mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.Order.AtTopBotIxx

Goal state before:
pp: case inr
X : Type u_1
inst✝² : LinearOrder X
inst✝¹ : TopologicalSpace X
inst✝ : OrderTopology X
s : Set X
b : X
hsb : s ⊆ Iio b
hs : ∀ a' < b, ∃ a < b, Ioo a b ⊆ s
hb : IsSuccPrelimit b
a : X
ha : a ∈ Iio b
⊢ s ∈ 𝓝[<] b

Tactic:
]

Goal state after:
case inr
X : Type u_1
inst✝² : LinearOrder X
inst✝¹ : TopologicalSpace X
inst✝ : OrderTopology X
s : Set X
b : X
hsb : s ⊆ Iio b
hs : ∀ a' < b, ∃ a < b, Ioo a b ⊆ s
hb : IsSuccPrelimit b
a : X
ha : a ∈ Iio b
⊢ s ∈ 𝓝[<] b
Module: Mathlib.Topology.Order.AtTopBotIxx

Goal state before:
pp: X : Type u_1
inst✝² : LinearOrder X
inst✝¹ : TopologicalSpace X
inst✝ : OrderTopology X
a b : X
α : Type u_2
l : Filter α
f : X → α
h : a < b
hb : IsSuccPrelimit b
⊢ Tendsto (fun x ↦ f ↑x) atTop l ↔ Tendsto (fun x ↦ f ↑x) atTop l

Tactic:
]

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

Goal state before:
pp: case refine_3
α : Type u_1
m : MeasurableSpace α
μ : Measure α
ι : Sort u_8
inst✝ : Countable ι
s t : ι → Set α
hsub : ∀ (i : ι), s i ⊆ t i
h_le : ∀ (i : ι), μ (t i) ≤ μ (s i)
htop : ∀ (i : ι), μ (t i) ≠ ∞
M : Set α → Set α := toMeasurable μ
b : ι
⊢ μ (t b) ≠ ∞

Tactic:
]

Goal state after:
case refine_3
α : Type u_1
m : MeasurableSpace α
μ : Measure α
ι : Sort u_8
inst✝ : Countable ι
s t : ι → Set α
hsub : ∀ (i : ι), s i ⊆ t i
h_le : ∀ (i : ι), μ (t i) ≤ μ (s i)
htop : ∀ (i : ι), μ (t i) ≠ ∞
M : Set α → Set α := toMeasurable μ
b : ι
⊢ μ (t b) ≠ ∞
Module: Mathlib.Topology.Order.AtTopBotIxx

Goal state before:
pp: X : Type u_1
inst✝² : LinearOrder X
inst✝¹ : TopologicalSpace X
inst✝ : OrderTopology X
a b : X
α : Type u_2
l : Filter α
f : X → α
h : a < b
ha : IsPredPrelimit a
⊢ Tendsto (fun x ↦ f ↑x) atBot l ↔ Tendsto (fun x ↦ f ↑x) atBot l

Tactic:
]

Goal state after:
Module: Mathlib.Topology.Order.AtTopBotIxx

Goal state before:
pp: X : Type u_1
inst✝² : LinearOrder X
inst✝¹ : TopologicalSpace X
inst✝ : OrderTopology X
a : X
α : Type u_2
l : Filter α
f : X → α
ha : IsPredPrelimit a
⊢ Tendsto (fun x ↦ f ↑x) atBot l ↔ Tendsto (fun x ↦ f ↑x) atBot l

Tactic:
]

Goal state after:
Module: Mathlib.Topology.Order.AtTopBotIxx

Goal state before:
pp: X : Type u_1
inst✝² : LinearOrder X
inst✝¹ : TopologicalSpace X
inst✝ : OrderTopology X
a : X
α : Type u_2
l : Filter α
f : X → α
ha : IsSuccPrelimit a
⊢ Tendsto (fun x ↦ f ↑x) atTop l ↔ Tendsto (fun x ↦ f ↑x) atTop l

Tactic:
]

Goal state after:
Module: Mathlib.Topology.Order.AtTopBotIxx

Goal state before:
pp: X : Type u_1
inst✝² : LinearOrder X
inst✝¹ : TopologicalSpace X
inst✝ : OrderTopology X
a b : X
α : Type u_2
l : Filter α
f : α → ↑(Ioo a b)
hb : IsSuccPrelimit b
⊢ Tendsto (fun x ↦ ↑(f x)) l (𝓝[<] b) ↔ Tendsto (fun x ↦ ↑(f x)) l (𝓝[<] b)

Tactic:
]

Goal state after:
Module: Mathlib.Topology.Order.AtTopBotIxx

Goal state before:
pp: X : Type u_1
inst✝² : LinearOrder X
inst✝¹ : TopologicalSpace X
inst✝ : OrderTopology X
a b : X
α : Type u_2
l : Filter α
f : α → ↑(Ioo a b)
ha : IsPredPrelimit a
⊢ Tendsto (fun x ↦ ↑(f x)) l (𝓝[>] a) ↔ Tendsto (fun x ↦ ↑(f x)) l (𝓝[>] a)

Tactic:
]

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
s₁ s₂ t₁ t₂ : Set α
hs : s₁ ⊆ s₂
hsμ : μ s₂ ≤ μ s₁
ht : t₁ ⊆ t₂
htμ : μ t₂ ≤ μ t₁
⊢ μ (⋃ b, bif b then s₁ else t₁) = μ (⋃ b, bif b then s₂ else t₂)

Tactic:
]

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
s₁ s₂ t₁ t₂ : Set α
hs : s₁ ⊆ s₂
hsμ : μ s₂ ≤ μ s₁
ht : t₁ ⊆ t₂
htμ : μ t₂ ≤ μ t₁
⊢ μ (⋃ b, bif b then s₁ else t₁) = μ (⋃ b, bif b then s₂ else t₂)
Module: Mathlib.Topology.Order.AtTopBotIxx

Goal state before:
pp: X : Type u_1
inst✝² : LinearOrder X
inst✝¹ : TopologicalSpace X
inst✝ : OrderTopology X
a : X
α : Type u_2
l : Filter α
f : α → ↑(Ioi a)
ha : IsPredPrelimit a
⊢ Tendsto (fun x ↦ ↑(f x)) l (𝓝[>] a) ↔ Tendsto (fun x ↦ ↑(f x)) l (𝓝[>] a)

Tactic:
]

Goal state after:
Module: Mathlib.Topology.Order.AtTopBotIxx

Goal state before:
pp: X : Type u_1
inst✝² : LinearOrder X
inst✝¹ : TopologicalSpace X
inst✝ : OrderTopology X
a : X
α : Type u_2
l : Filter α
f : α → ↑(Iio a)
ha : IsSuccPrelimit a
⊢ Tendsto (fun x ↦ ↑(f x)) l (𝓝[<] a) ↔ Tendsto (fun x ↦ ↑(f x)) l (𝓝[<] a)

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
s : Finset ι
t : ι → Set α
h : ∀ i ∈ s, NullMeasurableSet (t i) μ
H : (↑s).Pairwise (AEDisjoint μ on t)
⊢ μ (⋃ b ∈ s, t b) ≤ μ univ

Tactic:
]

Goal state after:
α : Type u_1
ι : Type u_5
m : MeasurableSpace α
μ : Measure α
s : Finset ι
t : ι → Set α
h : ∀ i ∈ s, NullMeasurableSet (t i) μ
H : (↑s).Pairwise (AEDisjoint μ on t)
⊢ μ (⋃ b ∈ s, t b) ≤ μ univ
Module: Mathlib.MeasureTheory.Measure.MeasureSpace

Goal state before:
pp: α : Type u_1
ι : Type u_5
m : MeasurableSpace α
μ : Measure α
s : ι → Set α
hs : ∀ (i : ι), NullMeasurableSet (s i) μ
H : Pairwise (AEDisjoint μ on s)
⊢ ⨆ s_1, ∑ a ∈ s_1, μ (s a) ≤ μ univ

Tactic:
]

Goal state after:
α : Type u_1
ι : Type u_5
m : MeasurableSpace α
μ : Measure α
s : ι → Set α
hs : ∀ (i : ι), NullMeasurableSet (s i) μ
H : Pairwise (AEDisjoint μ on s)
⊢ ⨆ s_1, ∑ a ∈ s_1, μ (s a) ≤ μ univ
Module: Mathlib.MeasureTheory.Measure.MeasureSpace

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
s t u : Set α
ht : MeasurableSet t
h's : s ⊆ u
h't : t ⊆ u
h : μ u < μ s + μ t
⊢ ¬Disjoint s t

Tactic:
]

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
s t u : Set α
ht : MeasurableSet t
h's : s ⊆ u
h't : t ⊆ u
h : μ u < μ s + μ t
⊢ ¬Disjoint s t
Module: Mathlib.MeasureTheory.Measure.MeasureSpace

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
s t u : Set α
hs : MeasurableSet s
h's : s ⊆ u
h't : t ⊆ u
h : μ u < μ t + μ s
⊢ (s ∩ t).Nonempty

Tactic:
]

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
s t u : Set α
hs : MeasurableSet s
h's : s ⊆ u
h't : t ⊆ u
h : μ u < μ t + μ s
⊢ (s ∩ t).Nonempty
Module: Mathlib.MeasureTheory.Measure.MeasureSpace

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
s t u : Set α
hs : MeasurableSet s
h's : s ⊆ u
h't : t ⊆ u
h : μ u < μ t + μ s
⊢ (t ∩ s).Nonempty

Tactic:
]

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
s t u : Set α
hs : MeasurableSet s
h's : s ⊆ u
h't : t ⊆ u
h : μ u < μ t + μ s
⊢ (t ∩ s).Nonempty
Module: Mathlib.MeasureTheory.Measure.Typeclasses.SFinite

Goal state before:
pp: α : Type u_1
β : Type u_2
ι : Type u_3
m0 : MeasurableSpace α
inst✝ : MeasurableSpace β
μ ν : Measure α
s t : Set α
a : α
⊢ 0 = 0

Tactic:
]

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

Goal state before:
pp: case inl
α : Type u_1
ι : Type u_3
m0 : MeasurableSpace α
inst✝¹ : Countable ι
m : ι → Measure α
inst✝ : ∀ (n : ι), IsFiniteMeasure (m n)
f : ι → ℕ
hf : Injective f
i : ι
⊢ IsFiniteMeasure (m i)

Tactic:
]

Goal state after:
case inl
α : Type u_1
ι : Type u_3
m0 : MeasurableSpace α
inst✝¹ : Countable ι
m : ι → Measure α
inst✝ : ∀ (n : ι), IsFiniteMeasure (m n)
f : ι → ℕ
hf : Injective f
i : ι
⊢ IsFiniteMeasure (m i)
Module: Mathlib.MeasureTheory.Measure.Typeclasses.SFinite

Goal state before:
pp: case inr
α : Type u_1
ι : Type u_3
m0 : MeasurableSpace α
inst✝¹ : Countable ι
m : ι → Measure α
inst✝ : ∀ (n : ι), IsFiniteMeasure (m n)
f : ι → ℕ
hf : Injective f
n : ℕ
hn : n ∉ range f
⊢ IsFiniteMeasure 0

Tactic:
]

Goal state after:
case inr
α : Type u_1
ι : Type u_3
m0 : MeasurableSpace α
inst✝¹ : Countable ι
m : ι → Measure α
inst✝ : ∀ (n : ι), IsFiniteMeasure (m n)
f : ι → ℕ
hf : Injective f
n : ℕ
hn : n ∉ range f
⊢ IsFiniteMeasure 0
Module: Mathlib.MeasureTheory.Measure.Typeclasses.SFinite

Goal state before:
pp: α : Type u_1
β : Type u_2
ι : Type u_3
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
μ ν : Measure α
s✝ t : Set α
a : α
inst✝ : SFinite μ
s : Set α
⊢ μ.restrict s = μ.restrict s

Tactic:
]

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
t : ℕ → Set α
hd : Directed (fun x1 x2 ↦ x1 ⊆ x2) t
T : ℕ → Set α := fun n ↦ toMeasurable μ (t n)
Td : ℕ → Set α := disjointed T
hm : ∀ (n : ℕ), MeasurableSet (Td n)
⊢ μ (⋃ n, t n) = μ (⋃ i, t i)

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
inst✝ : IsFiniteMeasure μ
hs : MeasurableSet s
ht : MeasurableSet t
ε : ℝ≥0∞
h : μ s ≤ μ t + ε
⊢ μ univ ≤ μ univ - μ s + ε + μ t

Tactic:
]

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
s t : Set α
inst✝ : IsFiniteMeasure μ
hs : MeasurableSet s
ht : MeasurableSet t
ε : ℝ≥0∞
h : μ s ≤ μ t + ε
⊢ μ univ ≤ μ univ - μ s + ε + μ t
Module: Mathlib.MeasureTheory.Measure.Typeclasses.Finite

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
s t : Set α
inst✝ : IsFiniteMeasure μ
hs : MeasurableSet s
ht : MeasurableSet t
ε : ℝ≥0∞
h : μ s ≤ μ t + ε
⊢ μ univ - μ s + (μ t + ε) = μ univ - μ s + (μ t + ε)

Tactic:
]

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

Goal state before:
pp: α : Type u_1
β : Type u_2
ι : Type u_3
m0 : MeasurableSpace α
inst✝ : MeasurableSpace β
μ✝ ν : Measure α
s t : Set α
a : α
μ : Measure α
h : SigmaFinite μ
n : ℕ
⊢ μ (⋯.some.set n) < ∞

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
ι : Type u_3
m0 : MeasurableSpace α
inst✝ : MeasurableSpace β
μ✝ ν : Measure α
s t : Set α
a : α
μ : Measure α
h : SigmaFinite μ
n : ℕ
⊢ μ (⋯.some.set n) < ∞
Module: Mathlib.MeasureTheory.Measure.Typeclasses.Finite

Goal state before:
pp: α : Type u_1
β : Type u_2
δ : Type u_3
ι : Type u_4
m0 : MeasurableSpace α
mβ : MeasurableSpace β
μ ν ν₁ ν₂ : Measure α
s t : Set α
inst✝ : IsEmpty α
⊢ IsFiniteMeasure 0

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
δ : Type u_3
ι : Type u_4
m0 : MeasurableSpace α
mβ : MeasurableSpace β
μ ν ν₁ ν₂ : Measure α
s t : Set α
inst✝ : IsEmpty α
⊢ IsFiniteMeasure 0
Module: Mathlib.MeasureTheory.Measure.Restrict

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

Tactic:
]

Goal state after:
α : Type u_2
m0 : MeasurableSpace α
μ : Measure α
s t : Set α
ht : NullMeasurableSet t ((liftLinear (OuterMeasure.restrict s) ⋯) μ)
⊢ (μ.restrict s) t = μ (t ∩ s)
Module: Mathlib.MeasureTheory.Measure.Typeclasses.Finite

Goal state before:
pp: α : Type u_1
β : Type u_2
δ : Type u_3
ι : Type u_4
m0 : MeasurableSpace α
mβ : MeasurableSpace β
μ ν ν₁ ν₂ : Measure α
s t : Set α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
⊢ μ univ < ∞ ∧ ν univ < ∞

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
δ : Type u_3
ι : Type u_4
m0 : MeasurableSpace α
mβ : MeasurableSpace β
μ ν ν₁ ν₂ : Measure α
s t : Set α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
⊢ μ univ < ∞ ∧ ν univ < ∞
Module: Mathlib.MeasureTheory.Measure.Restrict

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

Tactic:
]

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

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ ν : Measure α
inst✝ : SigmaFinite ν
⊢ μ = μ

Tactic:
]

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

Goal state before:
pp: α : Type u_1
β : Type u_2
δ : Type u_3
ι : Type u_4
m0 : MeasurableSpace α
mβ : MeasurableSpace β
μ ν ν₁ ν₂ : Measure α
s t : Set α
⊢ μ univ / μ univ < ∞

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
δ : Type u_3
ι : Type u_4
m0 : MeasurableSpace α
mβ : MeasurableSpace β
μ ν ν₁ ν₂ : Measure α
s t : Set α
⊢ μ univ / μ univ < ∞
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) = μ (t ∩ s)

Tactic:
]

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

Goal state before:
pp: case inr
α : Type u_1
ι : Type u_5
m : MeasurableSpace α
μ : Measure α
inst✝² : Preorder ι
inst✝¹ : IsDirectedOrder ι
inst✝ : atTop.IsCountablyGenerated
s : ι → Set α
hs : Monotone s
h✝ : Nonempty ι
x : ℕ → ι
hxm : Monotone x
hx : Tendsto x atTop atTop
⊢ μ (⋃ a, s (x a)) = ⨆ a, μ (s (x a))

pp: α : Type u_1
ι : Type u_5
m : MeasurableSpace α
μ : Measure α
inst✝² : Preorder ι
inst✝¹ : IsDirectedOrder ι
inst✝ : atTop.IsCountablyGenerated
s : ι → Set α
hs : Monotone s
h✝ : Nonempty ι
x : ℕ → ι
hxm : Monotone x
hx : Tendsto x atTop atTop
⊢ Monotone fun i ↦ μ (s i)

Tactic:
]

Goal state after:
case inr
α : Type u_1
ι : Type u_5
m : MeasurableSpace α
μ : Measure α
inst✝² : Preorder ι
inst✝¹ : IsDirectedOrder ι
inst✝ : atTop.IsCountablyGenerated
s : ι → Set α
hs : Monotone s
h✝ : Nonempty ι
x : ℕ → ι
hxm : Monotone x
hx : Tendsto x atTop atTop
⊢ μ (⋃ a, s (x a)) = ⨆ a, μ (s (x a))

α : Type u_1
ι : Type u_5
m : MeasurableSpace α
μ : Measure α
inst✝² : Preorder ι
inst✝¹ : IsDirectedOrder ι
inst✝ : atTop.IsCountablyGenerated
s : ι → Set α
hs : Monotone s
h✝ : Nonempty ι
x : ℕ → ι
hxm : Monotone x
hx : Tendsto x atTop atTop
⊢ Monotone fun i ↦ μ (s i)
Module: Mathlib.MeasureTheory.Measure.Typeclasses.Finite

Goal state before:
pp: α : Type u_1
β : Type u_2
δ : Type u_3
ι : Type u_4
m0 : MeasurableSpace α
mβ : MeasurableSpace β
μ ν ν₁ ν₂ : Measure α
s t : Set α
R : Type u_5
inst✝⁴ : SMul R ℝ≥0
inst✝³ : SMul R ℝ≥0∞
inst✝² : IsScalarTower R ℝ≥0 ℝ≥0∞
inst✝¹ : IsScalarTower R ℝ≥0∞ ℝ≥0∞
inst✝ : IsFiniteMeasure μ
r : R
⊢ IsFiniteMeasure ((r • 1) • μ)

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
δ : Type u_3
ι : Type u_4
m0 : MeasurableSpace α
mβ : MeasurableSpace β
μ ν ν₁ ν₂ : Measure α
s t : Set α
R : Type u_5
inst✝⁴ : SMul R ℝ≥0
inst✝³ : SMul R ℝ≥0∞
inst✝² : IsScalarTower R ℝ≥0 ℝ≥0∞
inst✝¹ : IsScalarTower R ℝ≥0∞ ℝ≥0∞
inst✝ : IsFiniteMeasure μ
r : R
⊢ IsFiniteMeasure ((r • 1) • μ)
Module: Mathlib.MeasureTheory.Measure.Typeclasses.SFinite

Goal state before:
pp: α : Type u_1
inst✝¹ : MeasurableSpace α
μ : Measure α
inst✝ : SigmaFinite μ
s : Set α
⊢ s = s

Tactic:
]

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

Goal state before:
pp: α : Type u_1
inst✝¹ : MeasurableSpace α
μ : Measure α
inst✝ : SigmaFinite μ
s : Set α
⊢ (∀ (n : ℕ), μ (s ∩ spanningSets μ n) = 0) ↔ μ (⋃ n, s ∩ spanningSets μ n) = 0

Tactic:
]

Goal state after:
α : Type u_1
inst✝¹ : MeasurableSpace α
μ : Measure α
inst✝ : SigmaFinite μ
s : Set α
⊢ (∀ (n : ℕ), μ (s ∩ spanningSets μ n) = 0) ↔ μ (⋃ n, s ∩ spanningSets μ n) = 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✝² : Preorder ι
inst✝¹ : IsDirectedOrder ι
inst✝ : atTop.IsCountablyGenerated
s : ι → Set α
hs : Monotone s
x✝ : IsEmpty ι ∨ Nonempty ι
⊢ μ (⋃ i, s i) = ⨆ i, μ (s i)

Tactic:
cases isEmpty_or_nonempty ι with

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

Goal state before:
pp: α : Type u_1
inst✝¹ : MeasurableSpace α
μ : Measure α
inst✝ : SigmaFinite μ
s : Set α
⊢ (∀ (n : ℕ), μ (s ∩ spanningSets μ n) = 0) ↔ ∀ (i : ℕ), μ (s ∩ spanningSets μ i) = 0

Tactic:
]

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

Goal state before:
pp: case pos
α : Type u_1
β : Type u_2
mβ : MeasurableSpace β
m : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
f : α → β
hf : AEMeasurable f μ
⊢ μ (f ⁻¹' univ) < ∞

Tactic:
]

Goal state after:
case pos
α : Type u_1
β : Type u_2
mβ : MeasurableSpace β
m : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
f : α → β
hf : AEMeasurable f μ
⊢ μ (f ⁻¹' univ) < ∞
Module: Mathlib.MeasureTheory.Measure.Restrict

Goal state before:
pp: α : Type u_2
m0 : MeasurableSpace α
μ : Measure α
s : Set α
C : Set (Set α)
ht : ∀ t ∈ C, μ (s ∩ t) = 0
t : ℕ → Set α
ht1 : ∀ (n : ℕ), t n ∈ C
ht2 : ⋃ n, t n = univ
⊢ s = 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 : IsDirectedOrder ι
inst✝ : atTop.IsCountablyGenerated
f : ι → Set α
⊢ μ (⋃ x, accumulate f x) = ⨆ i, μ (accumulate f 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 : IsDirectedOrder ι
inst✝ : atTop.IsCountablyGenerated
f : ι → Set α
⊢ μ (⋃ x, accumulate f x) = ⨆ i, μ (accumulate f i)
Module: Mathlib.MeasureTheory.Measure.Typeclasses.SFinite

Goal state before:
pp: α : Type u_1
inst✝¹ : MeasurableSpace α
μ : Measure α
inst✝ : SigmaFinite μ
s : Set α
⊢ (∀ (n : ℕ), μ (s ∩ spanningSets μ n) ≤ 0) ↔ ∀ t ∈ range (spanningSets μ), μ (s ∩ t) = 0

Tactic:
]

Goal state after:
α : Type u_1
inst✝¹ : MeasurableSpace α
μ : Measure α
inst✝ : SigmaFinite μ
s : Set α
⊢ (∀ (n : ℕ), μ (s ∩ spanningSets μ n) ≤ 0) ↔ ∀ t ∈ range (spanningSets μ), μ (s ∩ t) = 0
Module: Mathlib.MeasureTheory.Measure.Typeclasses.Finite

Goal state before:
pp: case neg
α : Type u_1
β : Type u_2
mβ : MeasurableSpace β
m : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
f : α → β
hf : ¬AEMeasurable f μ
⊢ IsFiniteMeasure 0

Tactic:
]

Goal state after:
case neg
α : Type u_1
β : Type u_2
mβ : MeasurableSpace β
m : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
f : α → β
hf : ¬AEMeasurable f μ
⊢ IsFiniteMeasure 0
Module: Mathlib.MeasureTheory.Measure.Typeclasses.Finite

Goal state before:
pp: α : Type u_1
β : Type u_2
m0 : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure α
f : α → β
hf : AEMeasurable f μ
inst✝ : IsFiniteMeasure (map f μ)
⊢ (map f μ) univ < ∞

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
m0 : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure α
f : α → β
hf : AEMeasurable f μ
inst✝ : IsFiniteMeasure (map f μ)
⊢ (map f μ) univ < ∞
Module: Mathlib.MeasureTheory.Measure.Typeclasses.Finite

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
⊢ measureUnivNNReal μ = 0 ↔ ↑(measureUnivNNReal μ) = 0

Tactic:
]

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
⊢ measureUnivNNReal μ = 0 ↔ ↑(measureUnivNNReal μ) = 0
Module: Mathlib.MeasureTheory.Measure.MeasureSpace

Goal state before:
pp: α : Type u_1
ι : Type u_5
m : MeasurableSpace α
μ : Measure α
s : ι → Set α
t : Set ι
ht : t.Countable
hd : DirectedOn ((fun x1 x2 ↦ x1 ⊆ x2) on s) t
this : Countable ↑t
⊢ ⨆ i, μ (s ↑i) = ⨆ i, μ (s ↑i)

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
C : Set (Set α)
ht : ∀ t ∈ C, μ (s ∩ t) = 0
t : ℕ → Set α
ht1 : ∀ (n : ℕ), t n ∈ C
ht2 : ⋃ n, t n = univ
⊢ ∀ (i : ℕ), μ (s ∩ t i) = 0

Tactic:
]

Goal state after:
α : Type u_2
m0 : MeasurableSpace α
μ : Measure α
s : Set α
C : Set (Set α)
ht : ∀ t ∈ C, μ (s ∩ t) = 0
t : ℕ → Set α
ht1 : ∀ (n : ℕ), t n ∈ C
ht2 : ⋃ n, t n = univ
⊢ ∀ (i : ℕ), μ (s ∩ t i) = 0
Module: Mathlib.MeasureTheory.Measure.Typeclasses.Finite

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ ν : Measure α
inst✝ : IsFiniteMeasure μ
hμν : μ ≤ ν
s : Set α
h_univ : μ s + μ sᶜ = ν s + ν sᶜ
hs : MeasurableSet s
x✝ : s.Nonempty
h_lt : μ s < ν s
h_disj : Disjoint s sᶜ
⊢ False

Tactic:
]

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ ν : Measure α
inst✝ : IsFiniteMeasure μ
hμν : μ ≤ ν
s : Set α
h_univ : μ s + μ sᶜ = ν s + ν sᶜ
hs : MeasurableSet s
x✝ : s.Nonempty
h_lt : μ s < ν s
h_disj : Disjoint s sᶜ
⊢ False
Module: Mathlib.MeasureTheory.Measure.Restrict

Goal state before:
pp: α : Type u_2
m0 : MeasurableSpace α
μ : Measure α
s : Set α
C : Set (Set α)
hC : IsCountablySpanning C
hm : ∀ t ∈ C, MeasurableSet t
ht : ∀ t ∈ C, (μ.restrict t) s = 0
⊢ ∀ t ∈ C, μ (s ∩ t) = 0

Tactic:
]

Goal state after:
α : Type u_2
m0 : MeasurableSpace α
μ : Measure α
s : Set α
C : Set (Set α)
hC : IsCountablySpanning C
hm : ∀ t ∈ C, MeasurableSet t
ht : ∀ t ∈ C, (μ.restrict t) s = 0
⊢ ∀ t ∈ C, μ (s ∩ t) = 0
Module: Mathlib.MeasureTheory.Measure.Typeclasses.Finite

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
hμ : IsFiniteMeasure μ
f : ℕ → Set α
hf₁ : ∀ (i : ℕ), MeasurableSet (f i)
hf₂ : Pairwise (Disjoint on f)
⊢ μ (⋃ i, f i) ≠ ∞

Tactic:
]

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
hμ : IsFiniteMeasure μ
f : ℕ → Set α
hf₁ : ∀ (i : ℕ), MeasurableSet (f i)
hf₂ : Pairwise (Disjoint on f)
⊢ μ (⋃ i, 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✝ : 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 ≠ ∞
⊢ μ (s k) - ⨆ i, μ (s k \ s i) = μ (s k) - ⨆ i, μ (s k) - μ (s i)

pp: 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 ≠ ∞
⊢ Directed (fun x1 x2 ↦ x1 ⊆ x2) fun i ↦ s k \ s i

Tactic:
]

Goal state after:
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 ≠ ∞
⊢ μ (s k) - ⨆ i, μ (s k \ s i) = μ (s k) - ⨆ i, μ (s k) - μ (s i)

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 ≠ ∞
⊢ Directed (fun x1 x2 ↦ x1 ⊆ x2) fun i ↦ s k \ s i
Module: Mathlib.MeasureTheory.Measure.Typeclasses.Finite

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
p : α → Prop
hp : NullMeasurableSet {a | p a} μ
⊢ {x | p x} ∈ ae μ ↔ {a | p a} ∈ ae μ

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
As : ι → Set α
As_mble : ∀ (i : ι), NullMeasurableSet (As i) μ
As_disj : Pairwise (AEDisjoint μ on As)
Union_As_finite : μ (⋃ i, As i) ≠ ∞
posmeas : Set ι := {i | 0 < μ (As i)}
posmeas_def : posmeas = {i | 0 < μ (As i)}
as : ℕ → ℝ≥0∞
left✝ : StrictAnti as
as_mem : ∀ (n : ℕ), as n ∈ Ioo 0 1
as_lim : Tendsto as atTop (𝓝 0)
fairmeas : ℕ → Set ι := fun n ↦ {i | as n ≤ μ (As i)}
countable_union : posmeas = ⋃ n, fairmeas n
⊢ (⋃ n, fairmeas n).Countable

Tactic:
]

Goal state after:
α : Type u_1
ι : Type u_4
x✝ : MeasurableSpace α
μ : Measure α
As : ι → Set α
As_mble : ∀ (i : ι), NullMeasurableSet (As i) μ
As_disj : Pairwise (AEDisjoint μ on As)
Union_As_finite : μ (⋃ i, As i) ≠ ∞
posmeas : Set ι := {i | 0 < μ (As i)}
posmeas_def : posmeas = {i | 0 < μ (As i)}
as : ℕ → ℝ≥0∞
left✝ : StrictAnti as
as_mem : ∀ (n : ℕ), as n ∈ Ioo 0 1
as_lim : Tendsto as atTop (𝓝 0)
fairmeas : ℕ → Set ι := fun n ↦ {i | as n ≤ μ (As i)}
countable_union : posmeas = ⋃ n, fairmeas n
⊢ (⋃ n, fairmeas n).Countable
Module: Mathlib.MeasureTheory.Measure.Restrict

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

Tactic:
]

Goal state after: