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

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
s : Set α
⊢ MeasurableSet
    (if h : ∃ t ⊇ s, MeasurableSet t ∧ t =ᵐ[μ] s then h.choose
    else
      if h' : ∃ t ⊇ s, MeasurableSet t ∧ ∀ (u : Set α), MeasurableSet u → μ (t ∩ u) = μ (s ∩ u) then h'.choose
      else ⋯.choose)

Tactic:
;

Goal state after:
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
s : Set α
⊢ MeasurableSet
    (if h : ∃ t ⊇ s, MeasurableSet t ∧ t =ᵐ[μ] s then h.choose
    else
      if h' : ∃ t ⊇ s, MeasurableSet t ∧ ∀ (u : Set α), MeasurableSet u → μ (t ∩ u) = μ (s ∩ u) then h'.choose
      else ⋯.choose)
Module: Mathlib.MeasureTheory.Measure.MeasureSpaceDef

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
s : Set α
⊢ μ
      (if h : ∃ t ⊇ s, MeasurableSet t ∧ t =ᵐ[μ] s then h.choose
      else
        if h' : ∃ t ⊇ s, MeasurableSet t ∧ ∀ (u : Set α), MeasurableSet u → μ (t ∩ u) = μ (s ∩ u) then h'.choose
        else ⋯.choose) =
    μ s

Tactic:
;

Goal state after:
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
s : Set α
⊢ μ
      (if h : ∃ t ⊇ s, MeasurableSet t ∧ t =ᵐ[μ] s then h.choose
      else
        if h' : ∃ t ⊇ s, MeasurableSet t ∧ ∀ (u : Set α), MeasurableSet u → μ (t ∩ u) = μ (s ∩ u) then h'.choose
        else ⋯.choose) =
    μ s
Module: Mathlib.MeasureTheory.OuterMeasure.AE

Goal state before:
pp: α : Type u_1
F : Type u_3
inst✝² : FunLike F (Set α) ℝ≥0∞
inst✝¹ : OuterMeasureClass F α
μ : F
M : Type u_4
inst✝ : One M
f : α → M
s : Set α
⊢ ({a | a ∈ s ∧ ¬f a = 1} ∈ fun x ↦ μ x = 0) ↔ μ (s ∩ Function.mulSupport f) = 0

Tactic:
;

Goal state after:
α : Type u_1
F : Type u_3
inst✝² : FunLike F (Set α) ℝ≥0∞
inst✝¹ : OuterMeasureClass F α
μ : F
M : Type u_4
inst✝ : One M
f : α → M
s : Set α
⊢ ({a | a ∈ s ∧ ¬f a = 1} ∈ fun x ↦ μ x = 0) ↔ μ (s ∩ Function.mulSupport f) = 0
Module: Mathlib.MeasureTheory.MeasurableSpace.MeasurablyGenerated

Goal state before:
pp: α : Type u_1
s : Set α
this : MeasurableSpace α := generateFrom {s}
u : Set Prop
⊢ MeasurableSet ((fun x ↦ x ∈ s) ⁻¹' u)

Tactic:
exact (show MeasurableSet s from GenerateMeasurable.basic _ <| mem_singleton s).mem trivial

Goal state after:
Module: Mathlib.MeasureTheory.OuterMeasure.Induced

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
m : (s : Set α) → MeasurableSet s → ℝ≥0∞
m0 : m ∅ ⋯ = 0
mU :
  ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), MeasurableSet (f i)),
    Pairwise (Disjoint on f) → m (⋃ i, f i) ⋯ = ∑' (i : ℕ), m (f i) ⋯
s₁ s₂ : Set α
h₁ : MeasurableSet s₁
hs : s₁ ⊆ s₂
⊢ ∀ (i : MeasurableSet s₂), extend m s₁ ≤ m s₂ i

Tactic:
;

Goal state after:
α : Type u_1
inst✝ : MeasurableSpace α
m : (s : Set α) → MeasurableSet s → ℝ≥0∞
m0 : m ∅ ⋯ = 0
mU :
  ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), MeasurableSet (f i)),
    Pairwise (Disjoint on f) → m (⋃ i, f i) ⋯ = ∑' (i : ℕ), m (f i) ⋯
s₁ s₂ : Set α
h₁ : MeasurableSet s₁
hs : s₁ ⊆ s₂
⊢ ∀ (i : MeasurableSet s₂), extend m s₁ ≤ m s₂ i
Module: Mathlib.MeasureTheory.OuterMeasure.Induced

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
m : (s : Set α) → MeasurableSet s → ℝ≥0∞
m0 : m ∅ ⋯ = 0
mU :
  ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), MeasurableSet (f i)),
    Pairwise (Disjoint on f) → m (⋃ i, f i) ⋯ = ∑' (i : ℕ), m (f i) ⋯
⊢ ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), MeasurableSet (f i)), m (⋃ i, f i) ⋯ ≤ ∑' (i : ℕ), m (f i) ⋯

Tactic:
;

Goal state after:
α : Type u_1
inst✝ : MeasurableSpace α
m : (s : Set α) → MeasurableSet s → ℝ≥0∞
m0 : m ∅ ⋯ = 0
mU :
  ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), MeasurableSet (f i)),
    Pairwise (Disjoint on f) → m (⋃ i, f i) ⋯ = ∑' (i : ℕ), m (f i) ⋯
⊢ ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), MeasurableSet (f i)), m (⋃ i, f i) ⋯ ≤ ∑' (i : ℕ), m (f i) ⋯
Module: Mathlib.MeasureTheory.OuterMeasure.Induced

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
m : OuterMeasure α
s : Set α
⊢ ∃ t, s ⊆ t ∧ MeasurableSet t ∧ m t = ⨅ t, ⨅ (_ : s ⊆ t), ⨅ (_ : MeasurableSet t), m t

Tactic:
;

Goal state after:
α : Type u_1
inst✝ : MeasurableSpace α
m : OuterMeasure α
s : Set α
⊢ ∃ t, s ⊆ t ∧ MeasurableSet t ∧ m t = ⨅ t, ⨅ (_ : s ⊆ t), ⨅ (_ : MeasurableSet t), m t
Module: Mathlib.MeasureTheory.OuterMeasure.Induced

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
m : OuterMeasure α
s : Set α
ms : ℝ≥0∞ := ⨅ t, ⨅ (_ : s ⊆ t), ⨅ (_ : MeasurableSet t), m t
hs : ¬ms = ∞
⊢ ∀ r > ms, ∃ t, s ⊆ t ∧ MeasurableSet t ∧ m t < r

Tactic:
intro r

Goal state after:
α : Type u_1
inst✝ : MeasurableSpace α
m : OuterMeasure α
s : Set α
ms : ℝ≥0∞ := ⨅ t, ⨅ (_ : s ⊆ t), ⨅ (_ : MeasurableSet t), m t
hs : ¬ms = ∞
r : ℝ≥0∞
⊢ r > ms → ∃ t, s ⊆ t ∧ MeasurableSet t ∧ m t < r
Module: Mathlib.MeasureTheory.MeasurableSpace.Embedding

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
δ' : Type u_5
ι : Sort uι
s t u : Set α
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
⊢ Measurable fun a ↦ (Equiv.prodAssoc α β γ) a

Tactic:
;

Goal state after:
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
δ' : Type u_5
ι : Sort uι
s t u : Set α
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
⊢ Measurable fun a ↦ (Equiv.prodAssoc α β γ) a
Module: Mathlib.MeasureTheory.MeasurableSpace.Embedding

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
δ' : Type u_5
ι : Sort uι
s t u : Set α
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
⊢ Measurable fun a ↦ (a.1.1, a.1.2, a.2)

Tactic:
;

Goal state after:
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
δ' : Type u_5
ι : Sort uι
s t u : Set α
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
⊢ Measurable fun a ↦ (a.1.1, a.1.2, a.2)
Module: Mathlib.MeasureTheory.MeasurableSpace.Embedding

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
δ' : Type u_5
ι : Sort uι
s t u : Set α
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
⊢ Measurable fun a ↦ (Equiv.prodAssoc α β γ).symm a

Tactic:
;

Goal state after:
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
δ' : Type u_5
ι : Sort uι
s t u : Set α
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
⊢ Measurable fun a ↦ (Equiv.prodAssoc α β γ).symm a
Module: Mathlib.MeasureTheory.MeasurableSpace.Embedding

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
δ' : Type u_5
ι : Sort uι
s t u : Set α
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
⊢ Measurable fun a ↦ ((a.1, a.2.1), a.2.2)

Tactic:
;

Goal state after:
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
δ' : Type u_5
ι : Sort uι
s t u : Set α
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
⊢ Measurable fun a ↦ ((a.1, a.2.1), a.2.2)
Module: Mathlib.MeasureTheory.MeasurableSpace.Embedding

Goal state before:
pp: α✝ : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
δ' : Type u_5
ι : Sort uι
s t u : Set α✝
inst✝⁵ : MeasurableSpace α✝
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace γ
π : δ' → Type u_6
π' : δ' → Type u_7
inst✝² : (x : δ') → MeasurableSpace (π x)
inst✝¹ : (x : δ') → MeasurableSpace (π' x)
α : δ ⊕ δ' → Type u_8
inst✝ : (i : δ ⊕ δ') → MeasurableSpace (α i)
⊢ Measurable fun a ↦ (Equiv.sumPiEquivProdPi fun a ↦ α a) fun i ↦ a i

Tactic:
;

Goal state after:
α✝ : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
δ' : Type u_5
ι : Sort uι
s t u : Set α✝
inst✝⁵ : MeasurableSpace α✝
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace γ
π : δ' → Type u_6
π' : δ' → Type u_7
inst✝² : (x : δ') → MeasurableSpace (π x)
inst✝¹ : (x : δ') → MeasurableSpace (π' x)
α : δ ⊕ δ' → Type u_8
inst✝ : (i : δ ⊕ δ') → MeasurableSpace (α i)
⊢ Measurable fun a ↦ (Equiv.sumPiEquivProdPi fun a ↦ α a) fun i ↦ a i
Module: Mathlib.MeasureTheory.MeasurableSpace.Embedding

Goal state before:
pp: α✝ : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
δ' : Type u_5
ι : Sort uι
s t u : Set α✝
inst✝⁵ : MeasurableSpace α✝
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace γ
π : δ' → Type u_6
π' : δ' → Type u_7
inst✝² : (x : δ') → MeasurableSpace (π x)
inst✝¹ : (x : δ') → MeasurableSpace (π' x)
α : δ ⊕ δ' → Type u_8
inst✝ : (i : δ ⊕ δ') → MeasurableSpace (α i)
⊢ Measurable fun a ↦ (fun i ↦ a (Sum.inl i), fun i' ↦ a (Sum.inr i'))

Tactic:
;

Goal state after:
α✝ : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
δ' : Type u_5
ι : Sort uι
s t u : Set α✝
inst✝⁵ : MeasurableSpace α✝
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace γ
π : δ' → Type u_6
π' : δ' → Type u_7
inst✝² : (x : δ') → MeasurableSpace (π x)
inst✝¹ : (x : δ') → MeasurableSpace (π' x)
α : δ ⊕ δ' → Type u_8
inst✝ : (i : δ ⊕ δ') → MeasurableSpace (α i)
⊢ Measurable fun a ↦ (fun i ↦ a (Sum.inl i), fun i' ↦ a (Sum.inr i'))
Module: Mathlib.MeasureTheory.MeasurableSpace.Embedding

Goal state before:
pp: α✝ : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
δ' : Type u_5
ι : Sort uι
s t u : Set α✝
inst✝⁵ : MeasurableSpace α✝
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace γ
π : δ' → Type u_6
π' : δ' → Type u_7
inst✝² : (x : δ') → MeasurableSpace (π x)
inst✝¹ : (x : δ') → MeasurableSpace (π' x)
α : δ ⊕ δ' → Type u_8
inst✝ : (i : δ ⊕ δ') → MeasurableSpace (α i)
⊢ ∀ (a : δ ⊕ δ'), Measurable fun x ↦ (Equiv.sumPiEquivProdPi α).symm x a

Tactic:
;

Goal state after:
α✝ : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
δ' : Type u_5
ι : Sort uι
s t u : Set α✝
inst✝⁵ : MeasurableSpace α✝
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace γ
π : δ' → Type u_6
π' : δ' → Type u_7
inst✝² : (x : δ') → MeasurableSpace (π x)
inst✝¹ : (x : δ') → MeasurableSpace (π' x)
α : δ ⊕ δ' → Type u_8
inst✝ : (i : δ ⊕ δ') → MeasurableSpace (α i)
⊢ ∀ (a : δ ⊕ δ'), Measurable fun x ↦ (Equiv.sumPiEquivProdPi α).symm x a
Module: Mathlib.MeasureTheory.MeasurableSpace.Embedding

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
f : α → β
hf : Injective f ∧ MeasurableSpace.comap f inst✝ = inst✝¹ ∧ MeasurableSet (range f)
⊢ Measurable f

Tactic:
;

Goal state after:
α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
f : α → β
hf : Injective f ∧ MeasurableSpace.comap f inst✝ = inst✝¹ ∧ MeasurableSet (range f)
⊢ Measurable f
Module: Mathlib.MeasureTheory.Measure.NullMeasurable

Goal state before:
pp: ι : Type u_1
α : Type u_2
m0 : MeasurableSpace α
μ : Measure α
inst✝ : Countable ι
f : ι → Set α
hn : Pairwise (Disjoint on f)
h : ∀ (i : ι), MeasurableSet (f i)
⊢ ∑' (i : ι), extend (fun t _ht ↦ μ t) (f i) = ∑' (i : ι), μ (f i)

Tactic:
simp [measure_eq_extend, h]

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

Goal state before:
pp: α : Type u_2
β : Type u_3
m0 : MeasurableSpace α
inst✝² : MeasurableSingletonClass α
mβ : MeasurableSpace β
inst✝¹ : MeasurableSingletonClass β
inst✝ : Countable α
μ : Measure (α × β)
y✝ : β
y : α × β
⊢ y ∈ Prod.snd ⁻¹' {y✝} ↔ y ∈ ⋃ x, {(x, y✝)}

Tactic:
;

Goal state after:
α : Type u_2
β : Type u_3
m0 : MeasurableSpace α
inst✝² : MeasurableSingletonClass α
mβ : MeasurableSpace β
inst✝¹ : MeasurableSingletonClass β
inst✝ : Countable α
μ : Measure (α × β)
y✝ : β
y : α × β
⊢ y ∈ Prod.snd ⁻¹' {y✝} ↔ y ∈ ⋃ x, {(x, y✝)}
Module: Mathlib.MeasureTheory.Measure.Map

Goal state before:
pp: 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∞), map f (c • μ) = c • map f μ

Tactic:
;

Goal state after:
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∞), map f (c • μ) = c • map f μ
Module: Mathlib.MeasureTheory.Measure.Map

Goal state before:
pp: case inl
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
⊢ map f (0 • μ) = 0 • map f μ

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

Tactic:
;

Goal state after:
case inl
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
⊢ map f (0 • μ) = 0 • map f μ

case inr
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
⊢ map f (c • μ) = c • map f μ
Module: Mathlib.MeasureTheory.Measure.Map

Goal state before:
pp: case h
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure α
f : α → β
hf : AEMeasurable f μ
s : Set β
⊢ s ⊆ toMeasurable (map f μ) s

Tactic:
;

Goal state after:
case h
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure α
f : α → β
hf : AEMeasurable f μ
s : Set β
⊢ s ⊆ toMeasurable (map f μ) s
Module: Mathlib.MeasureTheory.Measure.AbsolutelyContinuous

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ₁ μ₂ ν : Measure α
h : μ₁ + μ₂ ≪ ν
s : Set α
hs0 : ν s = 0
⊢ μ₁ s = 0 ∧ μ₂ s = 0

Tactic:
;

Goal state after:
α : Type u_1
mα : MeasurableSpace α
μ₁ μ₂ ν : Measure α
h : μ₁ + μ₂ ≪ ν
s : Set α
hs0 : ν s = 0
⊢ μ₁ s = 0 ∧ μ₂ s = 0
Module: Mathlib.MeasureTheory.OuterMeasure.BorelCantelli

Goal state before:
pp: α : Type u_1
F : Type u_3
inst✝¹ : FunLike F (Set α) ℝ≥0∞
inst✝ : OuterMeasureClass F α
μ : F
s : ℕ → Set α
h : ∑' (i : ℕ), μ (s i) ≠ ∞
⊢ μ (liminf s atTop) = 0

Tactic:
rw [← Nat.cofinite_eq_atTop, measure_liminf_cofinite_eq_zero h]

Goal state after:
Module: Mathlib.MeasureTheory.OuterMeasure.BorelCantelli

Goal state before:
pp: α : Type u_1
F : Type u_3
inst✝¹ : FunLike F (Set α) ℝ≥0∞
inst✝ : OuterMeasureClass F α
μ : F
s : ℕ → Set α
t : Set α
h : ∀ (n : ℕ), s n =ᵐ[μ] t
⊢ limsup s atTop =ᵐ[μ] t

Tactic:
simp only [eventuallyEq_set, ← eventually_countable_forall] at h

Goal state after:
α : Type u_1
F : Type u_3
inst✝¹ : FunLike F (Set α) ℝ≥0∞
inst✝ : OuterMeasureClass F α
μ : F
s : ℕ → Set α
t : Set α
h : ∀ᵐ (x : α) ∂μ, ∀ (i : ℕ), x ∈ s i ↔ x ∈ t
⊢ limsup s atTop =ᵐ[μ] t
Module: Mathlib.MeasureTheory.Measure.QuasiMeasurePreserving

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
μa : Measure α
μb : Measure β
f : α → β
hf : QuasiMeasurePreserving f μa μb
f' : α → β
hf' : Measurable f'
h : f =ᵐ[μa] f'
⊢ QuasiMeasurePreserving f' μa μb

Tactic:
refine ⟨hf', ?_⟩
  rw [Measure.map_congr h.symm]
  exact hf.absolutelyContinuous

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

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
μa : Measure α
μb : Measure β
f : α → β
R : Type u_5
inst✝¹ : SMul R ℝ≥0∞
inst✝ : IsScalarTower R ℝ≥0∞ ℝ≥0∞
hf : QuasiMeasurePreserving f μa μb
c : R
⊢ c • map f μa ≪ c • μb

Tactic:
;

Goal state after:
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
μa : Measure α
μb : Measure β
f : α → β
R : Type u_5
inst✝¹ : SMul R ℝ≥0∞
inst✝ : IsScalarTower R ℝ≥0∞ ℝ≥0∞
hf : QuasiMeasurePreserving f μa μb
c : R
⊢ c • map f μa ≪ c • μb
Module: Mathlib.MeasureTheory.OuterMeasure.BorelCantelli

Goal state before:
pp: α : Type u_1
F : Type u_3
inst✝¹ : FunLike F (Set α) ℝ≥0∞
inst✝ : OuterMeasureClass F α
μ : F
s : ℕ → Set α
t : Set α
h : ∀ (n : ℕ), s n =ᵐ[μ] t
⊢ liminf s atTop =ᵐ[μ] t

Tactic:
simp only [eventuallyEq_set, ← eventually_countable_forall] at h

Goal state after:
α : Type u_1
F : Type u_3
inst✝¹ : FunLike F (Set α) ℝ≥0∞
inst✝ : OuterMeasureClass F α
μ : F
s : ℕ → Set α
t : Set α
h : ∀ᵐ (x : α) ∂μ, ∀ (i : ℕ), x ∈ s i ↔ x ∈ t
⊢ liminf s atTop =ᵐ[μ] t
Module: Mathlib.MeasureTheory.Measure.MeasureSpace

Goal state before:
pp: case h
α : Type u_1
m : MeasurableSpace α
μ : Measure α
s₁ s₂ : Set α
⊢ s₁ ∩ s₂ ⊆ s₂

Tactic:
;

Goal state after:
case h
α : Type u_1
m : MeasurableSpace α
μ : Measure α
s₁ s₂ : Set α
⊢ s₁ ∩ s₂ ⊆ s₂
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
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: case h
α : Type u_1
m : MeasurableSpace α
μ : Measure α
ι : Sort u_8
inst✝ : Countable ι
s t : ι → Set α
hsub : ∀ (i : ι), s i ⊆ t i
h_le : ∀ (i : ι), μ (t i) ≤ μ (s i)
i : ι
⊢ s i ⊆ t i

Tactic:
;

Goal state after:
case h
α : Type u_1
m : MeasurableSpace α
μ : Measure α
ι : Sort u_8
inst✝ : Countable ι
s t : ι → Set α
hsub : ∀ (i : ι), s i ⊆ t i
h_le : ∀ (i : ι), μ (t i) ≤ μ (s i)
i : ι
⊢ s i ⊆ t i
Module: Mathlib.MeasureTheory.Measure.MeasureSpace

Goal state before:
pp: case refine_3
α : Type u_1
m : MeasurableSpace α
μ : Measure α
ι : Sort u_8
inst✝ : Countable ι
s t : ι → Set α
hsub : ∀ (i : ι), s i ⊆ t i
h_le : ∀ (i : ι), μ (t i) ≤ μ (s i)
htop : ∀ (i : ι), μ (t i) ≠ ∞
M : Set α → Set α := toMeasurable μ
b : ι
⊢ μ (M (t b)) ≠ ∞

Tactic:
rw [measure_toMeasurable]
      exact htop b

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

Goal state before:
pp: α : Type u_1
β : Type u_2
ι : Type u_3
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
μ ν : Measure α
s✝ t : Set α
a : α
inst✝ : SFinite μ
s : Set α
⊢ μ.restrict s = (sum (sfiniteSeq μ)).restrict s

Tactic:
sum_sfiniteSeq

Goal state after:
α : Type u_1
β : Type u_2
ι : Type u_3
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
μ ν : Measure α
s✝ t : Set α
a : α
inst✝ : SFinite μ
s : Set α
⊢ μ.restrict s = μ.restrict s
Module: Mathlib.MeasureTheory.Measure.Restrict

Goal state before:
pp: α : Type u_2
m0 : MeasurableSpace α
μ : Measure α
s t : Set α
h : s ⊆ t
⊢ μ (toMeasurable (μ.restrict t) s ∩ t) = (μ.restrict t) s

Tactic:
← restrict_apply (measurableSet_toMeasurable _ _),

Goal state after:
α : Type u_2
m0 : MeasurableSpace α
μ : Measure α
s t : Set α
h : s ⊆ t
⊢ (μ.restrict t) (toMeasurable (μ.restrict t) s) = (μ.restrict t) s
Module: Mathlib.MeasureTheory.Measure.Restrict

Goal state before:
pp: case insert
α : Type u_2
ι : Type u_6
m0 : MeasurableSpace α
μ ν : Measure α
t : ι → Set α
i : ι
s : Finset ι
a✝ : i ∉ s
hs : μ.restrict (⋃ i ∈ s, t i) = ν.restrict (⋃ i ∈ s, t i) ↔ ∀ i ∈ s, μ.restrict (t i) = ν.restrict (t i)
⊢ μ.restrict (t i ∪ ⋃ x ∈ s, t x) = ν.restrict (t i ∪ ⋃ x ∈ s, t x) ↔
    μ.restrict (t i) = ν.restrict (t i) ∧ ∀ a ∈ s, μ.restrict (t a) = ν.restrict (t a)

Tactic:
restrict_union_congr,

Goal state after:
case insert
α : Type u_2
ι : Type u_6
m0 : MeasurableSpace α
μ ν : Measure α
t : ι → Set α
i : ι
s : Finset ι
a✝ : i ∉ s
hs : μ.restrict (⋃ i ∈ s, t i) = ν.restrict (⋃ i ∈ s, t i) ↔ ∀ i ∈ s, μ.restrict (t i) = ν.restrict (t i)
⊢ μ.restrict (t i) = ν.restrict (t i) ∧ μ.restrict (⋃ x ∈ s, t x) = ν.restrict (⋃ x ∈ s, t x) ↔
    μ.restrict (t i) = ν.restrict (t i) ∧ ∀ a ∈ s, μ.restrict (t a) = ν.restrict (t a)
Module: Mathlib.MeasureTheory.Measure.Typeclasses.SFinite

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
C : Set (Set α)
hC : ∀ s ∈ C, MeasurableSet s
m : ℝ≥0∞ := ⨆ D ∈ {D | D ⊆ C ∧ D.Countable}, μ (⋃₀ D)
hm : ⊥ < m
u : ℕ → ℝ≥0∞
u_mem : ∀ (n : ℕ), u n ∈ Ioo 0 m
u_lim : Tendsto u atTop (𝓝 m)
D : ℕ → Set (Set α)
D_mem : ∀ (n : ℕ), D n ∈ {D | D ⊆ C ∧ D.Countable}
huD : ∀ (n : ℕ), u n < μ (⋃₀ D n)
⊢ (∀ (i : ℕ), D i ⊆ C) ∧ ∀ (i : ℕ), (D i).Countable

Tactic:
;

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
C : Set (Set α)
hC : ∀ s ∈ C, MeasurableSet s
m : ℝ≥0∞ := ⨆ D ∈ {D | D ⊆ C ∧ D.Countable}, μ (⋃₀ D)
hm : ⊥ < m
u : ℕ → ℝ≥0∞
u_mem : ∀ (n : ℕ), u n ∈ Ioo 0 m
u_lim : Tendsto u atTop (𝓝 m)
D : ℕ → Set (Set α)
D_mem : ∀ (n : ℕ), D n ∈ {D | D ⊆ C ∧ D.Countable}
huD : ∀ (n : ℕ), u n < μ (⋃₀ D n)
⊢ (∀ (i : ℕ), D i ⊆ C) ∧ ∀ (i : ℕ), (D i).Countable
Module: Mathlib.MeasureTheory.Measure.Typeclasses.SFinite

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
C : Set (Set α)
hC : ∀ s ∈ C, MeasurableSet s
m : ℝ≥0∞ := ⨆ D ∈ {D | D ⊆ C ∧ D.Countable}, μ (⋃₀ D)
hm : ⊥ < m
u : ℕ → ℝ≥0∞
u_mem : ∀ (n : ℕ), u n ∈ Ioo 0 m
u_lim : Tendsto u atTop (𝓝 m)
D : ℕ → Set (Set α)
D_mem : ∀ (n : ℕ), D n ∈ {D | D ⊆ C ∧ D.Countable}
huD : ∀ (n : ℕ), u n < μ (⋃₀ D n)
hD : ⋃ n, D n ∈ {D | D ⊆ C ∧ D.Countable}
n : ℕ
x : α
hx : ∃ t ∈ D n, x ∈ t
⊢ ∃ t, (∃ i, t ∈ D i) ∧ x ∈ t

Tactic:
;

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
C : Set (Set α)
hC : ∀ s ∈ C, MeasurableSet s
m : ℝ≥0∞ := ⨆ D ∈ {D | D ⊆ C ∧ D.Countable}, μ (⋃₀ D)
hm : ⊥ < m
u : ℕ → ℝ≥0∞
u_mem : ∀ (n : ℕ), u n ∈ Ioo 0 m
u_lim : Tendsto u atTop (𝓝 m)
D : ℕ → Set (Set α)
D_mem : ∀ (n : ℕ), D n ∈ {D | D ⊆ C ∧ D.Countable}
huD : ∀ (n : ℕ), u n < μ (⋃₀ D n)
hD : ⋃ n, D n ∈ {D | D ⊆ C ∧ D.Countable}
n : ℕ
x : α
hx : ∃ t ∈ D n, x ∈ t
⊢ ∃ t, (∃ i, t ∈ D i) ∧ x ∈ t
Module: Mathlib.MeasureTheory.Measure.Restrict

Goal state before:
pp: α : Type u_2
ι : Type u_6
m0 : MeasurableSpace α
μ ν : Measure α
S : Set ι
s : ι → Set α
hc : S.Countable
hs : ⋃ i ∈ S, s i = univ
⊢ μ = ν ↔ ∀ i ∈ S, μ.restrict (s i) = ν.restrict (s i)

Tactic:
rw [← restrict_biUnion_congr hc, hs, restrict_univ, restrict_univ]

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

Goal state before:
pp: α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : MeasurableSpace α
μ : Measure α
s : Set α
h : IsCompact s
hμ : ∀ x ∈ s, μ.FiniteAtFilter (𝓝 x)
⊢ ∃ U ⊇ s, IsOpen[inst✝¹] U ∧ μ U < ∞

Tactic:
refine IsCompact.induction_on h ?_ ?_ ?_ ?_
  · use ∅
    simp
  · rintro s t hst ⟨U, htU, hUo, hU⟩
    exact ⟨U, hst.trans htU, hUo, hU⟩
  · rintro s t ⟨U, hsU, hUo, hU⟩ ⟨V, htV, hVo, hV⟩
    refine
      ⟨U ∪ V, union_subset_union hsU htV, hUo.union hVo,
        (measure_union_le _ _).trans_lt <| ENNReal.add_lt_top.2 ⟨hU, hV⟩⟩
  · intro x hx
    rcases (hμ x hx).exists_mem_basis (nhds_basis_opens _) with ⟨U, ⟨hx, hUo⟩, hU⟩
    exact ⟨U, nhdsWithin_le_nhds (hUo.mem_nhds hx), U, Subset.rfl, hUo, hU⟩

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

Goal state before:
pp: case pos.hm
α : Type u_1
m : MeasurableSpace α
μ : Measure α
ι : Type u_8
inst✝³ : LinearOrder ι
inst✝² : TopologicalSpace ι
inst✝¹ : OrderTopology ι
inst✝ : FirstCountableTopology ι
s : ι → Set α
a : ι
hs : ∀ r > a, NullMeasurableSet (s r) μ
hm : ∀ (i j : ι), a < i → i ≤ j → s i ⊆ s j
hf : ∃ r > a, μ (s r) ≠ ∞
ha : Order.IsPredPrelimit a
this : atBot.IsCountablyGenerated
⊢ Monotone fun x ↦ s ↑x

Tactic:
exact fun i j h ↦ hm i j i.2 h

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

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
inst✝ : SFinite μ
C : Set (Set α)
hC : ∀ s ∈ C, MeasurableSet s
D : ℕ → Set (Set α)
DC : ∀ (n : ℕ), D n ⊆ C
D_count : ∀ (n : ℕ), (D n).Countable
hD : ∀ (n : ℕ), ∀ s ∈ C, s ≤ᵐ[sfiniteSeq μ n] ⋃₀ D n
s : Set α
hs : s ∈ C
n : ℕ
x : α
hx : ∃ t ∈ D n, x ∈ t
⊢ ∃ t, (∃ i, t ∈ D i) ∧ x ∈ t

Tactic:
;

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
inst✝ : SFinite μ
C : Set (Set α)
hC : ∀ s ∈ C, MeasurableSet s
D : ℕ → Set (Set α)
DC : ∀ (n : ℕ), D n ⊆ C
D_count : ∀ (n : ℕ), (D n).Countable
hD : ∀ (n : ℕ), ∀ s ∈ C, s ≤ᵐ[sfiniteSeq μ n] ⋃₀ D n
s : Set α
hs : s ∈ C
n : ℕ
x : α
hx : ∃ t ∈ D n, x ∈ t
⊢ ∃ t, (∃ i, t ∈ D i) ∧ x ∈ t