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

Goal state before:
pp: case pos
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
R : Type u_4
inst✝¹ : SMul R ℝ≥0∞
inst✝ : IsScalarTower R ℝ≥0∞ ℝ≥0∞
μ : Measure α
f : α → β
c : ℝ≥0∞
hc : c ≠ 0
hf : AEMeasurable f μ
⊢ map f (c • μ) = c • map f μ

pp: case neg
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
R : Type u_4
inst✝¹ : SMul R ℝ≥0∞
inst✝ : IsScalarTower R ℝ≥0∞ ℝ≥0∞
μ : Measure α
f : α → β
c : ℝ≥0∞
hc : c ≠ 0
hf : ¬AEMeasurable f μ
⊢ map f (c • μ) = c • map f μ

Tactic:
·

Goal state after:
case pos
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
R : Type u_4
inst✝¹ : SMul R ℝ≥0∞
inst✝ : IsScalarTower R ℝ≥0∞ ℝ≥0∞
μ : Measure α
f : α → β
c : ℝ≥0∞
hc : c ≠ 0
hf : AEMeasurable f μ
⊢ map f (c • μ) = c • map f μ
Module: Mathlib.MeasureTheory.Measure.Map

Goal state before:
pp: case neg
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
R : Type u_4
inst✝¹ : SMul R ℝ≥0∞
inst✝ : IsScalarTower R ℝ≥0∞ ℝ≥0∞
μ : Measure α
f : α → β
c : ℝ≥0∞
hc : c ≠ 0
hf : ¬AEMeasurable f μ
⊢ map f (c • μ) = c • map f μ

Tactic:
·

Goal state after:
case neg
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
R : Type u_4
inst✝¹ : SMul R ℝ≥0∞
inst✝ : IsScalarTower R ℝ≥0∞ ℝ≥0∞
μ : Measure α
f : α → β
c : ℝ≥0∞
hc : c ≠ 0
hf : ¬AEMeasurable f μ
⊢ map f (c • μ) = c • map f μ
Module: Mathlib.MeasureTheory.Measure.Map

Goal state before:
pp: case pos
α : Type u_1
β : Type u_2
mβ : MeasurableSpace β
m0 : MeasurableSpace α
f : α → β
hf : MeasurableSet (range f)
μ : Measure α
h : AEMeasurable f μ
⊢ ∀ᵐ (x : β) ∂map f μ, x ∈ range f

pp: case neg
α : Type u_1
β : Type u_2
mβ : MeasurableSpace β
m0 : MeasurableSpace α
f : α → β
hf : MeasurableSet (range f)
μ : Measure α
h : ¬AEMeasurable f μ
⊢ ∀ᵐ (x : β) ∂map f μ, x ∈ range f

Tactic:
·

Goal state after:
case pos
α : Type u_1
β : Type u_2
mβ : MeasurableSpace β
m0 : MeasurableSpace α
f : α → β
hf : MeasurableSet (range f)
μ : Measure α
h : AEMeasurable f μ
⊢ ∀ᵐ (x : β) ∂map f μ, x ∈ range f
Module: Mathlib.MeasureTheory.Measure.Map

Goal state before:
pp: case neg
α : Type u_1
β : Type u_2
mβ : MeasurableSpace β
m0 : MeasurableSpace α
f : α → β
hf : MeasurableSet (range f)
μ : Measure α
h : ¬AEMeasurable f μ
⊢ ∀ᵐ (x : β) ∂map f μ, x ∈ range f

Tactic:
·

Goal state after:
case neg
α : Type u_1
β : Type u_2
mβ : MeasurableSpace β
m0 : MeasurableSpace α
f : α → β
hf : MeasurableSet (range f)
μ : Measure α
h : ¬AEMeasurable f μ
⊢ ∀ᵐ (x : β) ∂map f μ, x ∈ range f
Module: Mathlib.MeasureTheory.Measure.AbsolutelyContinuous

Goal state before:
pp: case refine_1
α : Type u_1
mα : MeasurableSpace α
μ₁ μ₂ ν : Measure α
h : μ₁ + μ₂ ≪ ν
⊢ μ₁ ≪ ν ∧ μ₂ ≪ ν

pp: case refine_2
α : Type u_1
mα : MeasurableSpace α
μ₁ μ₂ ν : Measure α
h : μ₁ ≪ ν ∧ μ₂ ≪ ν
⊢ ν + ν ≪ ν

Tactic:
·

Goal state after:
case refine_1
α : Type u_1
mα : MeasurableSpace α
μ₁ μ₂ ν : Measure α
h : μ₁ + μ₂ ≪ ν
⊢ μ₁ ≪ ν ∧ μ₂ ≪ ν
Module: Mathlib.MeasureTheory.Measure.AbsolutelyContinuous

Goal state before:
pp: case refine_2
α : Type u_1
mα : MeasurableSpace α
μ₁ μ₂ ν : Measure α
h : μ₁ ≪ ν ∧ μ₂ ≪ ν
⊢ ν + ν ≪ ν

Tactic:
·

Goal state after:
case refine_2
α : Type u_1
mα : MeasurableSpace α
μ₁ μ₂ ν : Measure α
h : μ₁ ≪ ν ∧ μ₂ ≪ ν
⊢ ν + ν ≪ ν
Module: Mathlib.MeasureTheory.Measure.Comap

Goal state before:
pp: case pos
α : Type u_1
β : Type u_2
s : Set α
mα : MeasurableSpace α
mβ : MeasurableSpace β
f : α → β
μ : Measure β
hs : NullMeasurableSet s (comap f μ)
hf : Injective f ∧ ∀ (t : Set α), MeasurableSet t → NullMeasurableSet (f '' t) μ
⊢ (comap f μ) s ≤ μ (f '' s)

pp: case neg
α : Type u_1
β : Type u_2
s : Set α
mα : MeasurableSpace α
mβ : MeasurableSpace β
f : α → β
μ : Measure β
hs : NullMeasurableSet s (comap f μ)
hf : ¬(Injective f ∧ ∀ (t : Set α), MeasurableSet t → NullMeasurableSet (f '' t) μ)
⊢ (comap f μ) s ≤ μ (f '' s)

Tactic:
·

Goal state after:
case pos
α : Type u_1
β : Type u_2
s : Set α
mα : MeasurableSpace α
mβ : MeasurableSpace β
f : α → β
μ : Measure β
hs : NullMeasurableSet s (comap f μ)
hf : Injective f ∧ ∀ (t : Set α), MeasurableSet t → NullMeasurableSet (f '' t) μ
⊢ (comap f μ) s ≤ μ (f '' s)
Module: Mathlib.MeasureTheory.Measure.Comap

Goal state before:
pp: case neg
α : Type u_1
β : Type u_2
s : Set α
mα : MeasurableSpace α
mβ : MeasurableSpace β
f : α → β
μ : Measure β
hs : NullMeasurableSet s (comap f μ)
hf : ¬(Injective f ∧ ∀ (t : Set α), MeasurableSet t → NullMeasurableSet (f '' t) μ)
⊢ (comap f μ) s ≤ μ (f '' s)

Tactic:
·

Goal state after:
case neg
α : Type u_1
β : Type u_2
s : Set α
mα : MeasurableSpace α
mβ : MeasurableSpace β
f : α → β
μ : Measure β
hs : NullMeasurableSet s (comap f μ)
hf : ¬(Injective f ∧ ∀ (t : Set α), MeasurableSet t → NullMeasurableSet (f '' t) μ)
⊢ (comap f μ) s ≤ μ (f '' s)
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
he' : QuasiMeasurePreserving (⇑e.symm) μ μ
hs : ⇑e '' s =ᵐ[μ] s
k : ℕ
⊢ ⇑(e ^ ↑k).symm ⁻¹' s =ᵐ[μ] s

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)).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
he' : QuasiMeasurePreserving (⇑e.symm) μ μ
hs : ⇑e '' s =ᵐ[μ] s
k : ℕ
⊢ ⇑(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)).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)).symm ⁻¹' s =ᵐ[μ] s
Module: Mathlib.MeasureTheory.Measure.Comap

Goal state before:
pp: case refine_2
α : Type u_1
β : Type u_2
s : Set α
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
hs : NullMeasurableSet s (comap f μ)
⊢ NullMeasurableSet (f '' toMeasurable (comap f μ) s) μ

pp: case refine_1
α : Type u_1
β : Type u_2
s : Set α
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
hs : NullMeasurableSet s (comap f μ)
⊢ f '' s =ᵐ[μ] f '' toMeasurable (comap f μ) s

Tactic:
·

Goal state after:
case refine_2
α : Type u_1
β : Type u_2
s : Set α
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
hs : NullMeasurableSet s (comap f μ)
⊢ NullMeasurableSet (f '' toMeasurable (comap f μ) s) μ
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
hf : Injective f ∧ ∀ (s : Set α), MeasurableSet s → NullMeasurableSet (f '' s) 0
⊢ comap f 0 = 0

pp: case neg
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
f : α → β
hf : ¬(Injective f ∧ ∀ (s : Set α), MeasurableSet s → NullMeasurableSet (f '' s) 0)
⊢ comap f 0 = 0

Tactic:
·

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

Goal state before:
pp: case neg
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
f : α → β
hf : ¬(Injective f ∧ ∀ (s : Set α), MeasurableSet s → NullMeasurableSet (f '' s) 0)
⊢ comap f 0 = 0

Tactic:
·

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

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

Goal state before:
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:
all_goals simp [*]

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

Goal state before:
pp: case refine_1
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
s : Set α
f : α → α
h : QuasiMeasurePreserving f μ μ
hs' : f ⁻¹' s =ᵐ[μ] s
t : Set α
htm : MeasurableSet t
ht : s =ᵐ[μ] t
⊢ MeasurableSet (limsup (fun x ↦ f^[x] ⁻¹' t) atTop)

pp: case refine_2
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
s : Set α
f : α → α
h : QuasiMeasurePreserving f μ μ
hs' : f ⁻¹' s =ᵐ[μ] s
t : Set α
htm : MeasurableSet t
ht : s =ᵐ[μ] t
⊢ limsup (fun x ↦ f^[x] ⁻¹' t) atTop =ᵐ[μ] s

pp: case refine_3
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
s : Set α
f : α → α
h : QuasiMeasurePreserving f μ μ
hs' : f ⁻¹' s =ᵐ[μ] s
t : Set α
htm : MeasurableSet t
ht : s =ᵐ[μ] t
⊢ f ⁻¹' limsup (fun x ↦ f^[x] ⁻¹' t) atTop = limsup (fun x ↦ f^[x] ⁻¹' t) atTop

Tactic:
·

Goal state after:
case refine_1
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
s : Set α
f : α → α
h : QuasiMeasurePreserving f μ μ
hs' : f ⁻¹' s =ᵐ[μ] s
t : Set α
htm : MeasurableSet t
ht : s =ᵐ[μ] t
⊢ MeasurableSet (limsup (fun x ↦ f^[x] ⁻¹' t) atTop)
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
⊢ comap f (comap g μ) = comap (g ∘ f) μ

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
⊢ comap f (comap g μ) = comap (g ∘ f) μ

Tactic:
·

Goal state after:
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
⊢ comap f (comap g μ) = comap (g ∘ f) μ
Module: Mathlib.MeasureTheory.Measure.QuasiMeasurePreserving

Goal state before:
pp: case refine_2
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
s : Set α
f : α → α
h : QuasiMeasurePreserving f μ μ
hs' : f ⁻¹' s =ᵐ[μ] s
t : Set α
htm : MeasurableSet t
ht : s =ᵐ[μ] t
⊢ limsup (fun x ↦ f^[x] ⁻¹' t) atTop =ᵐ[μ] s

pp: case refine_3
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
s : Set α
f : α → α
h : QuasiMeasurePreserving f μ μ
hs' : f ⁻¹' s =ᵐ[μ] s
t : Set α
htm : MeasurableSet t
ht : s =ᵐ[μ] t
⊢ f ⁻¹' limsup (fun x ↦ f^[x] ⁻¹' t) atTop = limsup (fun x ↦ f^[x] ⁻¹' t) atTop

Tactic:
·

Goal state after:
case refine_2
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
s : Set α
f : α → α
h : QuasiMeasurePreserving f μ μ
hs' : f ⁻¹' s =ᵐ[μ] s
t : Set α
htm : MeasurableSet t
ht : s =ᵐ[μ] t
⊢ limsup (fun x ↦ f^[x] ⁻¹' t) atTop =ᵐ[μ] s
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
⊢ comap f (comap g μ) = comap (g ∘ f) μ

Tactic:
·

Goal state after:
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
⊢ comap f (comap g μ) = comap (g ∘ f) μ
Module: Mathlib.MeasureTheory.Measure.QuasiMeasurePreserving

Goal state before:
pp: case refine_3
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
s : Set α
f : α → α
h : QuasiMeasurePreserving f μ μ
hs' : f ⁻¹' s =ᵐ[μ] s
t : Set α
htm : MeasurableSet t
ht : s =ᵐ[μ] t
⊢ f ⁻¹' limsup (fun x ↦ f^[x] ⁻¹' t) atTop = limsup (fun x ↦ f^[x] ⁻¹' t) atTop

Tactic:
·

Goal state after:
case refine_3
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
s : Set α
f : α → α
h : QuasiMeasurePreserving f μ μ
hs' : f ⁻¹' s =ᵐ[μ] s
t : Set α
htm : MeasurableSet t
ht : s =ᵐ[μ] t
⊢ f ⁻¹' limsup (fun x ↦ f^[x] ⁻¹' t) atTop = limsup (fun x ↦ f^[x] ⁻¹' t) atTop
Module: Mathlib.MeasureTheory.Measure.Comap

Goal state before:
pp: case inl
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
f : α → β
μ : Measure β
⊢ comap f (0 • μ) = 0 • comap f μ

pp: case inr
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
⊢ comap f (c • μ) = c • comap f μ

Tactic:
·

Goal state after:
case inl
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
f : α → β
μ : Measure β
⊢ comap f (0 • μ) = 0 • comap f μ
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
⊢ comap f (c • μ) = c • comap f μ

pp: case neg
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)
⊢ comap f (c • μ) = c • comap f μ

Tactic:
·

Goal state after:
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
⊢ comap f (c • μ) = c • comap f μ
Module: Mathlib.MeasureTheory.Measure.Comap

Goal state before:
pp: case neg
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)
⊢ comap f (c • μ) = c • comap f μ

Tactic:
·

Goal state after:
case neg
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)
⊢ comap f (c • μ) = c • comap f μ
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
b : X
⊢ comap Subtype.val (𝓝[<] b) = atTop ↔ ∅ ⊆ Iio b ∧ (∅.Nonempty → ∀ a < b, (∅ ∩ Ioo a b).Nonempty)

pp: case inr
X : Type u_1
inst✝² : LinearOrder X
inst✝¹ : TopologicalSpace X
inst✝ : OrderTopology X
s : Set X
b : X
hsne : s.Nonempty
⊢ comap Subtype.val (𝓝[<] b) = atTop ↔ s ⊆ Iio b ∧ (s.Nonempty → ∀ a < b, (s ∩ Ioo a b).Nonempty)

Tactic:
·

Goal state after:
case inl
X : Type u_1
inst✝² : LinearOrder X
inst✝¹ : TopologicalSpace X
inst✝ : OrderTopology X
b : X
⊢ comap Subtype.val (𝓝[<] b) = atTop ↔ ∅ ⊆ Iio b ∧ (∅.Nonempty → ∀ a < b, (∅ ∩ Ioo a b).Nonempty)
Module: Mathlib.Topology.Order.AtTopBotIxx

Goal state before:
pp: case pos
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
⊢ comap Subtype.val (𝓝[<] b) = atTop ↔ s ⊆ Iio b ∧ ∀ a < b, (s ∩ Ioo a b).Nonempty

pp: case neg
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
⊢ comap Subtype.val (𝓝[<] b) = atTop ↔ s ⊆ Iio b ∧ ∀ a < b, (s ∩ Ioo a b).Nonempty

Tactic:
·

Goal state after:
case pos
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
⊢ comap Subtype.val (𝓝[<] b) = atTop ↔ s ⊆ Iio b ∧ ∀ a < b, (s ∩ Ioo a b).Nonempty
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
⊢ comap Subtype.val (𝓝[<] b) = atTop → ∀ a < b, (s ∩ Ioo a b).Nonempty

pp: case pos.mpr
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
⊢ (∀ a < b, (s ∩ Ioo a b).Nonempty) → comap Subtype.val (𝓝[<] b) = atTop

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
⊢ comap Subtype.val (𝓝[<] b) = atTop → ∀ 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 : Finset α
inst✝ : MeasurableSingletonClass α
⊢ ∑ x ∈ s, μ {x} = ∑ x ∈ s, μ (id ⁻¹' {x})

pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
s : Finset α
inst✝ : MeasurableSingletonClass α
⊢ ∑ x ∈ s, μ (id ⁻¹' {x}) = μ ↑s

Tactic:
·

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
s : Finset α
inst✝ : MeasurableSingletonClass α
⊢ ∑ x ∈ s, μ {x} = ∑ x ∈ s, μ (id ⁻¹' {x})
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
Module: Mathlib.MeasureTheory.Measure.MeasureSpace

Goal state before:
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:
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.Topology.Order.AtTopBotIxx

Goal state before:
pp: case pos.mpr
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
⊢ (∀ a < b, (s ∩ Ioo a b).Nonempty) → comap Subtype.val (𝓝[<] b) = atTop

Tactic:
·

Goal state after:
case pos.mpr
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
⊢ (∀ a < b, (s ∩ Ioo a b).Nonempty) → comap Subtype.val (𝓝[<] b) = atTop
Module: Mathlib.Topology.Order.AtTopBotIxx

Goal state before:
pp: case pos.mpr.refine_1
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 : ∀ a < b, (s ∩ Ioo a b).Nonempty
⊢ ∀ i < b, ∃ i', True ∧ Ici i' ⊆ Subtype.val ⁻¹' Ioo i b

pp: case pos.mpr.refine_2
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 : ∀ a < b, (s ∩ Ioo a b).Nonempty
⊢ ∀ (i' : { x // x ∈ s }), True → ∃ i < b, Subtype.val ⁻¹' Ioo i b ⊆ Ici i'

Tactic:
·

Goal state after:
case pos.mpr.refine_1
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 : ∀ a < b, (s ∩ Ioo a b).Nonempty
⊢ ∀ i < b, ∃ i', True ∧ Ici i' ⊆ Subtype.val ⁻¹' Ioo i b
Module: Mathlib.Topology.Order.AtTopBotIxx

Goal state before:
pp: case pos.mpr.refine_2
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 : ∀ a < b, (s ∩ Ioo a b).Nonempty
⊢ ∀ (i' : { x // x ∈ s }), True → ∃ i < b, Subtype.val ⁻¹' Ioo i b ⊆ Ici i'

Tactic:
·

Goal state after:
case pos.mpr.refine_2
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 : ∀ a < b, (s ∩ Ioo a b).Nonempty
⊢ ∀ (i' : { x // x ∈ s }), True → ∃ i < b, Subtype.val ⁻¹' Ioo i b ⊆ Ici i'
Module: Mathlib.Topology.Order.AtTopBotIxx

Goal state before:
pp: case neg
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
⊢ comap Subtype.val (𝓝[<] b) = atTop ↔ s ⊆ Iio b ∧ ∀ a < b, (s ∩ Ioo a b).Nonempty

Tactic:
·

Goal state after:
case neg
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
⊢ comap Subtype.val (𝓝[<] b) = atTop ↔ s ⊆ Iio b ∧ ∀ 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 t : Set α
ht : μ (s \ t) = 0
⊢ μ (s ∩ tᶜ) = 0

Tactic:
← sdiff_eq

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.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 = ∅
⊢ map Subtype.val atTop = 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
⊢ map Subtype.val atTop = 𝓝[<] b

Tactic:
·

Goal state after:
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 = ∅
⊢ map Subtype.val atTop = 𝓝[<] b
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
⊢ map Subtype.val atTop = 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
⊢ map Subtype.val atTop = 𝓝[<] b