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

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ ν : Measure α
s : Set α
hs : MeasurableSet s
t' : ℕ → Set α
ht' : s ⊆ iUnion t'
t : Set α := ⋃ n ∈ {k | μ (t' k) ≤ ν (t' k)}, t' n
ht : t = ⋃ n ∈ {k | μ (t' k) ≤ ν (t' k)}, t' n
hle₁ : μ (t ∩ s) ≤ ∑' (n : ↑{k | μ (t' k) ≤ ν (t' k)}), μ (t' ↑n)
hcap : tᶜ ∩ s ⊆ ⋃ n ∈ {k | ν (t' k) < μ (t' k)}, t' n
hle₂ : ν (tᶜ ∩ s) ≤ ∑' (n : ↑{k | ν (t' k) < μ (t' k)}), ν (t' ↑n)
heq : {k | μ (t' k) ≤ ν (t' k)} ∪ {k | ν (t' k) < μ (t' k)} = univ
k : ℕ
hk₁ : μ (t' k) ≤ ν (t' k)
hk₂ : ν (t' k) < μ (t' k)
⊢ k ∈ ∅

Tactic:
]

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ ν : Measure α
s : Set α
hs : MeasurableSet s
t' : ℕ → Set α
ht' : s ⊆ iUnion t'
t : Set α := ⋃ n ∈ {k | μ (t' k) ≤ ν (t' k)}, t' n
ht : t = ⋃ n ∈ {k | μ (t' k) ≤ ν (t' k)}, t' n
hle₁ : μ (t ∩ s) ≤ ∑' (n : ↑{k | μ (t' k) ≤ ν (t' k)}), μ (t' ↑n)
hcap : tᶜ ∩ s ⊆ ⋃ n ∈ {k | ν (t' k) < μ (t' k)}, t' n
hle₂ : ν (tᶜ ∩ s) ≤ ∑' (n : ↑{k | ν (t' k) < μ (t' k)}), ν (t' ↑n)
heq : {k | μ (t' k) ≤ ν (t' k)} ∪ {k | ν (t' k) < μ (t' k)} = univ
k : ℕ
hk₁ : μ (t' k) ≤ ν (t' k)
hk₂ : ν (t' k) < μ (t' k)
⊢ k ∈ ∅
Module: Mathlib.MeasureTheory.Measure.MeasureSpace

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Type u_5
R : Type u_6
R' : Type u_7
m0 : MeasurableSpace α
mβ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s s' t : Set α
⊢ m0 ≤ ⊤

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Type u_5
R : Type u_6
R' : Type u_7
m0 : MeasurableSpace α
mβ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s s' t : Set α
⊢ m0 ≤ ⊤
Module: Mathlib.MeasureTheory.Measure.AEMeasurable

Goal state before:
pp: case refine_1
α : Type u_2
β : Type u_3
m0 : MeasurableSpace α
μ : Measure α
f : α → β
a✝ : Nontrivial β
inhabited_h : Inhabited β
S : Set (Set β)
hSc : S.Countable
hc : MeasurableSpace.CountablyGenerated β
h : NullMeasurable f μ
T : Set β → Set α
hTf : ∀ s ∈ S, T s ⊆ f ⁻¹' s
hTm : ∀ s ∈ S, MeasurableSet (T s)
hTeq : ∀ s ∈ S, T s =ᵐ[μ] f ⁻¹' s
U : Set β → Set α
hUf : ∀ s ∈ S, U s ⊇ f ⁻¹' s
hUm : ∀ s ∈ S, MeasurableSet (U s)
hUeq : ∀ s ∈ S, U s =ᵐ[μ] f ⁻¹' s
v : Set α := ⋃ s ∈ S, U s \ T s
hvm : MeasurableSet v
hvμ : μ v = 0
⊢ Measurable (v.domRestrict fun x ↦ default)

Tactic:
]

Goal state after:
case refine_1
α : Type u_2
β : Type u_3
m0 : MeasurableSpace α
μ : Measure α
f : α → β
a✝ : Nontrivial β
inhabited_h : Inhabited β
S : Set (Set β)
hSc : S.Countable
hc : MeasurableSpace.CountablyGenerated β
h : NullMeasurable f μ
T : Set β → Set α
hTf : ∀ s ∈ S, T s ⊆ f ⁻¹' s
hTm : ∀ s ∈ S, MeasurableSet (T s)
hTeq : ∀ s ∈ S, T s =ᵐ[μ] f ⁻¹' s
U : Set β → Set α
hUf : ∀ s ∈ S, U s ⊇ f ⁻¹' s
hUm : ∀ s ∈ S, MeasurableSet (U s)
hUeq : ∀ s ∈ S, U s =ᵐ[μ] f ⁻¹' s
v : Set α := ⋃ s ∈ S, U s \ T s
hvm : MeasurableSet v
hvμ : μ v = 0
⊢ Measurable (v.domRestrict fun x ↦ default)
Module: Mathlib.MeasureTheory.Measure.MeasureSpace

Goal state before:
pp: α : Type u_1
ι : Type u_5
m0 : MeasurableSpace α
ι' : Type u_8
μ : ι → ι' → Measure α
s : Set α
hs : MeasurableSet s
⊢ ∑' (b : ι') (a : ι), (μ a b) s = ∑' (i : ι') (i_1 : ι), (μ i_1 i) s

Tactic:
]

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

Goal state before:
pp: case refine_2
α : Type u_2
β : Type u_3
m0 : MeasurableSpace α
μ : Measure α
f : α → β
a✝ : Nontrivial β
inhabited_h : Inhabited β
S : Set (Set β)
hSc : S.Countable
hc : MeasurableSpace.CountablyGenerated β
h : NullMeasurable f μ
T : Set β → Set α
hTf : ∀ s ∈ S, T s ⊆ f ⁻¹' s
hTm : ∀ s ∈ S, MeasurableSet (T s)
hTeq : ∀ s ∈ S, T s =ᵐ[μ] f ⁻¹' s
U : Set β → Set α
hUf : ∀ s ∈ S, U s ⊇ f ⁻¹' s
hUm : ∀ s ∈ S, MeasurableSet (U s)
hUeq : ∀ s ∈ S, U s =ᵐ[μ] f ⁻¹' s
v : Set α := ⋃ s ∈ S, U s \ T s
hvm : MeasurableSet v
hvμ : μ v = 0
⊢ Measurable (f ∘ Subtype.val)

Tactic:
]

Goal state after:
case refine_2
α : Type u_2
β : Type u_3
m0 : MeasurableSpace α
μ : Measure α
f : α → β
a✝ : Nontrivial β
inhabited_h : Inhabited β
S : Set (Set β)
hSc : S.Countable
hc : MeasurableSpace.CountablyGenerated β
h : NullMeasurable f μ
T : Set β → Set α
hTf : ∀ s ∈ S, T s ⊆ f ⁻¹' s
hTm : ∀ s ∈ S, MeasurableSet (T s)
hTeq : ∀ s ∈ S, T s =ᵐ[μ] f ⁻¹' s
U : Set β → Set α
hUf : ∀ s ∈ S, U s ⊇ f ⁻¹' s
hUm : ∀ s ∈ S, MeasurableSet (U s)
hUeq : ∀ s ∈ S, U s =ᵐ[μ] f ⁻¹' s
v : Set α := ⋃ s ∈ S, U s \ T s
hvm : MeasurableSet v
hvμ : μ v = 0
⊢ Measurable (f ∘ Subtype.val)
Module: Mathlib.MeasureTheory.Measure.MeasureSpace

Goal state before:
pp: α : Type u_1
ι : Type u_5
m0 : MeasurableSpace α
s : Finset ι
μ : ι → Measure α
⊢ ∑ i ∈ s, μ i = ∑ i ∈ s, μ i

Tactic:
]

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

Goal state before:
pp: α : Type u_1
ι : Type u_5
m0 : MeasurableSpace α
s : Finset ι
μ : ι → Measure α
⊢ (sum fun i ↦ μ ↑i) = ∑ i ∈ s, μ i

Tactic:
rw [sum_fintype, Finset.sum_coe_sort s μ]

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

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
f : Bool → Measure α
⊢ f true + f false = f true + f false

Tactic:
]

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

Goal state before:
pp: α : Type u_1
ι : Type u_5
m0 : MeasurableSpace α
inst✝ : IsEmpty ι
μ : ι → Measure α
⊢ 0 = 0

Tactic:
]

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

Goal state before:
pp: case refine_2
α : Type u_2
β : Type u_3
m0 : MeasurableSpace α
μ : Measure α
f : α → β
a✝ : Nontrivial β
inhabited_h : Inhabited β
S : Set (Set β)
hSc : S.Countable
hc : MeasurableSpace.CountablyGenerated β
h : NullMeasurable f μ
T : Set β → Set α
hTf : ∀ s ∈ S, T s ⊆ f ⁻¹' s
hTm : ∀ s ∈ S, MeasurableSet (T s)
hTeq : ∀ s ∈ S, T s =ᵐ[μ] f ⁻¹' s
U : Set β → Set α
hUf : ∀ s ∈ S, U s ⊇ f ⁻¹' s
hUm : ∀ s ∈ S, MeasurableSet (U s)
hUeq : ∀ s ∈ S, U s =ᵐ[μ] f ⁻¹' s
v : Set α := ⋃ s ∈ S, U s \ T s
hvm : MeasurableSet v
hvμ : μ v = 0
s : Set β
hs : s ∈ S
⊢ MeasurableSet (vᶜ ∩ f ⁻¹' s)

Tactic:
]

Goal state after:
case refine_2
α : Type u_2
β : Type u_3
m0 : MeasurableSpace α
μ : Measure α
f : α → β
a✝ : Nontrivial β
inhabited_h : Inhabited β
S : Set (Set β)
hSc : S.Countable
hc : MeasurableSpace.CountablyGenerated β
h : NullMeasurable f μ
T : Set β → Set α
hTf : ∀ s ∈ S, T s ⊆ f ⁻¹' s
hTm : ∀ s ∈ S, MeasurableSet (T s)
hTeq : ∀ s ∈ S, T s =ᵐ[μ] f ⁻¹' s
U : Set β → Set α
hUf : ∀ s ∈ S, U s ⊇ f ⁻¹' s
hUm : ∀ s ∈ S, MeasurableSet (U s)
hUeq : ∀ s ∈ S, U s =ᵐ[μ] f ⁻¹' s
v : Set α := ⋃ s ∈ S, U s \ T s
hvm : MeasurableSet v
hvμ : μ v = 0
s : Set β
hs : s ∈ S
⊢ MeasurableSet (vᶜ ∩ f ⁻¹' s)
Module: Mathlib.MeasureTheory.Measure.AEMeasurable

Goal state before:
pp: α : Type u_2
β : Type u_3
m0 : MeasurableSpace α
μ : Measure α
f : α → β
a✝ : Nontrivial β
inhabited_h : Inhabited β
S : Set (Set β)
hSc : S.Countable
hc : MeasurableSpace.CountablyGenerated β
h : NullMeasurable f μ
T : Set β → Set α
hTf : ∀ s ∈ S, T s ⊆ f ⁻¹' s
hTm : ∀ s ∈ S, MeasurableSet (T s)
hTeq : ∀ s ∈ S, T s =ᵐ[μ] f ⁻¹' s
U : Set β → Set α
hUf : ∀ s ∈ S, U s ⊇ f ⁻¹' s
hUm : ∀ s ∈ S, MeasurableSet (U s)
hUeq : ∀ s ∈ S, U s =ᵐ[μ] f ⁻¹' s
v : Set α := ⋃ s ∈ S, U s \ T s
hvm : MeasurableSet v
hvμ : μ v = 0
s : Set β
hs : s ∈ S
⊢ f ⁻¹' s ∩ vᶜ = T s \ v

Tactic:
]

Goal state after:
α : Type u_2
β : Type u_3
m0 : MeasurableSpace α
μ : Measure α
f : α → β
a✝ : Nontrivial β
inhabited_h : Inhabited β
S : Set (Set β)
hSc : S.Countable
hc : MeasurableSpace.CountablyGenerated β
h : NullMeasurable f μ
T : Set β → Set α
hTf : ∀ s ∈ S, T s ⊆ f ⁻¹' s
hTm : ∀ s ∈ S, MeasurableSet (T s)
hTeq : ∀ s ∈ S, T s =ᵐ[μ] f ⁻¹' s
U : Set β → Set α
hUf : ∀ s ∈ S, U s ⊇ f ⁻¹' s
hUm : ∀ s ∈ S, MeasurableSet (U s)
hUeq : ∀ s ∈ S, U s =ᵐ[μ] f ⁻¹' s
v : Set α := ⋃ s ∈ S, U s \ T s
hvm : MeasurableSet v
hvμ : μ v = 0
s : Set β
hs : s ∈ S
⊢ f ⁻¹' s ∩ vᶜ = T s \ v
Module: Mathlib.MeasureTheory.Measure.Restrict

Goal state before:
pp: α : Type u_2
ι : Type u_6
x✝ : MeasurableSpace α
μ : Measure α
s : ι → Set α
M : ℕ
hs_meas : ∀ (i : ι), MeasurableSet (s i)
hs : ∀ (y : α), {i | y ∈ s i}.encard ≤ ↑M
t : Set α
ht : MeasurableSet t
F : Finset ι
P : Finset ι → Set α := fun C ↦ (⋂ i ∈ C, s i) ∩ ⋂ i ∈ F \ C, (s i)ᶜ
Cs : Finset (Finset ι) := F.powerset \ {∅}
G : ι → Set (Finset ι) := fun i ↦ {C | C ∈ F.powerset ∧ i ∈ C}
P_meas : ∀ (C : Finset ι), MeasurableSet (P C)
P_cover : ∀ {i : ι}, i ∈ F → s i ⊆ ⋃ C ∈ G i, P C
iUnion_P : ⋃ C ∈ Cs, P C ⊆ ⋃ i, s i
P_subset_s : ∀ {i : ι} {C : Finset ι}, i ∈ C → P C ⊆ s i
mem_C : ∀ {i : ι}, i ∈ F → ∀ {C : Finset ι} {x : α}, x ∈ P C → x ∈ s i → i ∈ C
C_subset_C : ∀ {C₁ C₂ : Finset ι}, C₁ ∈ Cs → ∀ {x : α}, x ∈ P C₁ ∩ P C₂ → C₁ ⊆ C₂
⊢ M • ∑' (x : ↥Cs), (μ.restrict (P ↑x)) t = M • ∑' (i : ↥Cs), (μ.restrict (P ↑i)) t

pp: α : Type u_2
ι : Type u_6
x✝ : MeasurableSpace α
μ : Measure α
s : ι → Set α
M : ℕ
hs_meas : ∀ (i : ι), MeasurableSet (s i)
hs : ∀ (y : α), {i | y ∈ s i}.encard ≤ ↑M
t : Set α
ht : MeasurableSet t
F : Finset ι
P : Finset ι → Set α := fun C ↦ (⋂ i ∈ C, s i) ∩ ⋂ i ∈ F \ C, (s i)ᶜ
Cs : Finset (Finset ι) := F.powerset \ {∅}
G : ι → Set (Finset ι) := fun i ↦ {C | C ∈ F.powerset ∧ i ∈ C}
P_meas : ∀ (C : Finset ι), MeasurableSet (P C)
P_cover : ∀ {i : ι}, i ∈ F → s i ⊆ ⋃ C ∈ G i, P C
iUnion_P : ⋃ C ∈ Cs, P C ⊆ ⋃ i, s i
P_subset_s : ∀ {i : ι} {C : Finset ι}, i ∈ C → P C ⊆ s i
mem_C : ∀ {i : ι}, i ∈ F → ∀ {C : Finset ι} {x : α}, x ∈ P C → x ∈ s i → i ∈ C
C_subset_C : ∀ {C₁ C₂ : Finset ι}, C₁ ∈ Cs → ∀ {x : α}, x ∈ P C₁ ∩ P C₂ → C₁ ⊆ C₂
⊢ (↑Cs).Pairwise (Disjoint on P)

Tactic:
]

Goal state after:
α : Type u_2
ι : Type u_6
x✝ : MeasurableSpace α
μ : Measure α
s : ι → Set α
M : ℕ
hs_meas : ∀ (i : ι), MeasurableSet (s i)
hs : ∀ (y : α), {i | y ∈ s i}.encard ≤ ↑M
t : Set α
ht : MeasurableSet t
F : Finset ι
P : Finset ι → Set α := fun C ↦ (⋂ i ∈ C, s i) ∩ ⋂ i ∈ F \ C, (s i)ᶜ
Cs : Finset (Finset ι) := F.powerset \ {∅}
G : ι → Set (Finset ι) := fun i ↦ {C | C ∈ F.powerset ∧ i ∈ C}
P_meas : ∀ (C : Finset ι), MeasurableSet (P C)
P_cover : ∀ {i : ι}, i ∈ F → s i ⊆ ⋃ C ∈ G i, P C
iUnion_P : ⋃ C ∈ Cs, P C ⊆ ⋃ i, s i
P_subset_s : ∀ {i : ι} {C : Finset ι}, i ∈ C → P C ⊆ s i
mem_C : ∀ {i : ι}, i ∈ F → ∀ {C : Finset ι} {x : α}, x ∈ P C → x ∈ s i → i ∈ C
C_subset_C : ∀ {C₁ C₂ : Finset ι}, C₁ ∈ Cs → ∀ {x : α}, x ∈ P C₁ ∩ P C₂ → C₁ ⊆ C₂
⊢ (↑Cs).Pairwise (Disjoint on P)
Module: Mathlib.MeasureTheory.Measure.MeasureSpace

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
s : Set α
⊢ μ s < ∞ ↔ μ s < ∞

Tactic:
]

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

Goal state before:
pp: case inr
α : Type u_2
β : Type u_3
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
μ : Measure α
f : α → β
t : Set β
inst✝ : MeasurableSpace.CountablyGenerated ↑t
h : NullMeasurable f μ
hft : ∀ᵐ (x : α) ∂μ.completion, f x ∈ t
hne : t.Nonempty
⊢ AEMeasurable f μ

Tactic:
]

Goal state after:
case inr
α : Type u_2
β : Type u_3
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
μ : Measure α
f : α → β
t : Set β
inst✝ : MeasurableSpace.CountablyGenerated ↑t
h : NullMeasurable f μ
hft : ∀ᵐ (x : α) ∂μ.completion, f x ∈ t
hne : t.Nonempty
⊢ AEMeasurable f μ
Module: Mathlib.MeasureTheory.Measure.MeasureSpace

Goal state before:
pp: case refine_1
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Type u_5
R : Type u_6
R' : Type u_7
m0 : MeasurableSpace α
mβ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s✝ s' t s : Set α
hs : s ∈ μ.cofinite
⊢ μ sᶜ < ∞

Tactic:
]

Goal state after:
case refine_1
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Type u_5
R : Type u_6
R' : Type u_7
m0 : MeasurableSpace α
mβ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s✝ s' t s : Set α
hs : s ∈ μ.cofinite
⊢ μ sᶜ < ∞
Module: Mathlib.MeasureTheory.Measure.AEMeasurable

Goal state before:
pp: case inr
α : Type u_2
β : Type u_3
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
μ : Measure α
f : α → β
t : Set β
inst✝ : MeasurableSpace.CountablyGenerated ↑t
h : NullMeasurable f μ
hft : ∀ᵐ (x : α) ∂μ.completion, f x ∈ t
hne : t.Nonempty
f' : α → β
hf'm : NullMeasurable f' μ
hf't : ∀ (y : α), f' y ∈ t
hff' : f =ᵐ[μ] f'
⊢ AEMeasurable f μ

Tactic:
]

Goal state after:
case inr
α : Type u_2
β : Type u_3
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
μ : Measure α
f : α → β
t : Set β
inst✝ : MeasurableSpace.CountablyGenerated ↑t
h : NullMeasurable f μ
hft : ∀ᵐ (x : α) ∂μ.completion, f x ∈ t
hne : t.Nonempty
f' : α → β
hf'm : NullMeasurable f' μ
hf't : ∀ (y : α), f' y ∈ t
hff' : f =ᵐ[μ] f'
⊢ AEMeasurable f μ
Module: Mathlib.MeasureTheory.Measure.MeasureSpace

Goal state before:
pp: case refine_2
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Type u_5
R : Type u_6
R' : Type u_7
m0 : MeasurableSpace α
mβ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s✝ s' t s : Set α
hs : s ∈ μ.cofinite
⊢ sᶜ ⊆ toMeasurable μ sᶜ

Tactic:
]

Goal state after:
case refine_2
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Type u_5
R : Type u_6
R' : Type u_7
m0 : MeasurableSpace α
mβ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s✝ s' t s : Set α
hs : s ∈ μ.cofinite
⊢ sᶜ ⊆ toMeasurable μ sᶜ
Module: Mathlib.MeasureTheory.Measure.AEMeasurable

Goal state before:
pp: α : Type u_2
β : Type u_3
m0 : MeasurableSpace α
inst✝ : MeasurableSpace β
ι : Type u_7
m : ι → Measure α
f : α → β
hf : AEMeasurable f (sum m)
s : Set β
hs : MeasurableSet s
⊢ ∑' (i : ι), (m i) (f ⁻¹' s) = ∑' (i : ι), (map f (m i)) s

Tactic:
]

Goal state after:
α : Type u_2
β : Type u_3
m0 : MeasurableSpace α
inst✝ : MeasurableSpace β
ι : Type u_7
m : ι → Measure α
f : α → β
hf : AEMeasurable f (sum m)
s : Set β
hs : MeasurableSet s
⊢ ∑' (i : ι), (m i) (f ⁻¹' s) = ∑' (i : ι), (map f (m i)) s
Module: Mathlib.MeasureTheory.Measure.Restrict

Goal state before:
pp: α : Type u_2
ι : Type u_6
x✝ : MeasurableSpace α
μ : Measure α
s : ι → Set α
M : ℕ
hs_meas : ∀ (i : ι), MeasurableSet (s i)
hs : ∀ (y : α), {i | y ∈ s i}.encard ≤ ↑M
t : Set α
ht : MeasurableSet t
F : Finset ι
P : Finset ι → Set α := fun C ↦ (⋂ i ∈ C, s i) ∩ ⋂ i ∈ F \ C, (s i)ᶜ
Cs : Finset (Finset ι) := F.powerset \ {∅}
G : ι → Set (Finset ι) := fun i ↦ {C | C ∈ F.powerset ∧ i ∈ C}
P_meas : ∀ (C : Finset ι), MeasurableSet (P C)
P_cover : ∀ {i : ι}, i ∈ F → s i ⊆ ⋃ C ∈ G i, P C
iUnion_P : ⋃ C ∈ Cs, P C ⊆ ⋃ i, s i
P_subset_s : ∀ {i : ι} {C : Finset ι}, i ∈ C → P C ⊆ s i
mem_C : ∀ {i : ι}, i ∈ F → ∀ {C : Finset ι} {x : α}, x ∈ P C → x ∈ s i → i ∈ C
C_subset_C : ∀ {C₁ C₂ : Finset ι}, C₁ ∈ Cs → ∀ {x : α}, x ∈ P C₁ ∩ P C₂ → C₁ ⊆ C₂
⊢ M • (μ.restrict (⋃ C ∈ Cs, P C)) t ≤ M • (μ.restrict (⋃ i, s i)) t

Tactic:
]

Goal state after:
α : Type u_2
ι : Type u_6
x✝ : MeasurableSpace α
μ : Measure α
s : ι → Set α
M : ℕ
hs_meas : ∀ (i : ι), MeasurableSet (s i)
hs : ∀ (y : α), {i | y ∈ s i}.encard ≤ ↑M
t : Set α
ht : MeasurableSet t
F : Finset ι
P : Finset ι → Set α := fun C ↦ (⋂ i ∈ C, s i) ∩ ⋂ i ∈ F \ C, (s i)ᶜ
Cs : Finset (Finset ι) := F.powerset \ {∅}
G : ι → Set (Finset ι) := fun i ↦ {C | C ∈ F.powerset ∧ i ∈ C}
P_meas : ∀ (C : Finset ι), MeasurableSet (P C)
P_cover : ∀ {i : ι}, i ∈ F → s i ⊆ ⋃ C ∈ G i, P C
iUnion_P : ⋃ C ∈ Cs, P C ⊆ ⋃ i, s i
P_subset_s : ∀ {i : ι} {C : Finset ι}, i ∈ C → P C ⊆ s i
mem_C : ∀ {i : ι}, i ∈ F → ∀ {C : Finset ι} {x : α}, x ∈ P C → x ∈ s i → i ∈ C
C_subset_C : ∀ {C₁ C₂ : Finset ι}, C₁ ∈ Cs → ∀ {x : α}, x ∈ P C₁ ∩ P C₂ → C₁ ⊆ C₂
⊢ M • (μ.restrict (⋃ C ∈ Cs, P C)) t ≤ M • (μ.restrict (⋃ i, s i)) t
Module: Mathlib.MeasureTheory.Measure.MeasureSpace

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
⊢ μ = 0 ↔ μ = 0

Tactic:
]

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

Goal state before:
pp: α : Type u_2
m0 : MeasurableSpace α
ι : Type u_7
β : Type u_8
mβ : MeasurableSpace β
m : ι → Measure α
f : α → β
s : Finset ι
hf : AEMeasurable f (∑ i ∈ s, m i)
⊢ (sum fun i ↦ map f (m ↑i)) = sum fun i ↦ map f (m ↑i)

pp: α : Type u_2
m0 : MeasurableSpace α
ι : Type u_7
β : Type u_8
mβ : MeasurableSpace β
m : ι → Measure α
f : α → β
s : Finset ι
hf : AEMeasurable f (∑ i ∈ s, m i)
⊢ AEMeasurable f (sum fun i ↦ m ↑i)

Tactic:
]

Goal state after:
α : Type u_2
m0 : MeasurableSpace α
ι : Type u_7
β : Type u_8
mβ : MeasurableSpace β
m : ι → Measure α
f : α → β
s : Finset ι
hf : AEMeasurable f (∑ i ∈ s, m i)
⊢ AEMeasurable f (sum fun i ↦ m ↑i)
Module: Mathlib.MeasureTheory.Measure.AEMeasurable

Goal state before:
pp: α : Type u_2
m0 : MeasurableSpace α
ι : Type u_7
β : Type u_8
mβ : MeasurableSpace β
m : ι → Measure α
f : α → β
s : Finset ι
hf : AEMeasurable f (∑ i ∈ s, m i)
⊢ AEMeasurable f (∑ i ∈ s, m i)

Tactic:
]

Goal state after:
α : Type u_2
m0 : MeasurableSpace α
ι : Type u_7
β : Type u_8
mβ : MeasurableSpace β
m : ι → Measure α
f : α → β
s : Finset ι
hf : AEMeasurable f (∑ i ∈ s, m i)
⊢ AEMeasurable f (∑ i ∈ s, m i)
Module: Mathlib.MeasureTheory.Measure.MeasureSpace

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
inst✝ : Preorder α
s : Set α
hsc : s.Countable
hst : ∀ (x : α), ∃ y ∈ s, x ≤ y
hdir : DirectedOn (fun x1 x2 ↦ x1 ≤ x2) s
⊢ μ (⋃ i ∈ s, Iic i) = μ univ

pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
inst✝ : Preorder α
s : Set α
hsc : s.Countable
hst : ∀ (x : α), ∃ y ∈ s, x ≤ y
hdir : DirectedOn (fun x1 x2 ↦ x1 ≤ x2) s
⊢ DirectedOn ((fun x1 x2 ↦ x1 ⊆ x2) on Iic) s

Tactic:
]

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
inst✝ : Preorder α
s : Set α
hsc : s.Countable
hst : ∀ (x : α), ∃ y ∈ s, x ≤ y
hdir : DirectedOn (fun x1 x2 ↦ x1 ≤ x2) s
⊢ μ (⋃ i ∈ s, Iic i) = μ univ

α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
inst✝ : Preorder α
s : Set α
hsc : s.Countable
hst : ∀ (x : α), ∃ y ∈ s, x ≤ y
hdir : DirectedOn (fun x1 x2 ↦ x1 ≤ x2) s
⊢ DirectedOn ((fun x1 x2 ↦ x1 ⊆ x2) on Iic) s
Module: Mathlib.MeasureTheory.Measure.AEMeasurable

Goal state before:
pp: case pos
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✝³ : MeasurableSpace β
inst✝² : MeasurableSpace γ
inst✝¹ : MeasurableSpace δ
f✝ g : α → β
μ✝ ν μ : Measure α
f : α → β
inst✝ : SFinite μ
H : AEMeasurable f (sum (sfiniteSeq μ))
⊢ SFinite (map f (sum (sfiniteSeq μ)))

Tactic:
]

Goal state after:
case pos
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✝³ : MeasurableSpace β
inst✝² : MeasurableSpace γ
inst✝¹ : MeasurableSpace δ
f✝ g : α → β
μ✝ ν μ : Measure α
f : α → β
inst✝ : SFinite μ
H : AEMeasurable f (sum (sfiniteSeq μ))
⊢ SFinite (map f (sum (sfiniteSeq μ)))
Module: Mathlib.MeasureTheory.Measure.AEMeasurable

Goal state before:
pp: case pos
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✝³ : MeasurableSpace β
inst✝² : MeasurableSpace γ
inst✝¹ : MeasurableSpace δ
f✝ g : α → β
μ✝ ν μ : Measure α
f : α → β
inst✝ : SFinite μ
H : AEMeasurable f (sum (sfiniteSeq μ))
⊢ SFinite (sum fun i ↦ map f (sfiniteSeq μ i))

Tactic:
]

Goal state after:
case pos
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✝³ : MeasurableSpace β
inst✝² : MeasurableSpace γ
inst✝¹ : MeasurableSpace δ
f✝ g : α → β
μ✝ ν μ : Measure α
f : α → β
inst✝ : SFinite μ
H : AEMeasurable f (sum (sfiniteSeq μ))
⊢ SFinite (sum fun i ↦ map f (sfiniteSeq μ i))
Module: Mathlib.MeasureTheory.Measure.AEMeasurable

Goal state before:
pp: case neg
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✝³ : MeasurableSpace β
inst✝² : MeasurableSpace γ
inst✝¹ : MeasurableSpace δ
f✝ g : α → β
μ✝ ν μ : Measure α
f : α → β
inst✝ : SFinite μ
H : ¬AEMeasurable f μ
⊢ SFinite 0

Tactic:
]

Goal state after:
case neg
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✝³ : MeasurableSpace β
inst✝² : MeasurableSpace γ
inst✝¹ : MeasurableSpace δ
f✝ g : α → β
μ✝ ν μ : Measure α
f : α → β
inst✝ : SFinite μ
H : ¬AEMeasurable f μ
⊢ SFinite 0
Module: Mathlib.MeasureTheory.Measure.MeasureSpace

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
inst✝² : Preorder α
inst✝¹ : NoMaxOrder α
inst✝ : atTop.IsCountablyGenerated
μ : Measure α
a : α
⊢ Tendsto (fun x ↦ μ (Ico a x)) atTop (𝓝 (μ (⋃ b, Ico a b)))

Tactic:
]

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
inst✝² : Preorder α
inst✝¹ : NoMaxOrder α
inst✝ : atTop.IsCountablyGenerated
μ : Measure α
a : α
⊢ Tendsto (fun x ↦ μ (Ico a x)) atTop (𝓝 (μ (⋃ b, Ico a b)))
Module: Mathlib.MeasureTheory.Measure.MeasureSpace

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
inst✝² : Preorder α
inst✝¹ : NoMinOrder α
inst✝ : atBot.IsCountablyGenerated
μ : Measure α
a : α
⊢ Tendsto (fun x ↦ μ (Ioc x a)) atBot (𝓝 (μ (⋃ b, Ioc b a)))

Tactic:
]

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
inst✝² : Preorder α
inst✝¹ : NoMinOrder α
inst✝ : atBot.IsCountablyGenerated
μ : Measure α
a : α
⊢ Tendsto (fun x ↦ μ (Ioc x a)) atBot (𝓝 (μ (⋃ b, Ioc b a)))
Module: Mathlib.Topology.Separation.GDelta

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
s : Set X
inst✝ : T1Space X
hs : s.Countable
⊢ IsGδ (⋂ i ∈ s, {i}ᶜ)

Tactic:
]

Goal state after:
X : Type u_1
inst✝¹ : TopologicalSpace X
s : Set X
inst✝ : T1Space X
hs : s.Countable
⊢ IsGδ (⋂ i ∈ s, {i}ᶜ)
Module: Mathlib.Topology.Separation.GDelta

Goal state before:
pp: X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : FirstCountableTopology X
inst✝ : T1Space X
x : X
U : ℕ → Set X
hU : ∀ (i : ℕ), x ∈ U i ∧ IsOpen[inst✝²] (U i)
h_basis : (𝓝 x).HasAntitoneBasis fun i ↦ U i
⊢ IsGδ (⋂ i, ⋂ (_ : True), U i)

Tactic:
]

Goal state after:
X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : FirstCountableTopology X
inst✝ : T1Space X
x : X
U : ℕ → Set X
hU : ∀ (i : ℕ), x ∈ U i ∧ IsOpen[inst✝²] (U i)
h_basis : (𝓝 x).HasAntitoneBasis fun i ↦ U i
⊢ IsGδ (⋂ i, ⋂ (_ : True), U i)
Module: Mathlib.Topology.Separation.GDelta

Goal state before:
pp: case inl
X : Type u_1
inst✝¹ : TopologicalSpace X
s t : Set X
inst✝ : NormalSpace X
st_dis : Disjoint s univ
t_cl : IsClosed[inst✝¹] t
T_open : ∀ t ∈ ∅, IsOpen[inst✝¹] t
T_count : ∅.Countable
T_int : t = ⋂₀ ∅
⊢ HasSeparatingCover s t

Tactic:
]

Goal state after:
case inl
X : Type u_1
inst✝¹ : TopologicalSpace X
s t : Set X
inst✝ : NormalSpace X
st_dis : Disjoint s univ
t_cl : IsClosed[inst✝¹] t
T_open : ∀ t ∈ ∅, IsOpen[inst✝¹] t
T_count : ∅.Countable
T_int : t = ⋂₀ ∅
⊢ HasSeparatingCover s t
Module: Mathlib.Topology.Separation.GDelta

Goal state before:
pp: case inl
X : Type u_1
inst✝¹ : TopologicalSpace X
s t : Set X
inst✝ : NormalSpace X
st_dis : Disjoint s univ
t_cl : IsClosed[inst✝¹] t
T_open : ∀ t ∈ ∅, IsOpen[inst✝¹] t
T_count : ∅.Countable
T_int : t = ⋂₀ ∅
⊢ HasSeparatingCover ∅ t

Tactic:
]

Goal state after:
case inl
X : Type u_1
inst✝¹ : TopologicalSpace X
s t : Set X
inst✝ : NormalSpace X
st_dis : Disjoint s univ
t_cl : IsClosed[inst✝¹] t
T_open : ∀ t ∈ ∅, IsOpen[inst✝¹] t
T_count : ∅.Countable
T_int : t = ⋂₀ ∅
⊢ HasSeparatingCover ∅ t
Module: Mathlib.Topology.Separation.GDelta

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
s t : Set X
inst✝ : NormalSpace X
st_dis : Disjoint s t
t_cl : IsClosed[inst✝¹] t
T : Set (Set X)
T_open : ∀ t ∈ T, IsOpen[inst✝¹] t
T_count : T.Countable
T_int : t = ⋂₀ T
T_nonempty : T.Nonempty
g : ℕ → ↑T
g_surj : Surjective g
n : ℕ
⊢ ⋂₀ T ⊆ ↑(g n)

Tactic:
]

Goal state after:
X : Type u_1
inst✝¹ : TopologicalSpace X
s t : Set X
inst✝ : NormalSpace X
st_dis : Disjoint s t
t_cl : IsClosed[inst✝¹] t
T : Set (Set X)
T_open : ∀ t ∈ T, IsOpen[inst✝¹] t
T_count : T.Countable
T_int : t = ⋂₀ T
T_nonempty : T.Nonempty
g : ℕ → ↑T
g_surj : Surjective g
n : ℕ
⊢ ⋂₀ T ⊆ ↑(g n)
Module: Mathlib.MeasureTheory.Measure.MeasureSpace

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
inst✝¹ : Preorder α
inst✝ : atTop.IsCountablyGenerated
μ : Measure α
⊢ Tendsto (fun x ↦ μ (Iic x)) atTop (𝓝 (μ (⋃ a, Iic a)))

Tactic:
]

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
inst✝¹ : Preorder α
inst✝ : atTop.IsCountablyGenerated
μ : Measure α
⊢ Tendsto (fun x ↦ μ (Iic x)) atTop (𝓝 (μ (⋃ a, Iic a)))
Module: Mathlib.MeasureTheory.Measure.MeasureSpace

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
inst✝ : PartialOrder α
a : α
ha : μ {a} = 0
⊢ 0 = 0

Tactic:
]

Goal state after: