj : Pairwise (AEDisjoint (sum (sfiniteSeq μ)) on As)
⊢ {i | 0 < (sum (sfiniteSeq μ)) (As i)}.Countable
Module: Mathlib.MeasureTheory.Measure.Restrict

Goal state before:
pp: α : Type u_2
m0 : MeasurableSpace α
μ : Measure α
s t : Set α
h : s ⊆ t
u : Set α
hu : MeasurableSet u
⊢ μ (u ∩ s) = μ (u ∩ s)

pp: case h
α : Type u_2
m0 : MeasurableSpace α
μ : Measure α
s t : Set α
h : s ⊆ t
u : Set α
hu : MeasurableSet u
⊢ u ∩ s ⊆ t

Tactic:
]

Goal state after:
case h
α : Type u_2
m0 : MeasurableSpace α
μ : Measure α
s t : Set α
h : s ⊆ t
u : Set α
hu : MeasurableSet u
⊢ u ∩ s ⊆ t
Module: Mathlib.MeasureTheory.Measure.Restrict

Goal state before:
pp: α : Type u_2
m0 : MeasurableSpace α
μ : Measure α
s t : Set α
hs : MeasurableSet s
⊢ μ.restrict (t ∩ s) = μ.restrict (t ∩ s)

Tactic:
]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Restrict

Goal state before:
pp: α : Type u_2
m0 : MeasurableSpace α
μ : Measure α
s t : Set α
ht : MeasurableSet t
⊢ μ (t ∩ s) = 0 ↔ μ (t ∩ s) = 0

Tactic:
]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Restrict

Goal state before:
pp: α : Type u_2
m0 : MeasurableSpace α
μ : Measure α
s t : Set α
hs : MeasurableSet s
⊢ μ (t ∩ s) = 0 ↔ μ (t ∩ s) = 0

Tactic:
]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Typeclasses.SFinite

Goal state before:
pp: α : Type u_1
ι : Type u_4
x✝ : MeasurableSpace α
μ : Measure α
inst✝ : SFinite μ
As : ι → Set α
As_mble : ∀ (i : ι), NullMeasurableSet (As i) (sum (sfiniteSeq μ))
As_disj : Pairwise (AEDisjoint (sum (sfiniteSeq μ)) on As)
i : ι
hi : 0 < ∑' (i_1 : ℕ), (sfiniteSeq μ i_1) (As i)
con : ∀ (x : ℕ), (sfiniteSeq μ x) (As i) = 0
⊢ False

pp: case hs
α : Type u_1
ι : Type u_4
x✝ : MeasurableSpace α
μ : Measure α
inst✝ : SFinite μ
As : ι → Set α
As_mble : ∀ (i : ι), NullMeasurableSet (As i) (sum (sfiniteSeq μ))
As_disj : Pairwise (AEDisjoint (sum (sfiniteSeq μ)) on As)
i : ι
hi : 0 < (sum (sfiniteSeq μ)) (As i)
con : ∀ (x : ℕ), (sfiniteSeq μ x) (As i) = 0
⊢ NullMeasurableSet (As i) (sum (sfiniteSeq μ))

Tactic:
]

Goal state after:
α : Type u_1
ι : Type u_4
x✝ : MeasurableSpace α
μ : Measure α
inst✝ : SFinite μ
As : ι → Set α
As_mble : ∀ (i : ι), NullMeasurableSet (As i) (sum (sfiniteSeq μ))
As_disj : Pairwise (AEDisjoint (sum (sfiniteSeq μ)) on As)
i : ι
hi : 0 < ∑' (i_1 : ℕ), (sfiniteSeq μ i_1) (As i)
con : ∀ (x : ℕ), (sfiniteSeq μ x) (As i) = 0
⊢ False

case hs
α : Type u_1
ι : Type u_4
x✝ : MeasurableSpace α
μ : Measure α
inst✝ : SFinite μ
As : ι → Set α
As_mble : ∀ (i : ι), NullMeasurableSet (As i) (sum (sfiniteSeq μ))
As_disj : Pairwise (AEDisjoint (sum (sfiniteSeq μ)) on As)
i : ι
hi : 0 < (sum (sfiniteSeq μ)) (As i)
con : ∀ (x : ℕ), (sfiniteSeq μ x) (As i) = 0
⊢ NullMeasurableSet (As i) (sum (sfiniteSeq μ))
Module: Mathlib.MeasureTheory.Measure.Restrict

Goal state before:
pp: α : Type u_2
m0 : MeasurableSpace α
μ : Measure α
s : Set α
⊢ μ s = 0 ↔ μ s = 0

Tactic:
]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Restrict

Goal state before:
pp: α : Type u_2
m0 : MeasurableSpace α
μ : Measure α
t s : Set α
ht : NullMeasurableSet t μ
⊢ μ.restrict (s ∩ t) + μ.restrict (s \ t) + μ.restrict t = μ.restrict (s ∩ t) + μ.restrict (s \ t) + μ.restrict t

Tactic:
]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Typeclasses.Finite

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
s t : Set α
hs : NullMeasurableSet s μ
ht : NullMeasurableSet t μ
hs' : μ s ≠ ∞
ht' : μ t ≠ ∞
hst : μ (s \ t) ≠ ∞
hts : μ (t \ s) ≠ ∞
this : (μ s).toReal - (μ t).toReal = (μ (s \ t)).toReal - (μ (t \ s)).toReal
⊢ |(μ (s \ t)).toReal - (μ (t \ s)).toReal| ≤ (μ (s \ t)).toReal + (μ (t \ s)).toReal

Tactic:
]

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
s t : Set α
hs : NullMeasurableSet s μ
ht : NullMeasurableSet t μ
hs' : μ s ≠ ∞
ht' : μ t ≠ ∞
hst : μ (s \ t) ≠ ∞
hts : μ (t \ s) ≠ ∞
this : (μ s).toReal - (μ t).toReal = (μ (s \ t)).toReal - (μ (t \ s)).toReal
⊢ |(μ (s \ t)).toReal - (μ (t \ s)).toReal| ≤ (μ (s \ t)).toReal + (μ (t \ s)).toReal
Module: Mathlib.MeasureTheory.Measure.Restrict

Goal state before:
pp: α : Type u_2
m0 : MeasurableSpace α
μ : Measure α
s t : Set α
h : Disjoint s t
hs : MeasurableSet s
⊢ μ.restrict s + μ.restrict t = μ.restrict s + μ.restrict t

Tactic:
]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Restrict

Goal state before:
pp: α : Type u_2
m0 : MeasurableSpace α
μ : Measure α
s : Set α
hs : MeasurableSet s
⊢ μ = μ

Tactic:
]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Restrict

Goal state before:
pp: α : Type u_2
m0 : MeasurableSpace α
μ : Measure α
s : Set α
hs : MeasurableSet s
⊢ μ.restrict sᶜ + μ.restrict s = μ

Tactic:
rw [add_comm, restrict_add_restrict_compl hs]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Restrict

Goal state before:
pp: α : Type u_2
ι : Type u_6
m0 : MeasurableSpace α
μ : Measure α
inst✝ : Countable ι
s : ι → Set α
hd : Directed (fun x1 x2 ↦ x1 ⊆ x2) s
t : Set α
ht : MeasurableSet t
⊢ ⨆ i, μ (t ∩ s i) = ⨆ i, μ (t ∩ s i)

pp: α : Type u_2
ι : Type u_6
m0 : MeasurableSpace α
μ : Measure α
inst✝ : Countable ι
s : ι → Set α
hd : Directed (fun x1 x2 ↦ x1 ⊆ x2) s
t : Set α
ht : MeasurableSet t
⊢ Directed (fun x1 x2 ↦ x1 ⊆ x2) fun i ↦ t ∩ s i

Tactic:
]

Goal state after:
α : Type u_2
ι : Type u_6
m0 : MeasurableSpace α
μ : Measure α
inst✝ : Countable ι
s : ι → Set α
hd : Directed (fun x1 x2 ↦ x1 ⊆ x2) s
t : Set α
ht : MeasurableSet t
⊢ Directed (fun x1 x2 ↦ x1 ⊆ x2) fun i ↦ t ∩ s i
Module: Mathlib.MeasureTheory.Measure.MeasureSpace

Goal state before:
pp: α : Type u_1
ι : Type u_5
m : MeasurableSpace α
μ : Measure α
inst✝² : Preorder ι
inst✝¹ : IsCodirectedOrder ι
inst✝ : atBot.IsCountablyGenerated
s : ι → Set α
hs : ∀ᵐ (ω : α) ∂μ, Monotone fun x ↦ ω ∈ s x
hsm : ∀ (i : ι), NullMeasurableSet (s i) μ
i : ι
hi : μ (s i) ≠ ∞
this : Nonempty ι
t : ι → Set α := fun i ↦ s i ∩ {ω | Monotone fun x ↦ ω ∈ s x}
hst : ∀ (i : ι), s i =ᵐ[μ] t i
hMono : Monotone t
⊢ μ (s i) ≠ ∞

Tactic:
]

Goal state after:
α : Type u_1
ι : Type u_5
m : MeasurableSpace α
μ : Measure α
inst✝² : Preorder ι
inst✝¹ : IsCodirectedOrder ι
inst✝ : atBot.IsCountablyGenerated
s : ι → Set α
hs : ∀ᵐ (ω : α) ∂μ, Monotone fun x ↦ ω ∈ s x
hsm : ∀ (i : ι), NullMeasurableSet (s i) μ
i : ι
hi : μ (s i) ≠ ∞
this : Nonempty ι
t : ι → Set α := fun i ↦ s i ∩ {ω | Monotone fun x ↦ ω ∈ s x}
hst : ∀ (i : ι), s i =ᵐ[μ] t i
hMono : Monotone t
⊢ μ (s i) ≠ ∞
Module: Mathlib.MeasureTheory.Measure.MeasureSpace

Goal state before:
pp: α : Type u_1
ι : Type u_5
m : MeasurableSpace α
μ : Measure α
inst✝² : Preorder ι
inst✝¹ : IsCodirectedOrder ι
inst✝ : atBot.IsCountablyGenerated
s : ι → Set α
hs : ∀ᵐ (ω : α) ∂μ, Monotone fun x ↦ ω ∈ s x
hsm : ∀ (i : ι), NullMeasurableSet (s i) μ
i : ι
hi : μ (s i) ≠ ∞
this : Nonempty ι
t : ι → Set α := fun i ↦ s i ∩ {ω | Monotone fun x ↦ ω ∈ s x}
hst : ∀ (i : ι), s i =ᵐ[μ] t i
hMono : Monotone t
⊢ μ (⋂ i, s i) = μ (⋂ i, t i)

Tactic:
]

Goal state after:
α : Type u_1
ι : Type u_5
m : MeasurableSpace α
μ : Measure α
inst✝² : Preorder ι
inst✝¹ : IsCodirectedOrder ι
inst✝ : atBot.IsCountablyGenerated
s : ι → Set α
hs : ∀ᵐ (ω : α) ∂μ, Monotone fun x ↦ ω ∈ s x
hsm : ∀ (i : ι), NullMeasurableSet (s i) μ
i : ι
hi : μ (s i) ≠ ∞
this : Nonempty ι
t : ι → Set α := fun i ↦ s i ∩ {ω | Monotone fun x ↦ ω ∈ s x}
hst : ∀ (i : ι), s i =ᵐ[μ] t i
hMono : Monotone t
⊢ μ (⋂ i, s i) = μ (⋂ i, t i)
Module: Mathlib.MeasureTheory.Measure.Typeclasses.Finite

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
s t : Set α
hs : NullMeasurableSet s μ
ht : NullMeasurableSet t μ
hs' : μ s ≠ ∞
ht' : μ t ≠ ∞
hst : μ (s \ t) ≠ ∞
hts : μ (t \ s) ≠ ∞
⊢ (μ s).toReal - (μ t).toReal = (μ (s ∪ t)).toReal - (μ t).toReal - ((μ (s ∪ t)).toReal - (μ s).toReal)

Tactic:
]

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
s t : Set α
hs : NullMeasurableSet s μ
ht : NullMeasurableSet t μ
hs' : μ s ≠ ∞
ht' : μ t ≠ ∞
hst : μ (s \ t) ≠ ∞
hts : μ (t \ s) ≠ ∞
⊢ (μ s).toReal - (μ t).toReal = (μ (s ∪ t)).toReal - (μ t).toReal - ((μ (s ∪ t)).toReal - (μ s).toReal)
Module: Mathlib.MeasureTheory.Measure.Restrict

Goal state before:
pp: α : Type u_2
m0 : MeasurableSpace α
μ : Measure α
s t : Set α
h : μ s ≠ ∞
ht : MeasurableSet t
hst : s ⊆ t
u : Set α
hu : MeasurableSet u
⊢ μ (s ∩ (t ∩ u)) = μ (u ∩ s)

Tactic:
]

Goal state after:
α : Type u_2
m0 : MeasurableSpace α
μ : Measure α
s t : Set α
h : μ s ≠ ∞
ht : MeasurableSet t
hst : s ⊆ t
u : Set α
hu : MeasurableSet u
⊢ μ (s ∩ (t ∩ u)) = μ (u ∩ s)
Module: Mathlib.MeasureTheory.Measure.Restrict

Goal state before:
pp: α : Type u_2
m0 : MeasurableSpace α
μ ν : Measure α
s : Set α
hs : MeasurableSet s
H : μ.restrict s = ν.restrict s
t : Set α
hts : t ⊆ s
ht : MeasurableSet t
⊢ ν (t ∩ s) = ν (t ∩ s)

Tactic:
]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.MeasureSpace

Goal state before:
pp: α : Type u_1
ι : Type u_5
m : MeasurableSpace α
μ : Measure α
inst✝² : Preorder ι
inst✝¹ : IsCodirectedOrder ι
inst✝ : atBot.IsCountablyGenerated
s : ι → Set α
hs : ∀ᵐ (ω : α) ∂μ, Monotone fun x ↦ ω ∈ s x
hsm : ∀ (i : ι), NullMeasurableSet (s i) μ
i : ι
hi : μ (s i) ≠ ∞
this : Nonempty ι
t : ι → Set α := fun i ↦ s i ∩ {ω | Monotone fun x ↦ ω ∈ s x}
hst : ∀ (i : ι), s i =ᵐ[μ] t i
hMono : Monotone t
⊢ (⋂ i, s i) ∩ univ =ᵐ[μ] (⋂ i, s i) ∩ {ω | Monotone fun x ↦ ω ∈ s x}

Tactic:
]

Goal state after:
α : Type u_1
ι : Type u_5
m : MeasurableSpace α
μ : Measure α
inst✝² : Preorder ι
inst✝¹ : IsCodirectedOrder ι
inst✝ : atBot.IsCountablyGenerated
s : ι → Set α
hs : ∀ᵐ (ω : α) ∂μ, Monotone fun x ↦ ω ∈ s x
hsm : ∀ (i : ι), NullMeasurableSet (s i) μ
i : ι
hi : μ (s i) ≠ ∞
this : Nonempty ι
t : ι → Set α := fun i ↦ s i ∩ {ω | Monotone fun x ↦ ω ∈ s x}
hst : ∀ (i : ι), s i =ᵐ[μ] t i
hMono : Monotone t
⊢ (⋂ i, s i) ∩ univ =ᵐ[μ] (⋂ i, s i) ∩ {ω | Monotone fun x ↦ ω ∈ s x}
Module: Mathlib.MeasureTheory.Measure.Restrict

Goal state before:
pp: α : Type u_2
m0 : MeasurableSpace α
μ ν : Measure α
s : Set α
hs : MeasurableSet s
H : ∀ t ⊆ s, MeasurableSet t → μ t = ν t
t : Set α
ht : MeasurableSet t
⊢ ν (t ∩ s) = ν (t ∩ s)

Tactic:
]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Restrict

Goal state before:
pp: α : Type u_2
m0 : MeasurableSpace α
μ ν : Measure α
s t : Set α
hs : s ⊆ t
h : μ.restrict t = ν.restrict t
⊢ ν.restrict s = ν.restrict s

Tactic:
]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Typeclasses.Finite

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
γ : Type u_5
f g : α → γ
s : Set α
inst✝ : DecidablePred fun x ↦ x ∈ s
hs_zero : μ s = 0
h_ss : sᶜ ⊆ {a | (if a ∈ s then f a else g a) = g a}
⊢ sᶜ ⊆ {x | (fun x ↦ (fun x ↦ if x ∈ s then f x else g x) x = g x) x}

Tactic:
]

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
γ : Type u_5
f g : α → γ
s : Set α
inst✝ : DecidablePred fun x ↦ x ∈ s
hs_zero : μ s = 0
h_ss : sᶜ ⊆ {a | (if a ∈ s then f a else g a) = g a}
⊢ sᶜ ⊆ {x | (fun x ↦ (fun x ↦ if x ∈ s then f x else g x) x = g x) x}
Module: Mathlib.MeasureTheory.Measure.Typeclasses.Finite

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
γ : Type u_5
f g : α → γ
s : Set α
inst✝ : DecidablePred fun x ↦ x ∈ s
hs_zero : s ∈ ae μ
⊢ (fun x ↦ if x ∈ s then f x else g x) =ᵐ[μ] f

Tactic:
]

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
γ : Type u_5
f g : α → γ
s : Set α
inst✝ : DecidablePred fun x ↦ x ∈ s
hs_zero : s ∈ ae μ
⊢ (fun x ↦ if x ∈ s then f x else g x) =ᵐ[μ] f
Module: Mathlib.MeasureTheory.Measure.Restrict

Goal state before:
pp: α : Type u_2
m0 : MeasurableSpace α
μ ν : Measure α
s t : Set α
hs : μ.restrict s = ν.restrict s
ht : μ.restrict t = ν.restrict t
u : Set α
hu : MeasurableSet u
US : Set α
hsub : u ∩ s ⊆ US
hm : MeasurableSet US
hμ : μ US = μ (u ∩ s)
hν : ν US = ν (u ∩ s)
⊢ (ν.restrict s) u + (ν.restrict t) (u \ US) = (ν.restrict s) u + (ν.restrict t) (u \ US)

Tactic:
]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.MeasureSpace

Goal state before:
pp: α : Type u_8
ι : Type u_9
x✝ : MeasurableSpace α
μ : Measure α
inst✝² : Countable ι
inst✝¹ : Preorder ι
inst✝ : IsDirectedOrder ι
f : ι → Set α
h : ∀ (i : ι), NullMeasurableSet (f i) μ
hfin : ∃ i, μ (f i) ≠ ∞
⊢ μ (⋂ i, f i) = μ (⋂ i, ⋂ j, ⋂ (_ : j ≤ i), f j)

pp: case hs
α : Type u_8
ι : Type u_9
x✝ : MeasurableSpace α
μ : Measure α
inst✝² : Countable ι
inst✝¹ : Preorder ι
inst✝ : IsDirectedOrder ι
f : ι → Set α
h : ∀ (i : ι), NullMeasurableSet (f i) μ
hfin : ∃ i, μ (f i) ≠ ∞
⊢ Antitone fun i ↦ ⋂ j, ⋂ (_ : j ≤ i), f j

pp: case hsm
α : Type u_8
ι : Type u_9
x✝ : MeasurableSpace α
μ : Measure α
inst✝² : Countable ι
inst✝¹ : Preorder ι
inst✝ : IsDirectedOrder ι
f : ι → Set α
h : ∀ (i : ι), NullMeasurableSet (f i) μ
hfin : ∃ i, μ (f i) ≠ ∞
⊢ ∀ (i : ι), NullMeasurableSet (⋂ j, ⋂ (_ : j ≤ i), f j) μ

pp: case hfin
α : Type u_8
ι : Type u_9
x✝ : MeasurableSpace α
μ : Measure α
inst✝² : Countable ι
inst✝¹ : Preorder ι
inst✝ : IsDirectedOrder ι
f : ι → Set α
h : ∀ (i : ι), NullMeasurableSet (f i) μ
hfin : ∃ i, μ (f i) ≠ ∞
⊢ ∃ i, μ (⋂ j, ⋂ (_ : j ≤ i), f j) ≠ ∞

Tactic:
]

Goal state after:
α : Type u_8
ι : Type u_9
x✝ : MeasurableSpace α
μ : Measure α
inst✝² : Countable ι
inst✝¹ : Preorder ι
inst✝ : IsDirectedOrder ι
f : ι → Set α
h : ∀ (i : ι), NullMeasurableSet (f i) μ
hfin : ∃ i, μ (f i) ≠ ∞
⊢ μ (⋂ i, f i) = μ (⋂ i, ⋂ j, ⋂ (_ : j ≤ i), f j)

case hs
α : Type u_8
ι : Type u_9
x✝ : MeasurableSpace α
μ : Measure α
inst✝² : Countable ι
inst✝¹ : Preorder ι
inst✝ : IsDirectedOrder ι
f : ι → Set α
h : ∀ (i : ι), NullMeasurableSet (f i) μ
hfin : ∃ i, μ (f i) ≠ ∞
⊢ Antitone fun i ↦ ⋂ j, ⋂ (_ : j ≤ i), f j

case hsm
α : Type u_8
ι : Type u_9
x✝ : MeasurableSpace α
μ : Measure α
inst✝² : Countable ι
inst✝¹ : Preorder ι
inst✝ : IsDirectedOrder ι
f : ι → Set α
h : ∀ (i : ι), NullMeasurableSet (f i) μ
hfin : ∃ i, μ (f i) ≠ ∞
⊢ ∀ (i : ι), NullMeasurableSet (⋂ j, ⋂ (_ : j ≤ i), f j) μ

case hfin
α : Type u_8
ι : Type u_9
x✝ : MeasurableSpace α
μ : Measure α
inst✝² : Countable ι
inst✝¹ : Preorder ι
inst✝ : IsDirectedOrder ι
f : ι → Set α
h : ∀ (i : ι), NullMeasurableSet (f i) μ
hfin : ∃ i, μ (f i) ≠ ∞
⊢ ∃ i, μ (⋂ j, ⋂ (_ : j ≤ i), f j) ≠ ∞
Module: Mathlib.MeasureTheory.Measure.MeasureSpace

Goal state before:
pp: α : Type u_8
ι : Type u_9
x✝ : MeasurableSpace α
μ : Measure α
inst✝² : Countable ι
inst✝¹ : Preorder ι
inst✝ : IsDirectedOrder ι
f : ι → Set α
h : ∀ (i : ι), NullMeasurableSet (f i) μ
hfin : ∃ i, μ (f i) ≠ ∞
⊢ μ (⋂ i, f i) = μ (⋂ i', ⋂ i, ⋂ (_ : i' ≤ i), f i')

Tactic:
]

Goal state after:
α : Type u_8
ι : Type u_9
x✝ : MeasurableSpace α
μ : Measure α
inst✝² : Countable ι
inst✝¹ : Preorder ι
inst✝ : IsDirectedOrder ι
f : ι → Set α
h : ∀ (i : ι), NullMeasurableSet (f i) μ
hfin : ∃ i, μ (f i) ≠ ∞
⊢ μ (⋂ i, f i) = μ (⋂ i', ⋂ i, ⋂ (_ : i' ≤ i), f i')
Module: Mathlib.MeasureTheory.Measure.Restrict

Goal state before:
pp: case insert
α : Type u_2
ι : Type u_6
m0 : MeasurableSpace α
μ ν : Measure α
t : ι → Set α
i : ι
s : Finset ι
a✝ : i ∉ s
hs : μ.restrict (⋃ i ∈ s, t i) = ν.restrict (⋃ i ∈ s, t i) ↔ ∀ i ∈ s, μ.restrict (t i) = ν.restrict (t i)
⊢ μ.restrict (t i) = ν.restrict (t i) ∧ μ.restrict (⋃ x ∈ s, t x) = ν.restrict (⋃ x ∈ s, t x) ↔
    μ.restrict (t i) = ν.restrict (t i) ∧ μ.restrict (⋃ i ∈ s, t i) = ν.restrict (⋃ i ∈ s, t i)

Tactic:
]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Typeclasses.Finite

Goal state before:
pp: α : Type u_1
β : Type u_2
m0 : MeasurableSpace α
mβ : MeasurableSpace β
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
μ : Measure β
inst✝ : IsFiniteMeasureOnCompacts μ
f : α → β
f_cont : Continuous[inst✝², inst✝¹] f
f_me : MeasurableEmbedding f
K : Set α
hK : IsCompact K
⊢ μ (f '' K) < ∞

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
m0 : MeasurableSpace α
mβ : MeasurableSpace β
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
μ : Measure β
inst✝ : IsFiniteMeasureOnCompacts μ
f : α → β
f_cont : Continuous[inst✝², inst✝¹] f
f_me : MeasurableEmbedding f
K : Set α
hK : IsCompact K
⊢ μ (f '' K) < ∞
Module: Mathlib.MeasureTheory.Measure.Restrict

Goal state before:
pp: α : Type u_2
ι : Type u_6
m0 : MeasurableSpace α
μ ν : Measure α
inst✝ : Countable ι
s : ι → Set α
h : ∀ (i : ι), μ.restrict (s i) = ν.restrict (s i)
t : Set α
ht : MeasurableSet t
D : Directed (fun x1 x2 ↦ x1 ⊆ x2) fun t ↦ ⋃ i ∈ t, s i
⊢ (μ.restrict (⋃ t, ⋃ i ∈ t, s i)) t = (ν.restrict (⋃ t, ⋃ i ∈ t, s i)) t

Tactic:
]

Goal state after:
α : Type u_2
ι : Type u_6
m0 : MeasurableSpace α
μ ν : Measure α
inst✝ : Countable ι
s : ι → Set α
h : ∀ (i : ι), μ.restrict (s i) = ν.restrict (s i)
t : Set α
ht : MeasurableSet t
D : Directed (fun x1 x2 ↦ x1 ⊆ x2) fun t ↦ ⋃ i ∈ t, s i
⊢ (μ.restrict (⋃ t, ⋃ i ∈ t, s i)) t = (ν.restrict (⋃ t, ⋃ i ∈ t, s i)) t
Module: Mathlib.MeasureTheory.Measure.Typeclasses.Finite

Goal state before:
pp: α : Type u_1
ι : Type u_4
m0 : MeasurableSpace α
μ : Measure α
inst✝ : Countable ι
U : ι → Set α
hU : ⋃ i, U i = univ
H : ∀ (i : ι), μ (U i) ≤ 0
⊢ μ (⋃ i, U i) = 0

Tactic:
]

Goal state after:
α : Type u_1
ι : Type u_4
m0 : MeasurableSpace α
μ : Measure α
inst✝ : Countable ι
U : ι → Set α
hU : ⋃ i, U i = univ
H : ∀ (i : ι), μ (U i) ≤ 0
⊢ μ (⋃ i, U i) = 0
Module: Mathlib.MeasureTheory.Measure.MeasureSpace

Goal state before:
pp: α : Type u_1
ι : Type u_5
m : MeasurableSpace α
μ : Measure α
inst✝¹ : Preorder ι
inst✝ : atTop.IsCountablyGenerated
s : ι → Set α
hm : Monotone s
h : atTop.NeBot
this : IsDirectedOrder ι
⊢ Tendsto (⇑μ ∘ s) atTop (𝓝 (⨆ i, μ (s i)))

Tactic:
]

Goal state after:
α : Type u_1
ι : Type u_5
m : MeasurableSpace α
μ : Measure α
inst✝¹ : Preorder ι
inst✝ : atTop.IsCountablyGenerated
s : ι → Set α
hm : Monotone s
h : atTop.NeBot
this : IsDirectedOrder ι
⊢ Tendsto (⇑μ ∘ s) atTop (𝓝 (⨆ i, μ (s i)))
Module: Mathlib.MeasureTheory.Measure.Typeclasses.Finite

Goal state before:
pp: α : Type u_1
δ : Type u_3
m0 : MeasurableSpace α
μ : Measure α
inst✝ : PseudoMetricSpace δ
f : α → δ
x : δ
hμ : μ ≠ 0
⊢ univ = univ

Tactic:
]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Restrict

Goal state before:
pp: α : Type u_2
m0 : MeasurableSpace α
μ ν : Measure α
S : Set (Set α)
hc : S.Countable
⊢ (∀ i ∈ S, μ.restrict i = ν.restrict i) ↔ ∀ s ∈ S, μ.restrict s = ν.restrict s

Tactic:
]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.MeasureSpace

Goal state before:
pp: α : Type u_8
ι : Type u_9
inst✝¹ : Preorder ι
inst✝ : atTop.IsCountablyGenerated
x✝ : MeasurableSpace α
μ : Measure α
f : ι → Set α
h : atTop.NeBot
this : IsDirectedOrder ι
⊢ Tendsto (fun i ↦ μ (accumulate f i)) atTop (𝓝 (⨆ i, μ (accumulate f i)))

Tactic:
]

Goal state after:
α : Type u_8
ι : Type u_9
inst✝¹ : Preorder ι
inst✝ : atTop.IsCountablyGenerated
x✝ : MeasurableSpace α
μ : Measure α
f : ι → Set α
h : atTop.NeBot
this : IsDirectedOrder ι
⊢ Tendsto (fun i ↦ μ (accumulate f i)) atTop (𝓝 (⨆ i, μ (accumulate f i)))
Module: Mathlib.MeasureTheory.Measure.MeasureSpace

Goal state before:
pp: α : Type u_1
ι : Type u_5
m : MeasurableSpace α
μ : Measure α
inst✝¹ : Preorder ι
inst✝ : atTop.IsCountablyGenerated
s : ι → Set α
hs : ∀ (i : ι), NullMeasurableSet (s i) μ
hm : Antitone s
hf : ∃ i, μ (s i) ≠ ∞
h : atTop.NeBot
this : IsDirectedOrder ι
⊢ Tendsto (⇑μ ∘ s) atTop (𝓝 (⨅ i, μ (s i)))

Tactic:
]

Goal state after:
α : Type u_1
ι : Type u_5
m : MeasurableSpace α
μ : Measure α
inst✝¹ : Preorder ι
inst✝ : atTop.IsCountablyGenerated
s : ι → Set α
hs : ∀ (i : ι), NullMeasurableSet (s i) μ
hm : Antitone s
hf : ∃ i, μ (s i) ≠ ∞
h : atTop.NeBot
this : IsDirectedOrder ι
⊢ Tendsto (⇑μ ∘ s) atTop (𝓝 (⨅ i, μ (s i)))
Module: Mathlib.MeasureTheory.Measure.Typeclasses.Finite

Goal state before:
pp: α : Type u_5
m₀ : MeasurableSpace α
μ ν : Measure α
inst✝ : IsFiniteMeasure μ
C : Set (Set α)
hμν : ∀ s ∈ C, μ s = ν s
m : MeasurableSpace α
h : m ≤ m₀
hA : m = generateFrom C
hC : IsPiSystem C
h_univ : μ univ = ν univ
s : Set α
hs : MeasurableSet s
⊢ μ univ < ∞

Tactic:
]

Goal state after:
α : Type u_5
m₀ : MeasurableSpace α
μ ν : Measure α
inst✝ : IsFiniteMeasure μ
C : Set (Set α)
hμν : ∀ s ∈ C, μ s = ν s
m : MeasurableSpace α
h : m ≤ m₀
hA : m = generateFrom C
hC : IsPiSystem C
h_univ : μ univ = ν univ
s : Set α
hs : MeasurableSet s
⊢ μ univ < ∞
Module: Mathlib.MeasureTheory.Measure.MeasureSpace

Goal state before:
pp: case intro
α : Type u_8
ι : Type u_9
x✝ : MeasurableSpace α
μ : Measure α
inst✝¹ : Countable ι
inst✝ : Preorder ι
f : ι → Set α
hm : ∀ (i : ι), NullMeasurableSet (f i) μ
hf : ∃ i, μ (f i) ≠ ∞
hne : atTop.NeBot
left✝ : Nonempty ι
right✝ : IsDirectedOrder ι
⊢ Tendsto (fun i ↦ μ (⋂ j, ⋂ (_ : j ≤ i), f j)) atTop (𝓝 (⨅ i, μ (⋂ j, ⋂ (_ : j ≤ i), f j)))

Tactic:
]

Goal state after:
case intro
α : Type u_8
ι : Type u_9
x✝ : MeasurableSpace α
μ : Measure α
inst✝¹ : Countable ι
inst✝ : Preorder ι
f : ι → Set α
hm : ∀ (i : ι), NullMeasurableSet (f i) μ
hf : ∃ i, μ (f i) ≠ ∞
hne : atTop.NeBot
left✝ : Nonempty ι
right✝ : IsDirectedOrder ι
⊢ Tendsto (fun i ↦ μ (⋂ j, ⋂ (_ : j ≤ i), f j)) atTop (𝓝 (⨅ i, μ (⋂ j, ⋂ (_ : j ≤ i), f j)))
Module: Mathlib.MeasureTheory.Measure.Restrict

Goal state before:
pp: α : Type u_2
m0 : MeasurableSpace α
μ : Measure α
s : Set α
hs : ∃ᵐ (a : α) ∂μ.restrict s, a ∈ s
p : α → Prop
hp : ∀ᵐ (x : α) ∂μ.restrict s, p x
⊢ ∃ x ∈ s, p x

Tactic:
]

Goal state after:
α : Type u_2
m0 : MeasurableSpace α
μ : Measure α
s : Set α
hs : ∃ᵐ (a : α) ∂μ.restrict s, a ∈ s
p : α → Prop
hp : ∀ᵐ (x : α) ∂μ.restrict s, p x
⊢ ∃ x ∈ s, p x
Module: Mathlib.MeasureTheory.Measure.Typeclasses.Finite

Goal state before:
pp: case compl
α : Type u_5
m₀ : MeasurableSpace α
μ ν : Measure α
inst✝ : IsFiniteMeasure μ
C : Set (Set α)
hμν : ∀ s ∈ C, μ s = ν s
m : MeasurableSpace α
h : m ≤ m₀
hA : m = generateFrom C
hC : IsPiSystem C
h_univ : μ univ = ν univ
s : Set α
this : IsFiniteMeasure ν
t : Set α
htm : MeasurableSet t
iht : μ t = ν t
⊢ ν univ - ν t = ν univ - ν t

Tactic:
]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.MeasureSpace

Goal state before:
pp: α : Type u_8
ι : Type u_9
x✝¹ : MeasurableSpace α
μ : Measure α
inst✝¹ : SemilatticeSup ι
inst✝ : Countable ι
f : ι → Set α
hm : ∀ (i : ι), NullMeasurableSet (f i) μ
ε : ℝ≥0∞
hε : 0 < ε
hfin : ∃ i, μ (f i) ≠ ∞
hfem : ⋂ n, f n = ∅
F : ι → ℝ≥0∞ := fun m ↦ μ (⋂ n, ⋂ (_ : n ≤ m), f n)
hFAnti : Antitone F
this : ∀ (ε : ℝ≥0∞), 0 < ε → ∃ n, F n < ε
x✝ : Nonempty ι := Exists.nonempty hfin
⊢ ∃ m, μ (⋂ n, ⋂ (_ : n ≤ m), f n) < ε

Tactic:
]

Goal state after:
α : Type u_8
ι : Type u_9
x✝¹ : MeasurableSpace α
μ : Measure α
inst✝¹ : SemilatticeSup ι
inst✝ : Countable ι
f : ι → Set α
hm : ∀ (i : ι), NullMeasurableSet (f i) μ
ε : ℝ≥0∞
hε : 0 < ε
hfin : ∃ i, μ (f i) ≠ ∞
hfem : ⋂ n, f n = ∅
F : ι → ℝ≥0∞ := fun m ↦ μ (⋂ n, ⋂ (_ : n ≤ m), f n)
hFAnti : Antitone F
this : ∀ (ε : ℝ≥0∞), 0 < ε → ∃ n, F n < ε
x✝ : Nonempty ι := Exists.nonempty hfin
⊢ ∃ m, μ (⋂ n, ⋂ (_ : n ≤ m), f n) < ε