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

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
f : α → β
μ : Measure β
hfi : Injective f
hf : ∀ (s : Set α), MeasurableSet s → NullMeasurableSet (f '' s) μ
s : Set α
hs : (comap f μ) s = 0
⊢ μ (f '' s) ≤ 0

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
f : α → β
μ : Measure β
hfi : Injective f
hf : ∀ (s : Set α), MeasurableSet s → NullMeasurableSet (f '' s) μ
s : Set α
hs : (comap f μ) s = 0
⊢ μ (f '' s) ≤ 0
Module: Mathlib.MeasureTheory.Measure.Comap

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
f : α → β
μ : Measure β
hfi : Injective f
hf : ∀ (s : Set α), MeasurableSet s → NullMeasurableSet (f '' s) μ
s t : Set α
hst : (comap f μ) {a | ¬s a = t a} = 0
⊢ μ {a | ¬(f '' s) a = (f '' t) a} = 0

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
f : α → β
μ : Measure β
hfi : Injective f
hf : ∀ (s : Set α), MeasurableSet s → NullMeasurableSet (f '' s) μ
s t : Set α
hst : (comap f μ) {a | ¬s a = t a} = 0
⊢ μ {a | ¬(f '' s) a = (f '' t) a} = 0
Module: Mathlib.MeasureTheory.Measure.QuasiMeasurePreserving

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ : Measure α
s : Set α
e : α ≃ α
he : QuasiMeasurePreserving (⇑e) μ μ
he' : QuasiMeasurePreserving (⇑e.symm) μ μ
hs : ⇑e.symm ⁻¹' s =ᵐ[μ] s
k : ℕ
⊢ ⇑e⁻¹ ⁻¹' s =ᵐ[μ] s

Tactic:
]

Goal state after:
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
s : Set α
e : α ≃ α
he : QuasiMeasurePreserving (⇑e) μ μ
he' : QuasiMeasurePreserving (⇑e.symm) μ μ
hs : ⇑e.symm ⁻¹' s =ᵐ[μ] s
k : ℕ
⊢ ⇑e⁻¹ ⁻¹' s =ᵐ[μ] s
Module: Mathlib.MeasureTheory.Measure.Comap

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
f : α → β
μ : Measure β
hfi : Injective f
hf : ∀ (s : Set α), MeasurableSet s → NullMeasurableSet (f '' s) μ
s t : Set α
hst : (comap f μ) {a | ¬s a = t a} = 0
h_eq_α : {a | ¬s a = t a} = s \ t ∪ t \ s
h_eq_β : {a | ¬(f '' s) a = (f '' t) a} = f '' (s \ t ∪ t \ s)
⊢ μ {a | ¬(f '' s) a = (f '' t) a} = 0

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
f : α → β
μ : Measure β
hfi : Injective f
hf : ∀ (s : Set α), MeasurableSet s → NullMeasurableSet (f '' s) μ
s t : Set α
hst : (comap f μ) {a | ¬s a = t a} = 0
h_eq_α : {a | ¬s a = t a} = s \ t ∪ t \ s
h_eq_β : {a | ¬(f '' s) a = (f '' t) a} = f '' (s \ t ∪ t \ s)
⊢ μ {a | ¬(f '' s) a = (f '' t) a} = 0
Module: Mathlib.MeasureTheory.Measure.Comap

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
f : α → β
μ : Measure β
hfi : Injective f
hf : ∀ (s : Set α), MeasurableSet s → NullMeasurableSet (f '' s) μ
s t : Set α
hst : (comap f μ) {a | ¬s a = t a} = 0
h_eq_α : {a | ¬s a = t a} = s \ t ∪ t \ s
h_eq_β : {a | ¬(f '' s) a = (f '' t) a} = f '' (s \ t ∪ t \ s)
⊢ μ (f '' (s \ t ∪ t \ s)) = 0

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
f : α → β
μ : Measure β
hfi : Injective f
hf : ∀ (s : Set α), MeasurableSet s → NullMeasurableSet (f '' s) μ
s t : Set α
hst : (comap f μ) {a | ¬s a = t a} = 0
h_eq_α : {a | ¬s a = t a} = s \ t ∪ t \ s
h_eq_β : {a | ¬(f '' s) a = (f '' t) a} = f '' (s \ t ∪ t \ s)
⊢ μ (f '' (s \ t ∪ t \ s)) = 0
Module: Mathlib.MeasureTheory.Measure.Comap

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
f : α → β
μ : Measure β
hfi : Injective f
hf : ∀ (s : Set α), MeasurableSet s → NullMeasurableSet (f '' s) μ
s t : Set α
hst : (comap f μ) (s \ t ∪ t \ s) = 0
h_eq_α : {a | ¬s a = t a} = s \ t ∪ t \ s
h_eq_β : {a | ¬(f '' s) a = (f '' t) a} = f '' (s \ t ∪ t \ s)
⊢ μ (f '' (s \ t ∪ t \ s)) = 0

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
f : α → β
μ : Measure β
hfi : Injective f
hf : ∀ (s : Set α), MeasurableSet s → NullMeasurableSet (f '' s) μ
s t : Set α
hst : (comap f μ) (s \ t ∪ t \ s) = 0
h_eq_α : {a | ¬s a = t a} = s \ t ∪ t \ s
h_eq_β : {a | ¬(f '' s) a = (f '' t) a} = f '' (s \ t ∪ t \ s)
⊢ μ (f '' (s \ t ∪ t \ s)) = 0
Module: Mathlib.MeasureTheory.Measure.QuasiMeasurePreserving

Goal state before:
pp: case inl
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
s : Set α
e : α ≃ α
he : QuasiMeasurePreserving (⇑e) μ μ
k : ℕ
hs : ⇑e⁻¹ ⁻¹' s =ᵐ[μ] s
he' : ⇑(e ^ k)⁻¹ ⁻¹' s =ᵐ[μ] s
⊢ ⇑(e ^ ↑k).symm ⁻¹' s =ᵐ[μ] s

Tactic:
]

Goal state after:
case inl
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
s : Set α
e : α ≃ α
he : QuasiMeasurePreserving (⇑e) μ μ
k : ℕ
hs : ⇑e⁻¹ ⁻¹' s =ᵐ[μ] s
he' : ⇑(e ^ k)⁻¹ ⁻¹' s =ᵐ[μ] s
⊢ ⇑(e ^ ↑k).symm ⁻¹' s =ᵐ[μ] s
Module: Mathlib.MeasureTheory.Measure.QuasiMeasurePreserving

Goal state before:
pp: case inr
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
s : Set α
e : α ≃ α
he : QuasiMeasurePreserving (⇑e) μ μ
he' : QuasiMeasurePreserving (⇑e.symm) μ μ
hs : ⇑e '' s =ᵐ[μ] s
k : ℕ
⊢ ⇑(e ^ k)⁻¹.symm ⁻¹' s =ᵐ[μ] s

Tactic:
]

Goal state after:
case inr
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
s : Set α
e : α ≃ α
he : QuasiMeasurePreserving (⇑e) μ μ
he' : QuasiMeasurePreserving (⇑e.symm) μ μ
hs : ⇑e '' s =ᵐ[μ] s
k : ℕ
⊢ ⇑(e ^ k)⁻¹.symm ⁻¹' s =ᵐ[μ] s
Module: Mathlib.MeasureTheory.Measure.QuasiMeasurePreserving

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ : Measure α
s : Set α
e : α ≃ α
he : QuasiMeasurePreserving (⇑e) μ μ
he' : QuasiMeasurePreserving (⇑e.symm) μ μ
hs : ⇑e '' s =ᵐ[μ] s
k : ℕ
⊢ ⇑e ⁻¹' s =ᵐ[μ] s ↔ ⇑e ⁻¹' s =ᵐ[μ] s

Tactic:
]

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

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
f : α → β
μ : Measure β
hf : Injective f
hf' : Measurable f
h : ∀ (t : Set α), MeasurableSet t → NullMeasurableSet (f '' t) μ
s : Set β
hs : MeasurableSet s
⊢ μ (s ∩ range f) = μ (s ∩ range f)

Tactic:
]

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

Goal state before:
pp: case inr
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
s : Set α
e : α ≃ α
he' : QuasiMeasurePreserving (⇑e.symm) μ μ
k : ℕ
hs : ⇑e ⁻¹' s =ᵐ[μ] s
he : ⇑(e ^ k) ⁻¹' s =ᵐ[μ] s
⊢ ⇑(e ^ k)⁻¹.symm ⁻¹' s =ᵐ[μ] s

Tactic:
]

Goal state after:
case inr
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
s : Set α
e : α ≃ α
he' : QuasiMeasurePreserving (⇑e.symm) μ μ
k : ℕ
hs : ⇑e ⁻¹' s =ᵐ[μ] s
he : ⇑(e ^ k) ⁻¹' s =ᵐ[μ] s
⊢ ⇑(e ^ k)⁻¹.symm ⁻¹' s =ᵐ[μ] s
Module: Mathlib.MeasureTheory.Measure.Comap

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

pp: case hfi
β : Type u_2
mβ : MeasurableSpace β
μ : Measure β
s : Set β
hs : MeasurableSet s
⊢ Injective fun x ↦ x

pp: case hf
β : Type u_2
mβ : MeasurableSpace β
μ : Measure β
s : Set β
hs : MeasurableSet s
⊢ ∀ (s : Set β), MeasurableSet s → MeasurableSet ((fun x ↦ x) '' s)

pp: case hs
β : Type u_2
mβ : MeasurableSpace β
μ : Measure β
s : Set β
hs : MeasurableSet s
⊢ MeasurableSet s

Tactic:
]

Goal state after:
case hfi
β : Type u_2
mβ : MeasurableSpace β
μ : Measure β
s : Set β
hs : MeasurableSet s
⊢ Injective fun x ↦ x

case hf
β : Type u_2
mβ : MeasurableSpace β
μ : Measure β
s : Set β
hs : MeasurableSet s
⊢ ∀ (s : Set β), MeasurableSet s → MeasurableSet ((fun x ↦ x) '' s)

case hs
β : Type u_2
mβ : MeasurableSpace β
μ : Measure β
s : Set β
hs : MeasurableSet s
⊢ MeasurableSet s
Module: Mathlib.MeasureTheory.Measure.Comap

Goal state before:
pp: case pos
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
f : α → β
g : β → γ
hf' : ∀ (s : Set α), MeasurableSet s → MeasurableSet (f '' s)
hg : Injective g
hg' : ∀ (s : Set β), MeasurableSet s → MeasurableSet (g '' s)
μ : Measure γ
hf : Injective f
s : Set α
hs : MeasurableSet s
⊢ μ (g '' f '' s) = μ (g '' f '' s)

Tactic:
]

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

Goal state before:
pp: case neg
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
f : α → β
g : β → γ
hf' : ∀ (s : Set α), MeasurableSet s → MeasurableSet (f '' s)
hg : Injective g
hg' : ∀ (s : Set β), MeasurableSet s → MeasurableSet (g '' s)
μ : Measure γ
hf : ¬Injective f
⊢ 0 = 0

Tactic:
]

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

Goal state before:
pp: G : Type u_5
α : Type u_6
inst✝¹ : Group G
inst✝ : MulAction G α
x✝ : MeasurableSpace α
μ : Measure α
s : Set α
h_ae_disjoint : ∀ (g : G), g ≠ 1 → AEDisjoint μ (g • s) s
h_qmp : ∀ (g : G), QuasiMeasurePreserving (fun x ↦ g • x) μ μ
g₁ g₂ : G
hg : g₁ ≠ g₂
g : G := g₂⁻¹ * g₁
⊢ ¬g₂ = g₁

Tactic:
]

Goal state after:
G : Type u_5
α : Type u_6
inst✝¹ : Group G
inst✝ : MulAction G α
x✝ : MeasurableSpace α
μ : Measure α
s : Set α
h_ae_disjoint : ∀ (g : G), g ≠ 1 → AEDisjoint μ (g • s) s
h_qmp : ∀ (g : G), QuasiMeasurePreserving (fun x ↦ g • x) μ μ
g₁ g₂ : G
hg : g₁ ≠ g₂
g : G := g₂⁻¹ * g₁
⊢ ¬g₂ = g₁
Module: Mathlib.MeasureTheory.Measure.Comap

Goal state before:
pp: case pos
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
f : α → β
μ : Measure β
c : ℝ≥0∞
hc : c ≠ 0
h : Injective f ∧ ∀ (s : Set α), MeasurableSet s → NullMeasurableSet (f '' s) μ
s : Set α
hs : MeasurableSet s
⊢ c • μ (f '' s) = c • μ (f '' s)

pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
f : α → β
μ : Measure β
c : ℝ≥0∞
hc : c ≠ 0
h : Injective f ∧ ∀ (s : Set α), MeasurableSet s → NullMeasurableSet (f '' s) μ
s : Set α
hs : MeasurableSet s
⊢ ∀ (s : Set α), MeasurableSet s → NullMeasurableSet (f '' s) (c • μ)

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
f : α → β
μ : Measure β
c : ℝ≥0∞
hc : c ≠ 0
h : Injective f ∧ ∀ (s : Set α), MeasurableSet s → NullMeasurableSet (f '' s) μ
s : Set α
hs : MeasurableSet s
⊢ ∀ (s : Set α), MeasurableSet s → NullMeasurableSet (f '' s) (c • μ)
Module: Mathlib.MeasureTheory.Measure.QuasiMeasurePreserving

Goal state before:
pp: G : Type u_5
α : Type u_6
inst✝¹ : Group G
inst✝ : MulAction G α
x✝ : MeasurableSpace α
μ : Measure α
s : Set α
h_ae_disjoint : ∀ (g : G), g ≠ 1 → AEDisjoint μ (g • s) s
h_qmp : ∀ (g : G), QuasiMeasurePreserving (fun x ↦ g • x) μ μ
g₁ g₂ : G
g : G := g₂⁻¹ * g₁
hg : g ≠ 1
⊢ g • s ∩ s = (g₂⁻¹ * g₁) • s ∩ s

Tactic:
]

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

Goal state before:
pp: α : Type u_1
β : Type u_2
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
e : α ≃ᵐ β
⊢ μ ≪ μ

pp: case hg
α : Type u_1
β : Type u_2
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
e : α ≃ᵐ β
⊢ Measurable ⇑e.symm

pp: case hf
α : Type u_1
β : Type u_2
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
e : α ≃ᵐ β
⊢ Measurable ⇑e

Tactic:
]

Goal state after:
case hg
α : Type u_1
β : Type u_2
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
e : α ≃ᵐ β
⊢ Measurable ⇑e.symm

case hf
α : Type u_1
β : Type u_2
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
e : α ≃ᵐ β
⊢ Measurable ⇑e
Module: Mathlib.MeasureTheory.Measure.Comap

Goal state before:
pp: α : Type u_1
β : Type u_2
ma : MeasurableSpace α
mb : MeasurableSpace β
μ : Measure α
e : α ≃ᵐ β
s : Set β
hs : MeasurableSet s
⊢ μ (⇑e ⁻¹' s) = μ (⇑e ⁻¹' s)

pp: α : Type u_1
β : Type u_2
ma : MeasurableSpace α
mb : MeasurableSpace β
μ : Measure α
e : α ≃ᵐ β
s : Set β
hs : MeasurableSet s
⊢ ∀ (s : Set β), MeasurableSet s → MeasurableSet (⇑e.symm '' s)

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
ma : MeasurableSpace α
mb : MeasurableSpace β
μ : Measure α
e : α ≃ᵐ β
s : Set β
hs : MeasurableSet s
⊢ ∀ (s : Set β), MeasurableSet s → MeasurableSet (⇑e.symm '' s)
Module: Mathlib.MeasureTheory.Measure.Comap

Goal state before:
pp: α : Type u_1
β : Type u_2
ma : MeasurableSpace α
mb : MeasurableSpace β
μ : Measure α
e : β ≃ᵐ α
⊢ Measure.comap (⇑e) μ = Measure.comap (⇑e) μ

Tactic:
]

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
t s : Set α
ht : MeasurableSet t
⊢ μ t + μ (s \ t) + μ (s ∩ t) = μ (s ∩ t) + μ (s \ t) + μ t

Tactic:
]

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
t s : Set α
ht : MeasurableSet t
⊢ μ t + μ (s \ t) + μ (s ∩ t) = μ (s ∩ t) + μ (s \ t) + μ t
Module: Mathlib.MeasureTheory.Measure.MeasureSpace

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
s : Set α
hs : MeasurableSet s
t : Set α
⊢ μ s + μ t = μ s + μ t

Tactic:
]

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

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
s : Set β
f : β → Set α
hs : s.Countable
hd : s.Pairwise (AEDisjoint μ on f)
h : ∀ b ∈ s, NullMeasurableSet (f b) μ
this : Encodable ↑s
⊢ μ (⋃ x, f ↑x) = ∑' (p : ↑s), μ (f ↑p)

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
s : Set β
f : β → Set α
hs : s.Countable
hd : s.Pairwise (AEDisjoint μ on f)
h : ∀ b ∈ s, NullMeasurableSet (f b) μ
this : Encodable ↑s
⊢ μ (⋃ x, f ↑x) = ∑' (p : ↑s), μ (f ↑p)
Module: Mathlib.MeasureTheory.Measure.MeasureSpace

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
S : Set (Set α)
hs : S.Countable
hd : S.Pairwise (AEDisjoint μ)
h : ∀ s ∈ S, NullMeasurableSet s μ
⊢ ∑' (p : ↑S), μ ↑p = ∑' (s : ↑S), μ ↑s

Tactic:
]

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
S : Set (Set α)
hs : S.Countable
hd : S.Pairwise Disjoint
h : ∀ s ∈ S, MeasurableSet s
⊢ ∑' (p : ↑S), μ ↑p = ∑' (s : ↑S), μ ↑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
s : Finset ι
f : ι → Set α
hd : (↑s).Pairwise (AEDisjoint μ on f)
hm : ∀ b ∈ s, NullMeasurableSet (f b) μ
⊢ μ (⋃ b ∈ s, f b) = ∑' (b : ↥s), μ (f ↑b)

Tactic:
]

Goal state after:
α : Type u_1
ι : Type u_5
m : MeasurableSpace α
μ : Measure α
s : Finset ι
f : ι → Set α
hd : (↑s).Pairwise (AEDisjoint μ on f)
hm : ∀ b ∈ s, NullMeasurableSet (f b) μ
⊢ μ (⋃ b ∈ s, f b) = ∑' (b : ↥s), μ (f ↑b)
Module: Mathlib.MeasureTheory.Measure.MeasureSpace

Goal state before:
pp: α : Type u_1
ι : Type u_8
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)
⊢ ∀ (i : Finset ι), ∑ a ∈ i, μ (As a) ≤ μ (⋃ i, As i)

Tactic:
]

Goal state after:
α : Type u_1
ι : Type u_8
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)
⊢ ∀ (i : Finset ι), ∑ a ∈ i, μ (As a) ≤ μ (⋃ i, As i)
Module: Mathlib.MeasureTheory.Measure.MeasureSpace

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
s : Set β
hs : s.Countable
f : α → β
hf : ∀ y ∈ s, MeasurableSet (f ⁻¹' {y})
⊢ ∑' (b : ↑s), μ (f ⁻¹' {↑b}) = ∑' (p : ↑s), μ (f ⁻¹' {↑p})

Tactic:
]

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

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
s : Set β
f : α → β
hs : s.Countable
⊢ (∀ i ∈ s, μ (f ⁻¹' {i}) = 0) ↔ ∀ x ∈ s, μ (f ⁻¹' {x}) = 0

Tactic:
]

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

Goal state before:
pp: case pos.mp
X : Type u_1
inst✝² : LinearOrder X
inst✝¹ : TopologicalSpace X
inst✝ : OrderTopology X
s : Set X
b : X
hsne : s.Nonempty
this✝ : Nonempty ↑s
hsub : s ⊆ Iio b
h : comap Subtype.val (𝓝[<] b) = atTop
a : X
ha : a < b
this : Subtype.val ⁻¹' Ioo a b ∈ atTop
⊢ (s ∩ Ioo a b).Nonempty

Tactic:
]

Goal state after:
case pos.mp
X : Type u_1
inst✝² : LinearOrder X
inst✝¹ : TopologicalSpace X
inst✝ : OrderTopology X
s : Set X
b : X
hsne : s.Nonempty
this✝ : Nonempty ↑s
hsub : s ⊆ Iio b
h : comap Subtype.val (𝓝[<] b) = atTop
a : X
ha : a < b
this : Subtype.val ⁻¹' Ioo a b ∈ atTop
⊢ (s ∩ Ioo a b).Nonempty
Module: Mathlib.MeasureTheory.Measure.MeasureSpace

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
s : Finset α
inst✝ : MeasurableSingletonClass α
⊢ μ (id ⁻¹' ↑s) = μ ↑s

pp: case hf
α : Type u_1
m : MeasurableSpace α
μ : Measure α
s : Finset α
inst✝ : MeasurableSingletonClass α
⊢ ∀ y ∈ s, MeasurableSet (id ⁻¹' {y})

Tactic:
]

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
s : Finset α
inst✝ : MeasurableSingletonClass α
⊢ μ (id ⁻¹' ↑s) = μ ↑s

case hf
α : Type u_1
m : MeasurableSpace α
μ : Measure α
s : Finset α
inst✝ : MeasurableSingletonClass α
⊢ ∀ y ∈ s, MeasurableSet (id ⁻¹' {y})
Module: Mathlib.MeasureTheory.Measure.MeasureSpace

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
s : Set α
hs : NullMeasurableSet s μ
t : Set α
⊢ μ (s ∪ t) = μ (s ∪ t)

Tactic:
]

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
t s : Set α
hm : NullMeasurableSet t μ
h_fin : μ t ≠ ∞
⊢ μ (s ∪ t) = μ (s ∪ t)

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 : Set α
h : s₂ ⊆ s₁
h₂ : NullMeasurableSet s₂ μ
h_fin : μ s₂ ≠ ∞
⊢ μ s₁ - μ s₂ = μ s₁ - μ s₂

Tactic:
]

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
s t : Set α
hμst : μ (s ∆ t) ≠ ∞
u v : Set α
hμuv : μ (u ∆ v) ≠ ∞
hμu : μ u = ∞
hμv : μ v ≠ ∞
⊢ ∞ ≤ μ (u \ v ∪ v \ u)

Tactic:
]

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
s t : Set α
hμst : μ (s ∆ t) ≠ ∞
u v : Set α
hμuv : μ (u ∆ v) ≠ ∞
hμu : μ u = ∞
hμv : μ v ≠ ∞
⊢ ∞ ≤ μ (u \ v ∪ v \ u)
Module: Mathlib.MeasureTheory.Measure.MeasureSpace

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
s t : Set α
hμst : μ (s ∆ t) ≠ ∞
u v : Set α
hμuv : μ (u ∆ v) ≠ ∞
hμu : μ u = ∞
hμv : μ v ≠ ∞
⊢ ∞ = ∞

Tactic:
]

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
s t : Set α
hs : NullMeasurableSet s μ
hst : s ⊆ t
hs' : μ s ≠ ∞
ε : ℝ≥0∞
h : μ t < μ s + ε
⊢ μ t - μ s < ε

Tactic:
]

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
s t : Set α
hs : NullMeasurableSet s μ
hst : s ⊆ t
hs' : μ s ≠ ∞
ε : ℝ≥0∞
h : μ t < μ s + ε
⊢ μ t - μ s < ε
Module: Mathlib.MeasureTheory.Measure.MeasureSpace

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
s t : Set α
hs : NullMeasurableSet s μ
hst : s ⊆ t
hs' : μ s ≠ ∞
ε : ℝ≥0∞
h : μ t < ε + μ s
⊢ μ t - μ s < ε

Tactic:
]

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
s t : Set α
hs : NullMeasurableSet s μ
hst : s ⊆ t
hs' : μ s ≠ ∞
ε : ℝ≥0∞
h : μ t < ε + μ s
⊢ μ t - μ s < ε
Module: Mathlib.MeasureTheory.Measure.MeasureSpace

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
s t : Set α
hs : NullMeasurableSet s μ
hst : s ⊆ t
hs' : μ s ≠ ∞
ε : ℝ≥0∞
⊢ μ t ≤ μ s + ε ↔ μ t ≤ μ s + ε

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 s₃ : Set α
h12 : s₁ ⊆ s₂
h23 : s₂ ⊆ s₃
h_nullsdiff : μ (s₃ \ s₁) = 0
le12 : μ s₁ ≤ μ s₂
le23 : μ s₂ ≤ μ s₃
⊢ μ s₃ = μ s₃

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
s : Set X
b : X
hsb : s ⊆ Iio b
hs : s.Nonempty → ∃ a < b, Ioo a b ⊆ s
hb : IsSuccPrelimit b
⊢ s ⊆ Iio b ∧ (s.Nonempty → ∀ a < b, (s ∩ Ioo a b).Nonempty)

Tactic:
]

Goal state after:
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 : s.Nonempty → ∃ a < b, Ioo a b ⊆ s
hb : IsSuccPrelimit b
⊢ s ⊆ Iio b ∧ (s.Nonempty → ∀ a < b, (s ∩ Ioo a b).Nonempty)
Module: Mathlib.MeasureTheory.Measure.MeasureSpace

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
s : Set α
h : NullMeasurableSet s μ
hs : μ s ≠ ∞
⊢ μ sᶜ = μ sᶜ

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
s : Set X
b : X
hsb : s ⊆ Iio b
hs : s.Nonempty → ∃ a < b, Ioo a b ⊆ s
hb : IsSuccPrelimit b
hsne : s.Nonempty
a : X
ha : a < b
c : X
hcb : c < b
hcs : Ioo c b ⊆ s
x : X
hxb : x < b
hacx : a < x ∧ c < x
⊢ (s ∩ Ioo a b).Nonempty

Tactic:
]

Goal state after:
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 : s.Nonempty → ∃ a < b, Ioo a b ⊆ s
hb : IsSuccPrelimit b
hsne : s.Nonempty
a : X
ha : a < b
c : X
hcb : c < b
hcs : Ioo c b ⊆ s
x : X
hxb : x < b
hacx : a < x ∧ c < x
⊢ (s ∩ Ioo a b).Nonempty
Module: Mathlib.MeasureTheory.Measure.MeasureSpace

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
s t : Set α
ht : μ (s \ t) = 0
⊢ μ s = μ s

pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
s t : Set α
ht : μ (s \ t) = 0
⊢ μ (s ∩ tᶜ) = 0

Tactic:
]

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
s t : Set α
ht : μ (s \ t) = 0
⊢ μ (s ∩ tᶜ) = 0
Module: Mathlib.MeasureTheory.Measure.MeasureSpace

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
s t : Set α
ht : μ (s \ t) = 0
⊢ μ (s \ t) = 0

Tactic:
]

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
s t : Set α
ht : μ (s \ t) = 0
⊢ μ (s \ t) = 0
Module: Mathlib.MeasureTheory.Measure.MeasureSpace

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
s t : Set α
ht : μ tᶜ = 0
⊢ μ s = μ s

Tactic:
]

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
s t : Set α
⊢ s ∪ t =ᵐ[μ] s ↔ μ (t \ s) = 0

Tactic:
]

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
s t : Set α
⊢ s ∪ t =ᵐ[μ] s ↔ μ (t \ s) = 0
Module: Mathlib.MeasureTheory.Measure.MeasureSpace

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
s t : Set α
h : μ (t \ s) = 0
⊢ μ (t \ s) = 0

Tactic:
]

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
s t : Set α
h : μ (t \ s) = 0
⊢ μ (t \ s) = 0
Module: Mathlib.MeasureTheory.Measure.MeasureSpace

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
s t : Set α
⊢ s ≤ᵐ[μ] t ↔ s ≤ᵐ[μ] t

Tactic:
]

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

Goal state before:
pp: case inl
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
hb' : Iio b = ∅
this : IsEmpty ↑s
⊢ ⊥ = ⊥

Tactic:
]

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
s t : Set α
h₁ : s ≤ᵐ[μ] t
hsm : NullMeasurableSet s μ
h₂ : μ t = μ s
ht : μ s ≠ ∞
⊢ 0 = 0

Tactic:
]

Goal state after:
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
⊢ 𝓝[<] b = 𝓝[<] b

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
⊢ range Subtype.val ∈ 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
⊢ range Subtype.val ∈ 𝓝[<] b