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

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
m : Set (Measure α)
h : ∀ μ' ∈ m, μ ≤ μ'
this : μ.toOuterMeasure ≤ sInf (toOuterMeasure '' m)
s : Set α
hs : MeasurableSet s
⊢ μ s ≤ (sInf (toOuterMeasure '' m)) s

Tactic:
]

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
m : Set (Measure α)
h : ∀ μ' ∈ m, μ ≤ μ'
this : μ.toOuterMeasure ≤ sInf (toOuterMeasure '' m)
s : Set α
hs : MeasurableSet s
⊢ μ s ≤ (sInf (toOuterMeasure '' m)) s
Module: Mathlib.MeasureTheory.Measure.Restrict

Goal state before:
pp: α : Type u_2
m0 : MeasurableSpace α
μ : Measure α
β : Type u_7
inst✝ : MeasurableSpace β
b : β
f : α → β
s : Set α
f_mble : NullMeasurable f (μ.restrict s)
hs : f =ᵐ[μ.restrict sᶜ] fun x ↦ b
t : Set β
t_mble : MeasurableSet t
ht : b ∉ t
⊢ μ (f ⁻¹' t) = μ (f ⁻¹' t ∩ s)

Tactic:
]

Goal state after:
α : Type u_2
m0 : MeasurableSpace α
μ : Measure α
β : Type u_7
inst✝ : MeasurableSpace β
b : β
f : α → β
s : Set α
f_mble : NullMeasurable f (μ.restrict s)
hs : f =ᵐ[μ.restrict sᶜ] fun x ↦ b
t : Set β
t_mble : MeasurableSet t
ht : b ∉ t
⊢ μ (f ⁻¹' t) = μ (f ⁻¹' t ∩ s)
Module: Mathlib.MeasureTheory.Measure.Restrict

Goal state before:
pp: α : Type u_2
m0 : MeasurableSpace α
μ : Measure α
β : Type u_7
inst✝ : MeasurableSpace β
b : β
f : α → β
s : Set α
f_mble : NullMeasurable f (μ.restrict s)
hs : μ ({a | ¬f a = b} ∩ sᶜ) = 0
t : Set β
t_mble : MeasurableSet t
ht : b ∉ t
⊢ μ (f ⁻¹' t) = μ (f ⁻¹' t ∩ s)

pp: α : Type u_2
m0 : MeasurableSpace α
μ : Measure α
β : Type u_7
inst✝ : MeasurableSpace β
b : β
f : α → β
s : Set α
f_mble : NullMeasurable f (μ.restrict s)
hs : (μ.restrict sᶜ) {a | ¬f a = b} = 0
t : Set β
t_mble : MeasurableSet t
ht : b ∉ t
⊢ NullMeasurableSet {a | ¬f a = b} (μ.restrict sᶜ)

Tactic:
]

Goal state after:
α : Type u_2
m0 : MeasurableSpace α
μ : Measure α
β : Type u_7
inst✝ : MeasurableSpace β
b : β
f : α → β
s : Set α
f_mble : NullMeasurable f (μ.restrict s)
hs : μ ({a | ¬f a = b} ∩ sᶜ) = 0
t : Set β
t_mble : MeasurableSet t
ht : b ∉ t
⊢ μ (f ⁻¹' t) = μ (f ⁻¹' t ∩ s)

α : Type u_2
m0 : MeasurableSpace α
μ : Measure α
β : Type u_7
inst✝ : MeasurableSpace β
b : β
f : α → β
s : Set α
f_mble : NullMeasurable f (μ.restrict s)
hs : (μ.restrict sᶜ) {a | ¬f a = b} = 0
t : Set β
t_mble : MeasurableSet t
ht : b ∉ t
⊢ NullMeasurableSet {a | ¬f a = b} (μ.restrict sᶜ)
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
m : Set (Measure α)
x✝ : MeasurableSpace α
f : ℕ → Set α
a✝¹ : ∀ (i : ℕ), MeasurableSet (f i)
a✝ : Pairwise (Disjoint on f)
hne : (⋃ i, f i).Nonempty
⊢ ∑' (i : ℕ), ⊤ (f i) ≤ ∞

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
m : Set (Measure α)
x✝ : MeasurableSpace α
f : ℕ → Set α
a✝¹ : ∀ (i : ℕ), MeasurableSet (f i)
a✝ : Pairwise (Disjoint on f)
hne : (⋃ i, f i).Nonempty
⊢ ∑' (i : ℕ), ⊤ (f i) ≤ ∞
Module: Mathlib.MeasureTheory.Measure.Typeclasses.SFinite

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
s : Set α
hs : MeasurableSet s
t : Set α
v : ℕ → Set α
hv : t ⊆ ⋃ n, v n
h'v : ∀ (n : ℕ), μ (t ∩ v n) ≠ ∞
w : ℕ → Set α := fun n ↦ toMeasurable μ (t ∩ v n)
hw : ∀ (n : ℕ), μ (w n) < ∞
t' : Set α := ⋃ n, toMeasurable μ (t ∩ disjointed w n)
ht' : t' = ⋃ n, toMeasurable μ (t ∩ disjointed w n)
tt' : t ⊆ t'
u : Set α
hu : MeasurableSet u
⊢ μ (t ∩ u) = μ (t ∩ u)

Tactic:
]

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

Goal state before:
pp: α : Type u_2
m0 : MeasurableSpace α
μ : Measure α
β : Type u_7
inst✝ : MeasurableSpace β
b : β
f : α → β
s : Set α
f_mble : NullMeasurable f (μ.restrict s)
hs : μ ({a | ¬f a = b} ∩ sᶜ) = 0
t : Set β
t_mble : MeasurableSet t
ht : b ∉ t
⊢ μ (f ⁻¹' t ∩ sᶜ) ≤ μ ({a | ¬f a = b} ∩ sᶜ)

Tactic:
]

Goal state after:
α : Type u_2
m0 : MeasurableSpace α
μ : Measure α
β : Type u_7
inst✝ : MeasurableSpace β
b : β
f : α → β
s : Set α
f_mble : NullMeasurable f (μ.restrict s)
hs : μ ({a | ¬f a = b} ∩ sᶜ) = 0
t : Set β
t_mble : MeasurableSet t
ht : b ∉ t
⊢ μ (f ⁻¹' t ∩ sᶜ) ≤ μ ({a | ¬f a = b} ∩ sᶜ)
Module: Mathlib.MeasureTheory.Measure.Typeclasses.SFinite

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
s : Set α
hs : MeasurableSet s
t : Set α
v : ℕ → Set α
hv : t ⊆ ⋃ n, v n
h'v : ∀ (n : ℕ), μ (t ∩ v n) ≠ ∞
A : ∃ t' ⊇ t, MeasurableSet t' ∧ ∀ (u : Set α), MeasurableSet u → μ (t' ∩ u) = μ (t ∩ u)
⊢ μ
      ((if h : ∃ t_1 ⊇ t, MeasurableSet t_1 ∧ t_1 =ᵐ[μ] t then h.choose
        else
          if h' : ∃ t_1 ⊇ t, MeasurableSet t_1 ∧ ∀ (u : Set α), MeasurableSet u → μ (t_1 ∩ u) = μ (t ∩ u) then h'.choose
          else ⋯.choose) ∩
        s) =
    μ (t ∩ s)

Tactic:
]

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
s : Set α
hs : MeasurableSet s
t : Set α
v : ℕ → Set α
hv : t ⊆ ⋃ n, v n
h'v : ∀ (n : ℕ), μ (t ∩ v n) ≠ ∞
A : ∃ t' ⊇ t, MeasurableSet t' ∧ ∀ (u : Set α), MeasurableSet u → μ (t' ∩ u) = μ (t ∩ u)
⊢ μ
      ((if h : ∃ t_1 ⊇ t, MeasurableSet t_1 ∧ t_1 =ᵐ[μ] t then h.choose
        else
          if h' : ∃ t_1 ⊇ t, MeasurableSet t_1 ∧ ∀ (u : Set α), MeasurableSet u → μ (t_1 ∩ u) = μ (t ∩ u) then h'.choose
          else ⋯.choose) ∩
        s) =
    μ (t ∩ s)
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, ⨅ (_ : s ⊆ iUnion t), ∑' (n : ℕ), ⨅ μ_1 ∈ toOuterMeasure '' {μ, ν}, μ_1 (t n) =
    sInf {m | ∃ t, m = μ (t ∩ s) + ν (tᶜ ∩ s)}

Tactic:
]

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ ν : Measure α
s : Set α
hs : MeasurableSet s
⊢ ⨅ t, ⨅ (_ : s ⊆ iUnion t), ∑' (n : ℕ), ⨅ μ_1 ∈ toOuterMeasure '' {μ, ν}, μ_1 (t n) =
    sInf {m | ∃ t, m = μ (t ∩ s) + ν (tᶜ ∩ s)}
Module: Mathlib.MeasureTheory.Measure.Restrict

Goal state before:
pp: α : Type u_2
m0 : MeasurableSpace α
μ : Measure α
s : Set α
hs : NullMeasurableSet s μ
t : Set α
h : NullMeasurableSet (t ∩ s) μ
⊢ μ (t \ s ∩ s) = 0

Tactic:
]

Goal state after:
α : Type u_2
m0 : MeasurableSpace α
μ : Measure α
s : Set α
hs : NullMeasurableSet s μ
t : Set α
h : NullMeasurableSet (t ∩ s) μ
⊢ μ (t \ s ∩ s) = 0
Module: Mathlib.MeasureTheory.Measure.Typeclasses.SFinite

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
inst✝ : SFinite μ
s t : Set α
ht : MeasurableSet t
⊢ μ (s ∩ t) = μ (s ∩ t)

Tactic:
]

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

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
s : Set α
inst✝ : SigmaFinite μ
hs : MeasurableSet s
⊢ μ s = μ s

Tactic:
]

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

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
inst✝ : SigmaFinite μ
s : Set α
⊢ ⨆ i, (μ.restrict (spanningSets μ i)) s = ⨆ i, (μ.restrict (spanningSets μ i)) (toMeasurable μ s)

Tactic:
]

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
inst✝ : SigmaFinite μ
s : Set α
⊢ ⨆ i, (μ.restrict (spanningSets μ i)) s = ⨆ i, (μ.restrict (spanningSets μ i)) (toMeasurable μ s)
Module: Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated

Goal state before:
pp: α : Type u_1
inst✝¹ : MeasurableSpace α
inst✝ : CountablyGenerated α
s : Set α
hs : s ∈ countableGeneratingSet α
⊢ MeasurableSet s

Tactic:
]

Goal state after:
α : Type u_1
inst✝¹ : MeasurableSpace α
inst✝ : CountablyGenerated α
s : Set α
hs : s ∈ countableGeneratingSet α
⊢ MeasurableSet s
Module: Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated

Goal state before:
pp: α : Type u_3
m : MeasurableSpace α
inst✝ : CountablyGenerated α
⊢ m = m

Tactic:
]

Goal state after:
Module: Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated

Goal state before:
pp: α : Type u_1
β : Type u_2
f : α → β
b : Set (Set β)
hbc : b.Countable
⊢ CountablyGenerated α

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
f : α → β
b : Set (Set β)
hbc : b.Countable
⊢ CountablyGenerated α
Module: Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : Countable α
s : Set α
hs : MeasurableSet s
this : s = ⋃ y ∈ s, measurableAtom y
⊢ MeasurableSet (⋃ y ∈ s, measurableAtom y)

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : Countable α
s : Set α
hs : MeasurableSet s
this : s = ⋃ y ∈ s, measurableAtom y
⊢ MeasurableSet (⋃ y ∈ s, measurableAtom y)
Module: Mathlib.MeasureTheory.Measure.Typeclasses.SFinite

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
s : Set α
inst✝ : SigmaFinite μ
r : ℝ≥0∞
hs : MeasurableSet s
h's : ∃ i, r < (μ.restrict (spanningSets μ i)) s
⊢ ∃ t, MeasurableSet t ∧ t ⊆ s ∧ r < μ t ∧ μ t < ∞

Tactic:
]

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
s : Set α
inst✝ : SigmaFinite μ
r : ℝ≥0∞
hs : MeasurableSet s
h's : ∃ i, r < (μ.restrict (spanningSets μ i)) s
⊢ ∃ t, MeasurableSet t ∧ t ⊆ s ∧ r < μ t ∧ μ t < ∞
Module: Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
inst✝ : CountablyGenerated α
⊢ Pairwise (Function.onFun Disjoint (countablyGeneratedAtom α))

Tactic:
intro p q hpq s hsp hsq

Goal state after:
α : Type u_1
mα : MeasurableSpace α
inst✝ : CountablyGenerated α
p q : ℕ → Prop
hpq : p ≠ q
s : Set α
hsp : s ⊆ countablyGeneratedAtom α p
hsq : s ⊆ countablyGeneratedAtom α q
⊢ s ⊆ ⊥
Module: Mathlib.MeasureTheory.Measure.Restrict

Goal state before:
pp: α : Type u_2
m0 : MeasurableSpace α
μ : Measure α
s : Set α
t : Set ↑s
hs : NullMeasurableSet s μ
ht : MeasurableSet t
⊢ NullMeasurableSet (Subtype.val '' t) μ

Tactic:
comap_eq_generateFrom

Goal state after:
α : Type u_2
m0 : MeasurableSpace α
μ : Measure α
s : Set α
t : Set ↑s
hs : NullMeasurableSet s μ
ht : MeasurableSet t
⊢ NullMeasurableSet (Subtype.val '' t) μ
Module: Mathlib.MeasureTheory.Measure.Restrict

Goal state before:
pp: α : Type u_2
m0 : MeasurableSpace α
μ : Measure α
s : Set α
t : Set ↑s
hs : NullMeasurableSet s μ
ht : MeasurableSet t
⊢ NullMeasurableSet (Subtype.val '' t) μ

Tactic:
]

Goal state after:
α : Type u_2
m0 : MeasurableSpace α
μ : Measure α
s : Set α
t : Set ↑s
hs : NullMeasurableSet s μ
ht : MeasurableSet t
⊢ NullMeasurableSet (Subtype.val '' t) μ
Module: Mathlib.MeasureTheory.Measure.Restrict

Goal state before:
pp: case hC
α : Type u_2
m0 : MeasurableSpace α
μ : Measure α
s : Set α
t : Set ↑s
hs : NullMeasurableSet s μ
s' : Set α
hs' : MeasurableSet s'
ht✝ : MeasurableSet (Subtype.val ⁻¹' s')
⊢ NullMeasurableSet (s ∩ s') μ

Tactic:
]

Goal state after:
case hC
α : Type u_2
m0 : MeasurableSpace α
μ : Measure α
s : Set α
t : Set ↑s
hs : NullMeasurableSet s μ
s' : Set α
hs' : MeasurableSet s'
ht✝ : MeasurableSet (Subtype.val ⁻¹' s')
⊢ NullMeasurableSet (s ∩ s') μ
Module: Mathlib.MeasureTheory.Measure.Restrict

Goal state before:
pp: case iUnion
α : Type u_2
m0 : MeasurableSpace α
μ : Measure α
s : Set α
t : Set ↑s
hs : NullMeasurableSet s μ
f : ℕ → Set ↑s
hs✝ : ∀ (n : ℕ), MeasurableSet (f n)
hf : ∀ (n : ℕ), NullMeasurableSet (Subtype.val '' f n) μ
⊢ NullMeasurableSet (⋃ i, Subtype.val '' f i) μ

Tactic:
]

Goal state after:
case iUnion
α : Type u_2
m0 : MeasurableSpace α
μ : Measure α
s : Set α
t : Set ↑s
hs : NullMeasurableSet s μ
f : ℕ → Set ↑s
hs✝ : ∀ (n : ℕ), MeasurableSet (f n)
hf : ∀ (n : ℕ), NullMeasurableSet (Subtype.val '' f n) μ
⊢ NullMeasurableSet (⋃ i, Subtype.val '' f i) μ
Module: Mathlib.MeasureTheory.Measure.Restrict

Goal state before:
pp: δ : Type u_4
u : Set δ
inst✝ : MeasureSpace δ
hu : NullMeasurableSet u volume
⊢ volume (Subtype.val '' univ) = volume u

pp: δ : Type u_4
u : Set δ
inst✝ : MeasureSpace δ
hu : NullMeasurableSet u volume
⊢ Injective Subtype.val

pp: δ : Type u_4
u : Set δ
inst✝ : MeasureSpace δ
hu : NullMeasurableSet u volume
⊢ ∀ (s : Set { x // x ∈ u }), MeasurableSet s → NullMeasurableSet (Subtype.val '' s) volume

Tactic:
]

Goal state after:
δ : Type u_4
u : Set δ
inst✝ : MeasureSpace δ
hu : NullMeasurableSet u volume
⊢ volume (Subtype.val '' univ) = volume u

δ : Type u_4
u : Set δ
inst✝ : MeasureSpace δ
hu : NullMeasurableSet u volume
⊢ Injective Subtype.val

δ : Type u_4
u : Set δ
inst✝ : MeasureSpace δ
hu : NullMeasurableSet u volume
⊢ ∀ (s : Set { x // x ∈ u }), MeasurableSet s → NullMeasurableSet (Subtype.val '' s) volume
Module: Mathlib.MeasureTheory.Measure.Typeclasses.SFinite

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ ν₁ ν₂ : Measure α
inst✝ : SigmaFinite μ
h : ν₁ = ν₂
⊢ μ + ν₂ = μ + ν₂

Tactic:
]

Goal state after:
Module: Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
inst✝ : CountablyGenerated α
s : Set α
hs : MeasurableSet s
⊢ ∃ q, s = ⋃ p, if q p then countablyGeneratedAtom α p else ∅

Tactic:
]

Goal state after:
α : Type u_1
mα : MeasurableSpace α
inst✝ : CountablyGenerated α
s : Set α
hs : MeasurableSet s
⊢ ∃ q, s = ⋃ p, if q p then countablyGeneratedAtom α p else ∅
Module: Mathlib.MeasureTheory.Measure.Typeclasses.SFinite

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ ν₁ ν₂ : Measure α
inst✝ : SigmaFinite μ
h : μ + ν₁ = μ + ν₂
⊢ ∀ (i : ℕ), ν₁.restrict (spanningSets μ i) = ν₂.restrict (spanningSets μ i)

Tactic:
]

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ ν₁ ν₂ : Measure α
inst✝ : SigmaFinite μ
h : μ + ν₁ = μ + ν₂
⊢ ∀ (i : ℕ), ν₁.restrict (spanningSets μ i) = ν₂.restrict (spanningSets μ i)
Module: Mathlib.MeasureTheory.Measure.Restrict

Goal state before:
pp: α : Type u_2
β : Type u_3
m0 : MeasurableSpace α
m1 : MeasurableSpace β
f : α → β
hf : MeasurableEmbedding f
μ : Measure β
t : Set β
ht : MeasurableSet t
⊢ μ (t ∩ range f) = μ (t ∩ range f)

Tactic:
]

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

Goal state before:
pp: α : Type u_2
β : Type u_3
m0 : MeasurableSpace α
m1 : MeasurableSpace β
f : α → β
hf : MeasurableEmbedding f
μ : Measure β
s : Set α
⊢ (Measure.comap f μ) s = (Measure.comap f μ) s

Tactic:
]

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

Goal state before:
pp: α : Type u_2
β : Type u_3
m0 : MeasurableSpace α
m1 : MeasurableSpace β
f : α → β
hf : MeasurableEmbedding f
μ : Measure β
s : Set α
⊢ μ (f '' s) = μ (f '' s)

Tactic:
]

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

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ ν₁ ν₂ : Measure α
inst✝ : SigmaFinite μ
h : μ + ν₁ = μ + ν₂
i : ℕ
s : Set α
hs : MeasurableSet s
⊢ μ (s ∩ spanningSets μ i) + (ν₁.restrict (spanningSets μ i)) s =
    μ (s ∩ spanningSets μ i) + (ν₂.restrict (spanningSets μ i)) s

Tactic:
]

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ ν₁ ν₂ : Measure α
inst✝ : SigmaFinite μ
h : μ + ν₁ = μ + ν₂
i : ℕ
s : Set α
hs : MeasurableSet s
⊢ μ (s ∩ spanningSets μ i) + (ν₁.restrict (spanningSets μ i)) s =
    μ (s ∩ spanningSets μ i) + (ν₂.restrict (spanningSets μ i)) s
Module: Mathlib.MeasureTheory.Measure.Restrict

Goal state before:
pp: α : Type u_2
β : Type u_3
m0 : MeasurableSpace α
m1 : MeasurableSpace β
f : α → β
hf : MeasurableEmbedding f
μ : Measure α
t : Set α
a✝ : MeasurableSet t
⊢ μ t = μ t

Tactic:
]

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

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ ν₁ ν₂ : Measure α
inst✝ : SigmaFinite μ
⊢ ν₁ = ν₂ ↔ ν₁ = ν₂

Tactic:
]

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

Goal state before:
pp: α : Type u_2
β : Type u_3
m0 : MeasurableSpace α
m1 : MeasurableSpace β
f : α → β
hf : MeasurableEmbedding f
μ : Measure β
s : Set β
⊢ μ (s ∩ range f) = μ (s ∩ range f)

Tactic:
]

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

Goal state before:
pp: case inl
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
inst✝ : Countable α
hμ : ∀ (a : α), μ {a} < ∞
h✝ : IsEmpty α
⊢ SigmaFinite 0

Tactic:
]

Goal state after:
case inl
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
inst✝ : Countable α
hμ : ∀ (a : α), μ {a} < ∞
h✝ : IsEmpty α
⊢ SigmaFinite 0
Module: Mathlib.MeasureTheory.Measure.Restrict

Goal state before:
pp: α : Type u_2
β : Type u_3
m0 : MeasurableSpace α
m1 : MeasurableSpace β
f : α → β
hf : MeasurableEmbedding f
μ : Measure β
s : Set β
t : Set α
ht : MeasurableSet t
⊢ μ (f '' t ∩ s) = μ (f '' t ∩ s)

Tactic:
]

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

Goal state before:
pp: α : Type u_2
β : Type u_3
m0 : MeasurableSpace α
m1 : MeasurableSpace β
f : α → β
hf : MeasurableEmbedding f
μ : Measure β
s : Set α
⊢ (Measure.comap f μ).restrict s = (Measure.comap f μ).restrict s

Tactic:
]

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

Goal state before:
pp: α : Type u_2
β : Type u_3
m0 : MeasurableSpace α
m1 : MeasurableSpace β
e : α ≃ᵐ β
μ : Measure β
s : Set α
⊢ μ (⇑e.symm ⁻¹' s) = μ (⇑e.symm ⁻¹' s)

Tactic:
]

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

Goal state before:
pp: α : Type u_2
β : Type u_3
x✝¹ : MeasurableSpace α
x✝ : MeasurableSpace β
f : α → β
g : β → α
μ : Measure α
hf : Measurable f
hg : MeasurableEmbedding g
hμg : ∀ᵐ (a : α) ∂μ, a ∈ range g
hfg : ∀ (a : β), f (g a) = a
s : Set β
hs : MeasurableSet s
⊢ μ (f ⁻¹' s \ {x | (fun a ↦ a ∈ range g) x}ᶜ) = μ (g '' s)

Tactic:
]

Goal state after:
α : Type u_2
β : Type u_3
x✝¹ : MeasurableSpace α
x✝ : MeasurableSpace β
f : α → β
g : β → α
μ : Measure α
hf : Measurable f
hg : MeasurableEmbedding g
hμg : ∀ᵐ (a : α) ∂μ, a ∈ range g
hfg : ∀ (a : β), f (g a) = a
s : Set β
hs : MeasurableSet s
⊢ μ (f ⁻¹' s \ {x | (fun a ↦ a ∈ range g) x}ᶜ) = μ (g '' s)
Module: Mathlib.MeasureTheory.Measure.MeasureSpace

Goal state before:
pp: case refine_1.refine_2
α : Type u_1
m0 : MeasurableSpace α
μ ν : Measure α
s : Set α
hs : MeasurableSet s
t : Set α
x✝ : μ (t ∩ s) + ν (tᶜ ∩ s) ∈ {m | ∃ t, m = μ (t ∩ s) + ν (tᶜ ∩ s)}
t' : ℕ → Set α := fun n ↦ if n = 0 then t ∩ s else if n = 1 then tᶜ ∩ s else ∅
ht' : t' = fun n ↦ if n = 0 then t ∩ s else if n = 1 then tᶜ ∩ s else ∅
⊢ min (μ (t' 0)) (ν (t' 0)) + min (μ (t' 1)) (ν (t' 1)) ≤ μ (t ∩ s) + ν (tᶜ ∩ s)

pp: α : Type u_1
m0 : MeasurableSpace α
μ ν : Measure α
s : Set α
hs : MeasurableSet s
t : Set α
x✝ : μ (t ∩ s) + ν (tᶜ ∩ s) ∈ {m | ∃ t, m = μ (t ∩ s) + ν (tᶜ ∩ s)}
t' : ℕ → Set α := fun n ↦ if n = 0 then t ∩ s else if n = 1 then tᶜ ∩ s else ∅
ht' : t' = fun n ↦ if n = 0 then t ∩ s else if n = 1 then tᶜ ∩ s else ∅
⊢ ∀ (x : ℕ), (if x = 1 then 0 else if x = 0 then 0 else min (μ (t' x)) (ν (t' x))) = 0

Tactic:
]

Goal state after:
case refine_1.refine_2
α : Type u_1
m0 : MeasurableSpace α
μ ν : Measure α
s : Set α
hs : MeasurableSet s
t : Set α
x✝ : μ (t ∩ s) + ν (tᶜ ∩ s) ∈ {m | ∃ t, m = μ (t ∩ s) + ν (tᶜ ∩ s)}
t' : ℕ → Set α := fun n ↦ if n = 0 then t ∩ s else if n = 1 then tᶜ ∩ s else ∅
ht' : t' = fun n ↦ if n = 0 then t ∩ s else if n = 1 then tᶜ ∩ s else ∅
⊢ min (μ (t' 0)) (ν (t' 0)) + min (μ (t' 1)) (ν (t' 1)) ≤ μ (t ∩ s) + ν (tᶜ ∩ s)

α : Type u_1
m0 : MeasurableSpace α
μ ν : Measure α
s : Set α
hs : MeasurableSet s
t : Set α
x✝ : μ (t ∩ s) + ν (tᶜ ∩ s) ∈ {m | ∃ t, m = μ (t ∩ s) + ν (tᶜ ∩ s)}
t' : ℕ → Set α := fun n ↦ if n = 0 then t ∩ s else if n = 1 then tᶜ ∩ s else ∅
ht' : t' = fun n ↦ if n = 0 then t ∩ s else if n = 1 then tᶜ ∩ s else ∅
⊢ ∀ (x : ℕ), (if x = 1 then 0 else if x = 0 then 0 else min (μ (t' x)) (ν (t' x))) = 0
Module: Mathlib.MeasureTheory.Measure.Restrict

Goal state before:
pp: α : Type u_2
m0 : MeasurableSpace α
s : Set α
hs : MeasurableSet s
μ : Measure α
⊢ μ.restrict s = μ.restrict s

Tactic:
]

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

Goal state before:
pp: α : Type u_2
m0 : MeasurableSpace α
μ : Measure α
s : Set α
hs : MeasurableSet s
p : α → Prop
⊢ (∀ᵐ (x : { x // x ∈ s }) ∂Measure.comap Subtype.val μ, p ↑x) ↔ ∀ᵐ (x : ↑s) ∂Measure.comap Subtype.val μ, p ↑x

Tactic:
]

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

Goal state before:
pp: α : Type u_2
inst✝ : MeasureSpace α
s : Set α
hs : MeasurableSet s
⊢ volume.restrict s = volume.restrict s

Tactic:
]

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

Goal state before:
pp: α : Type u_2
inst✝ : MeasureSpace α
s t : Set α
hs : NullMeasurableSet s volume
ht : MeasurableSet t
⊢ volume (t ∩ s) = volume (t ∩ s)

Tactic:
]

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

Goal state before:
pp: α : Type u_2
β : Type u_3
inst✝¹ : MeasurableSpace α
μ : Measure α
s : Set α
f g : α → β
inst✝ : DecidablePred fun x ↦ x ∈ s
hs : MeasurableSet s
⊢ s.piecewise f g =ᶠ[ae μ ⊓ 𝓟 s] f

Tactic:
]

Goal state after:
α : Type u_2
β : Type u_3
inst✝¹ : MeasurableSpace α
μ : Measure α
s : Set α
f g : α → β
inst✝ : DecidablePred fun x ↦ x ∈ s
hs : MeasurableSet s
⊢ s.piecewise f g =ᶠ[ae μ ⊓ 𝓟 s] f
Module: Mathlib.MeasureTheory.Measure.Restrict

Goal state before:
pp: α : Type u_2
β : Type u_3
inst✝¹ : MeasurableSpace α
μ : Measure α
s : Set α
f g : α → β
inst✝ : DecidablePred fun x ↦ x ∈ s
hs : MeasurableSet s
⊢ s.piecewise f g =ᶠ[ae μ ⊓ 𝓟 sᶜ] g

Tactic:
]

Goal state after:
α : Type u_2
β : Type u_3
inst✝¹ : MeasurableSpace α
μ : Measure α
s : Set α
f g : α → β
inst✝ : DecidablePred fun x ↦ x ∈ s
hs : MeasurableSet s
⊢ s.piecewise f g =ᶠ[ae μ ⊓ 𝓟 sᶜ] g
Module: Mathlib.MeasureTheory.Measure.Restrict

Goal state before:
pp: α : Type u_2
β : Type u_3
inst✝¹ : MeasurableSpace α
μ : Measure α
s : Set α
f : α → β
inst✝ : Zero β
t : Set β
ht : 0 ∈ t
hs : MeasurableSet s
⊢ μ (f ⁻¹' t ∩ s ∪ 0 ⁻¹' t \ s)ᶜ = 0 ↔ μ ((f ⁻¹' t)ᶜ ∩ s) = 0

Tactic:
]

Goal state after:
α : Type u_2
β : Type u_3
inst✝¹ : MeasurableSpace α
μ : Measure α
s : Set α
f : α → β
inst✝ : Zero β
t : Set β
ht : 0 ∈ t
hs : MeasurableSet s
⊢ μ (f ⁻¹' t ∩ s ∪ 0 ⁻¹' t \ s)ᶜ = 0 ↔ μ ((f ⁻¹' t)ᶜ ∩ s) = 0
Module: Mathlib.MeasureTheory.Measure.Typeclasses.SFinite

Goal state before:
pp: case pos
α : Type u_1
μ : Measure α
h this : SigmaFinite μ
s : ℕ → Set α := spanningSets μ
hs_univ : ⋃ i, s i = univ
hs_meas : ∀ (i : ℕ), s i = ∅ ∨ s i = univ
h_univ_empty : univ = ∅
⊢ 0 < ∞

Tactic:
]

Goal state after:
case pos
α : Type u_1
μ : Measure α
h this : SigmaFinite μ
s : ℕ → Set α := spanningSets μ
hs_univ : ⋃ i, s i = univ
hs_meas : ∀ (i : ℕ), s i = ∅ ∨ s i = univ
h_univ_empty : univ = ∅
⊢ 0 < ∞
Module: Mathlib.MeasureTheory.Measure.Restrict

Goal state before:
pp: α : Type u_2
β : Type u_3
inst✝¹ : MeasurableSpace α
μ : Measure α
s : Set α
f : α → β
inst✝ : Zero β
t : Set β
ht : 0 ∉ t
⊢ μ (((f ⁻¹' t)ᶜ ∪ sᶜ) ∩ (0 ⁻¹' t \ s)ᶜ) = 0 ↔ μ ((f ⁻¹' t)ᶜ ∪ sᶜ) = 0

Tactic:
]

Goal state after:
α : Type u_2
β : Type u_3
inst✝¹ : MeasurableSpace α
μ : Measure α
s : Set α
f : α → β
inst✝ : Zero β
t : Set β
ht : 0 ∉ t
⊢ μ (((f ⁻¹' t)ᶜ ∪ sᶜ) ∩ (0 ⁻¹' t \ s)ᶜ) = 0 ↔ μ ((f ⁻¹' t)ᶜ ∪ sᶜ) = 0
Module: Mathlib.MeasureTheory.Measure.Restrict

Goal state before:
pp: case pos
α : Type u_2
β : Type u_3
inst✝¹ : MeasurableSpace α
μ : Measure α
s : Set α
f : α → β
inst✝ : Zero β
hs : MeasurableSet s
t : Set β
ht : 0 ∈ t
⊢ t ∈ Filter.map f (ae (μ.restrict s)) → t ∈ Filter.map f (ae (μ.restrict s))

Tactic:
]

Goal state after:
case pos
α : Type u_2
β : Type u_3
inst✝¹ : MeasurableSpace α
μ : Measure α
s : Set α
f : α → β
inst✝ : Zero β
hs : MeasurableSet s
t : Set β
ht : 0 ∈ t
⊢ t ∈ Filter.map f (ae (μ.restrict s)) → t ∈ Filter.map f (ae (μ.restrict s))
Module: Mathlib.MeasureTheory.Measure.Restrict

Goal state before:
pp: case neg
α : Type u_2
β : Type u_3
inst✝¹ : MeasurableSpace α
μ : Measure α
s : Set α
f : α → β
inst✝ : Zero β
hs : MeasurableSet s
t : Set β
ht : 0 ∉ t
⊢ μ ((f ⁻¹' t)ᶜ ∪ sᶜ) = 0 → μ ((f ⁻¹' t)ᶜ ∩ s) = 0

Tactic:
]

Goal state after:
case neg
α : Type u_2
β : Type u_3
inst✝¹ : MeasurableSpace α
μ : Measure α
s : Set α
f : α → β
inst✝ : Zero β
hs : MeasurableSet s
t : Set β
ht : 0 ∉ t
⊢ μ ((f ⁻¹' t)ᶜ ∪ sᶜ) = 0 → μ ((f ⁻¹' t)ᶜ ∩ s) = 0