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

Goal state before:
pp: case pos
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, μ (t i) = ∞
⊢ μ (⋃ i, t i) ≤ μ (⋃ i, s i)

pp: case neg
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) ≠ ∞
⊢ μ (⋃ i, t i) ≤ μ (⋃ i, s i)

Tactic:
·

Goal state after:
case pos
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, μ (t i) = ∞
⊢ μ (⋃ i, t i) ≤ μ (⋃ i, s i)
Module: Mathlib.MeasureTheory.Measure.MeasureSpace

Goal state before:
pp: case refine_1
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 (M (t b)) ≤ μ (M (t b) ∩ M (⋃ b, s b))

pp: case refine_2
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
⊢ NullMeasurableSet (M (t b) ∩ M (⋃ b, s b)) μ

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 (M (t b)) ≠ ∞

Tactic:
·

Goal state after:
case refine_1
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 (M (t b)) ≤ μ (M (t b) ∩ M (⋃ b, s b))
Module: Mathlib.MeasureTheory.Measure.MeasureSpace

Goal state before:
pp: case refine_2
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
⊢ NullMeasurableSet (M (t b) ∩ M (⋃ b, s b)) μ

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 (M (t b)) ≠ ∞

Tactic:
·

Goal state after:
case refine_2
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
⊢ NullMeasurableSet (M (t b) ∩ M (⋃ b, s b)) μ
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 (M (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 (M (t b)) ≠ ∞
Module: Mathlib.Topology.Order.AtTopBotIxx

Goal state before:
pp: case inl
X : Type u_1
inst✝⁶ : LinearOrder X
inst✝⁵ : TopologicalSpace X
inst✝⁴ : OrderTopology X
α : Type u_2
inst✝³ : LinearOrder α
inst✝² : LocallyFiniteOrder α
inst✝¹ : NoMaxOrder X
inst✝ : NoMinOrder X
f g : α → X
hl : Tendsto f atTop atTop
hu : Tendsto g atBot atBot
x : X
h✝ : IsEmpty α
⊢ ∃ t ∈ 𝓝 x, {i | ((fun n ↦ Icc (f n) (g n)) i ∩ t).Nonempty}.Finite

pp: case inr
X : Type u_1
inst✝⁶ : LinearOrder X
inst✝⁵ : TopologicalSpace X
inst✝⁴ : OrderTopology X
α : Type u_2
inst✝³ : LinearOrder α
inst✝² : LocallyFiniteOrder α
inst✝¹ : NoMaxOrder X
inst✝ : NoMinOrder X
f g : α → X
hl : Tendsto f atTop atTop
hu : Tendsto g atBot atBot
x : X
h✝ : Nonempty α
⊢ ∃ t ∈ 𝓝 x, {i | ((fun n ↦ Icc (f n) (g n)) i ∩ t).Nonempty}.Finite

Tactic:
·

Goal state after:
case inl
X : Type u_1
inst✝⁶ : LinearOrder X
inst✝⁵ : TopologicalSpace X
inst✝⁴ : OrderTopology X
α : Type u_2
inst✝³ : LinearOrder α
inst✝² : LocallyFiniteOrder α
inst✝¹ : NoMaxOrder X
inst✝ : NoMinOrder X
f g : α → X
hl : Tendsto f atTop atTop
hu : Tendsto g atBot atBot
x : X
h✝ : IsEmpty α
⊢ ∃ t ∈ 𝓝 x, {i | ((fun n ↦ Icc (f n) (g n)) i ∩ t).Nonempty}.Finite
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)
⊢ ∑' (i : ι), μ (s i) ≤ μ univ

Tactic:
rw [ENNReal.tsum_eq_iSup_sum]

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.Topology.Order.AtTopBotIxx

Goal state before:
pp: case inr.left
X : Type u_1
inst✝⁶ : LinearOrder X
inst✝⁵ : TopologicalSpace X
inst✝⁴ : OrderTopology X
α : Type u_2
inst✝³ : LinearOrder α
inst✝² : LocallyFiniteOrder α
inst✝¹ : NoMaxOrder X
inst✝ : NoMinOrder X
f g : α → X
hl : Tendsto f atTop atTop
hu : Tendsto g atBot atBot
x : X
h✝ : Nonempty α
x_L : X
hx_L : x_L < x
x_R : X
hx_R : x < x_R
a_L : α
ha_L : ∀ a ≤ a_L, g a ≤ x_L
a_R : α
ha_R : ∀ a ≥ a_R, x_R ≤ f a
n : α
y : X
hf : f n ≤ y
hg : y ≤ g n
hxL : x_L < y
hxR : y < x_R
⊢ a_L ≤ n

pp: case inr.right
X : Type u_1
inst✝⁶ : LinearOrder X
inst✝⁵ : TopologicalSpace X
inst✝⁴ : OrderTopology X
α : Type u_2
inst✝³ : LinearOrder α
inst✝² : LocallyFiniteOrder α
inst✝¹ : NoMaxOrder X
inst✝ : NoMinOrder X
f g : α → X
hl : Tendsto f atTop atTop
hu : Tendsto g atBot atBot
x : X
h✝ : Nonempty α
x_L : X
hx_L : x_L < x
x_R : X
hx_R : x < x_R
a_L : α
ha_L : ∀ a ≤ a_L, g a ≤ x_L
a_R : α
ha_R : ∀ a ≥ a_R, x_R ≤ f a
n : α
y : X
hf : f n ≤ y
hg : y ≤ g n
hxL : x_L < y
hxR : y < x_R
⊢ n ≤ a_R

Tactic:
·

Goal state after:
case inr.left
X : Type u_1
inst✝⁶ : LinearOrder X
inst✝⁵ : TopologicalSpace X
inst✝⁴ : OrderTopology X
α : Type u_2
inst✝³ : LinearOrder α
inst✝² : LocallyFiniteOrder α
inst✝¹ : NoMaxOrder X
inst✝ : NoMinOrder X
f g : α → X
hl : Tendsto f atTop atTop
hu : Tendsto g atBot atBot
x : X
h✝ : Nonempty α
x_L : X
hx_L : x_L < x
x_R : X
hx_R : x < x_R
a_L : α
ha_L : ∀ a ≤ a_L, g a ≤ x_L
a_R : α
ha_R : ∀ a ≥ a_R, x_R ≤ f a
n : α
y : X
hf : f n ≤ y
hg : y ≤ g n
hxL : x_L < y
hxR : y < x_R
⊢ a_L ≤ n
Module: Mathlib.Topology.Order.AtTopBotIxx

Goal state before:
pp: case inr.right
X : Type u_1
inst✝⁶ : LinearOrder X
inst✝⁵ : TopologicalSpace X
inst✝⁴ : OrderTopology X
α : Type u_2
inst✝³ : LinearOrder α
inst✝² : LocallyFiniteOrder α
inst✝¹ : NoMaxOrder X
inst✝ : NoMinOrder X
f g : α → X
hl : Tendsto f atTop atTop
hu : Tendsto g atBot atBot
x : X
h✝ : Nonempty α
x_L : X
hx_L : x_L < x
x_R : X
hx_R : x < x_R
a_L : α
ha_L : ∀ a ≤ a_L, g a ≤ x_L
a_R : α
ha_R : ∀ a ≥ a_R, x_R ≤ f a
n : α
y : X
hf : f n ≤ y
hg : y ≤ g n
hxL : x_L < y
hxR : y < x_R
⊢ n ≤ a_R

Tactic:
·

Goal state after:
case inr.right
X : Type u_1
inst✝⁶ : LinearOrder X
inst✝⁵ : TopologicalSpace X
inst✝⁴ : OrderTopology X
α : Type u_2
inst✝³ : LinearOrder α
inst✝² : LocallyFiniteOrder α
inst✝¹ : NoMaxOrder X
inst✝ : NoMinOrder X
f g : α → X
hl : Tendsto f atTop atTop
hu : Tendsto g atBot atBot
x : X
h✝ : Nonempty α
x_L : X
hx_L : x_L < x
x_R : X
hx_R : x < x_R
a_L : α
ha_L : ∀ a ≤ a_L, g a ≤ x_L
a_R : α
ha_R : ∀ a ≥ a_R, x_R ≤ f a
n : α
y : X
hf : f n ≤ y
hg : y ≤ g n
hxL : x_L < y
hxR : y < x_R
⊢ n ≤ a_R
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 (Function.extend f m 0 (f i))

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 (Function.extend f m 0 n)

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 (Function.extend f m 0 (f 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 (Function.extend f m 0 n)

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 (Function.extend f m 0 n)
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 (⋃ i, s i) = ⨆ i, μ (s i)

Tactic:
← hs.iUnion_comp_tendsto_atTop hx,

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)) = ⨆ i, μ (s i)
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
⊢ IsFiniteMeasure (map f μ)

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 (map f μ)

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
⊢ IsFiniteMeasure (map f μ)
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 (map f μ)

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 (map f μ)
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)
Module: Mathlib.MeasureTheory.Measure.MeasureSpace

Goal state before:
pp: case e_a
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 : ι
⊢ ∃ i', μ (s k \ s i) ≤ μ (s k) - μ (s i')

Tactic:
rcases hd i k with ⟨j, hji, hjk⟩

Goal state after:
case e_a
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
⊢ ∃ i', μ (s k \ s i) ≤ μ (s k) - 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✝ : 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 ≠ ∞
⊢ Directed (fun x1 x2 ↦ x1 ⊆ x2) fun i ↦ s k \ s i
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
Module: Mathlib.MeasureTheory.Measure.Typeclasses.SFinite

Goal state before:
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:
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.Typeclasses.SFinite

Goal state before:
pp: case As_mble
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)
obs : {i | 0 < (sum (sfiniteSeq μ)) (As i)} ⊆ ⋃ n, {i | 0 < (sfiniteSeq μ n) (As i)}
n : ℕ
⊢ ∀ (i : ι), NullMeasurableSet (As i) (sfiniteSeq μ n)

pp: case As_disj
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)
obs : {i | 0 < (sum (sfiniteSeq μ)) (As i)} ⊆ ⋃ n, {i | 0 < (sfiniteSeq μ n) (As i)}
n : ℕ
⊢ Pairwise (AEDisjoint (sfiniteSeq μ n) on As)

pp: case Union_As_finite
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)
obs : {i | 0 < (sum (sfiniteSeq μ)) (As i)} ⊆ ⋃ n, {i | 0 < (sfiniteSeq μ n) (As i)}
n : ℕ
⊢ (sfiniteSeq μ n) (⋃ i, As i) ≠ ∞

Tactic:
·

Goal state after:
case As_mble
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)
obs : {i | 0 < (sum (sfiniteSeq μ)) (As i)} ⊆ ⋃ n, {i | 0 < (sfiniteSeq μ n) (As i)}
n : ℕ
⊢ ∀ (i : ι), NullMeasurableSet (As i) (sfiniteSeq μ n)
Module: Mathlib.MeasureTheory.Measure.Typeclasses.SFinite

Goal state before:
pp: case As_disj
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)
obs : {i | 0 < (sum (sfiniteSeq μ)) (As i)} ⊆ ⋃ n, {i | 0 < (sfiniteSeq μ n) (As i)}
n : ℕ
⊢ Pairwise (AEDisjoint (sfiniteSeq μ n) on As)

pp: case Union_As_finite
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)
obs : {i | 0 < (sum (sfiniteSeq μ)) (As i)} ⊆ ⋃ n, {i | 0 < (sfiniteSeq μ n) (As i)}
n : ℕ
⊢ (sfiniteSeq μ n) (⋃ i, As i) ≠ ∞

Tactic:
·

Goal state after:
case As_disj
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)
obs : {i | 0 < (sum (sfiniteSeq μ)) (As i)} ⊆ ⋃ n, {i | 0 < (sfiniteSeq μ n) (As i)}
n : ℕ
⊢ Pairwise (AEDisjoint (sfiniteSeq μ n) on As)
Module: Mathlib.MeasureTheory.Measure.Typeclasses.SFinite

Goal state before:
pp: case Union_As_finite
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)
obs : {i | 0 < (sum (sfiniteSeq μ)) (As i)} ⊆ ⋃ n, {i | 0 < (sfiniteSeq μ n) (As i)}
n : ℕ
⊢ (sfiniteSeq μ n) (⋃ i, As i) ≠ ∞

Tactic:
·

Goal state after:
case Union_As_finite
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)
obs : {i | 0 < (sum (sfiniteSeq μ)) (As i)} ⊆ ⋃ n, {i | 0 < (sfiniteSeq μ n) (As i)}
n : ℕ
⊢ (sfiniteSeq μ n) (⋃ i, As i) ≠ ∞
Module: Mathlib.MeasureTheory.Measure.Typeclasses.SFinite

Goal state before:
pp: case inl
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
C : Set (Set α)
hC : ∀ s ∈ C, MeasurableSet s
m : ℝ≥0∞ := ⨆ D ∈ {D | D ⊆ C ∧ D.Countable}, μ (⋃₀ D)
hm : m = ⊥
⊢ ∃ D ∈ {D | D ⊆ C ∧ D.Countable}, μ (⋃₀ D) = m

pp: case inr
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
C : Set (Set α)
hC : ∀ s ∈ C, MeasurableSet s
m : ℝ≥0∞ := ⨆ D ∈ {D | D ⊆ C ∧ D.Countable}, μ (⋃₀ D)
hm : ⊥ < m
⊢ ∃ D ∈ {D | D ⊆ C ∧ D.Countable}, μ (⋃₀ D) = m

Tactic:
·

Goal state after:
case inl
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
C : Set (Set α)
hC : ∀ s ∈ C, MeasurableSet s
m : ℝ≥0∞ := ⨆ D ∈ {D | D ⊆ C ∧ D.Countable}, μ (⋃₀ D)
hm : m = ⊥
⊢ ∃ D ∈ {D | D ⊆ C ∧ D.Countable}, μ (⋃₀ D) = m
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)
Module: Mathlib.MeasureTheory.Measure.MeasureSpace

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

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

Goal state before:
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:
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) ≠ ∞