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

Goal state before:
pp: α : Type u_1
S : Set (Set α)
h_pi : IsPiSystem S
⊢ IsPiSystem (insert univ S)

Tactic:
intro s

Goal state after:
α : Type u_1
S : Set (Set α)
h_pi : IsPiSystem S
s : Set α
⊢ s ∈ insert univ S → ∀ t ∈ insert univ S, (s ∩ t).Nonempty → s ∩ t ∈ insert univ S
Module: Mathlib.MeasureTheory.PiSystem

Goal state before:
pp: case cons
α : Type u_1
S : Set (Set α)
h_pi : IsPiSystem S
t✝ : Finset (Set α)
a : Set α
t : Finset (Set α)
hat : a ∉ t
t_ne : t.Nonempty
ih : (∀ s ∈ t, s ∈ S) → (⋂ s ∈ t, s).Nonempty → ⋂ s ∈ t, s ∈ S
ht : ∀ s ∈ Finset.cons a t hat, s ∈ S
h' : (⋂ s ∈ Finset.cons a t hat, s).Nonempty
⊢ ⋂ s ∈ Finset.cons a t hat, s ∈ S

Tactic:
simp only [Finset.cons_eq_insert, Finset.mem_insert, iInter_iInter_eq_or_left] at h' ht ⊢

Goal state after:
case cons
α : Type u_1
S : Set (Set α)
h_pi : IsPiSystem S
t✝ : Finset (Set α)
a : Set α
t : Finset (Set α)
hat : a ∉ t
t_ne : t.Nonempty
ih : (∀ s ∈ t, s ∈ S) → (⋂ s ∈ t, s).Nonempty → ⋂ s ∈ t, s ∈ S
h' : (a ∩ ⋂ x ∈ t, x).Nonempty
ht : ∀ (s : Set α), s = a ∨ s ∈ t → s ∈ S
⊢ a ∩ ⋂ x ∈ t, x ∈ S
Module: Mathlib.MeasureTheory.OuterMeasure.Caratheodory

Goal state before:
pp: α : Type u_1
a : α
s : Set α
x✝ : MeasurableSet s
t : Set α
⊢ (dirac a) t = (dirac a) (t ∩ s) + (dirac a) (t \ s)

Tactic:
by_cases ht : a ∈ t; swap; · simp [ht]
    by_cases hs : a ∈ s <;> simp [*]

Goal state after:
Module: Mathlib.MeasureTheory.PiSystem

Goal state before:
pp: case refine_1.mp
α : Type u_3
β : Type u_4
g : β → Set (Set α)
s : Set β
h_pi : ∀ b ∈ s, IsPiSystem (g b)
T : Finset ↑s
f : ↑s → Set α
h_t' : ∀ b ∈ T, f b ∈ (g ∘ Subtype.val) b
h_t : ⋂ b ∈ T, f b ∈ generatePiSystem (⋃ b ∈ s, g b)
this : ⋂ b ∈ T, f b ∈ generatePiSystem (⋃ b, (g ∘ Subtype.val) b)
a : α
h1 : ∀ (a_1 : β) (b : a_1 ∈ s), ⟨a_1, b⟩ ∈ T → a ∈ f ⟨a_1, b⟩
b : β
h_b : b ∈ s
h_b_in_T : ⟨b, h_b⟩ ∈ T
⊢ a ∈ f ⟨b, h_b⟩ → a ∈ Function.extend (fun x ↦ ↑x) f (fun x ↦ ∅) ↑⟨b, h_b⟩

Tactic:
rw [Subtype.val_injective.extend_apply]

Goal state after:
case refine_1.mp
α : Type u_3
β : Type u_4
g : β → Set (Set α)
s : Set β
h_pi : ∀ b ∈ s, IsPiSystem (g b)
T : Finset ↑s
f : ↑s → Set α
h_t' : ∀ b ∈ T, f b ∈ (g ∘ Subtype.val) b
h_t : ⋂ b ∈ T, f b ∈ generatePiSystem (⋃ b ∈ s, g b)
this : ⋂ b ∈ T, f b ∈ generatePiSystem (⋃ b, (g ∘ Subtype.val) b)
a : α
h1 : ∀ (a_1 : β) (b : a_1 ∈ s), ⟨a_1, b⟩ ∈ T → a ∈ f ⟨a_1, b⟩
b : β
h_b : b ∈ s
h_b_in_T : ⟨b, h_b⟩ ∈ T
⊢ a ∈ f ⟨b, h_b⟩ → a ∈ f ⟨b, h_b⟩
Module: Mathlib.MeasureTheory.PiSystem

Goal state before:
pp: case refine_1.mpr
α : Type u_3
β : Type u_4
g : β → Set (Set α)
s : Set β
h_pi : ∀ b ∈ s, IsPiSystem (g b)
T : Finset ↑s
f : ↑s → Set α
h_t' : ∀ b ∈ T, f b ∈ (g ∘ Subtype.val) b
h_t : ⋂ b ∈ T, f b ∈ generatePiSystem (⋃ b ∈ s, g b)
this : ⋂ b ∈ T, f b ∈ generatePiSystem (⋃ b, (g ∘ Subtype.val) b)
a : α
h1 : ∀ (a_1 : β) (b : a_1 ∈ s), ⟨a_1, b⟩ ∈ T → a ∈ Function.extend (fun x ↦ ↑x) f (fun x ↦ ∅) ↑⟨a_1, b⟩
b : β
h_b : b ∈ s
h_b_in_T : ⟨b, h_b⟩ ∈ T
⊢ a ∈ Function.extend (fun x ↦ ↑x) f (fun x ↦ ∅) ↑⟨b, h_b⟩ → a ∈ f ⟨b, h_b⟩

Tactic:
rw [Subtype.val_injective.extend_apply]

Goal state after:
case refine_1.mpr
α : Type u_3
β : Type u_4
g : β → Set (Set α)
s : Set β
h_pi : ∀ b ∈ s, IsPiSystem (g b)
T : Finset ↑s
f : ↑s → Set α
h_t' : ∀ b ∈ T, f b ∈ (g ∘ Subtype.val) b
h_t : ⋂ b ∈ T, f b ∈ generatePiSystem (⋃ b ∈ s, g b)
this : ⋂ b ∈ T, f b ∈ generatePiSystem (⋃ b, (g ∘ Subtype.val) b)
a : α
h1 : ∀ (a_1 : β) (b : a_1 ∈ s), ⟨a_1, b⟩ ∈ T → a ∈ Function.extend (fun x ↦ ↑x) f (fun x ↦ ∅) ↑⟨a_1, b⟩
b : β
h_b : b ∈ s
h_b_in_T : ⟨b, h_b⟩ ∈ T
⊢ a ∈ f ⟨b, h_b⟩ → a ∈ f ⟨b, h_b⟩
Module: Mathlib.MeasureTheory.OuterMeasure.AE

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 t : Set α
H : s ≤ᵐ[μ] t
⊢ μ (s ∪ t) = μ (t ∪ s)

Tactic:
Set.union_comm

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 t : Set α
H : s ≤ᵐ[μ] t
⊢ μ (t ∪ s) = μ (t ∪ s)
Module: Mathlib.MeasureTheory.Function.AEMeasurableSequence

Goal state before:
pp: ι : Sort u_1
α : Type u_2
β : Type u_3
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
f : ι → α → β
μ : Measure α
p : α → (ι → β) → Prop
hf : ∀ (i : ι), AEMeasurable (f i) μ
x : α
hx : x ∈ aeSeqSet hf p
i : ι
⊢ aeSeqSet hf p ⊆ {x | ∀ (i : ι), f i x = AEMeasurable.mk (f i) ⋯ x}

Tactic:
by
    rw [aeSeqSet, ← compl_compl { x | ∀ i, f i x = (hf i).mk (f i) x }, Set.compl_subset_compl]
    refine Set.Subset.trans (Set.compl_subset_compl.mpr fun x h => ?_) (subset_toMeasurable _ _)
    exact h.1

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

Goal state before:
pp: α : Type u_2
m : OuterMeasure α
m0 m1 : MeasurableSpace α
h : m0 ≤ m1
⊢ ∀ (s : Set α), MeasurableSet s → m.trim s ≤ m s

Tactic:
intro s

Goal state after:
α : Type u_2
m : OuterMeasure α
m0 m1 : MeasurableSpace α
h : m0 ≤ m1
s : Set α
⊢ MeasurableSet s → m.trim s ≤ m s
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
f : α → β
g : β → α
hf : MeasurableEmbedding f
⊢ ↑(f '' univ) = ↑(range f)

Tactic:
image_univ

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
f : α → β
g : β → α
hf : MeasurableEmbedding f
⊢ ↑(range f) = ↑(range f)
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
f : α → β
g : β → α
hf : MeasurableEmbedding f
⊢ Subtype.instMeasurableSpace ≍ Subtype.instMeasurableSpace

Tactic:
image_univ

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
f : α → β
g : β → α
hf : MeasurableEmbedding f
⊢ Subtype.instMeasurableSpace ≍ Subtype.instMeasurableSpace
Module: Mathlib.MeasureTheory.Measure.NullMeasurable

Goal state before:
pp: α : Type u_2
m0 : MeasurableSpace α
μ : Measure α
s t : Set α
ht : NullMeasurableSet t μ
hd : AEDisjoint μ s t
⊢ μ (s ∪ t) = μ s + μ t

Tactic:
rw [← measure_union_add_inter₀ s ht, hd, add_zero]

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

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
μ ν : Measure α
s✝ : Set α
inst✝ : MeasurableSpace β
f : OuterMeasure α →ₗ[ℝ≥0∞] OuterMeasure β
hf : ∀ (μ : Measure α), inst✝ ≤ (f μ.toOuterMeasure).caratheodory
μ₁ μ₂ : Measure α
s : Set β
hs : MeasurableSet s
⊢ ((f (μ₁ + μ₂).toOuterMeasure).toMeasure ⋯) s =
    ((f μ₁.toOuterMeasure).toMeasure ⋯ + (f μ₂.toOuterMeasure).toMeasure ⋯) s

Tactic:
by
    simp only [map_add, coe_add, Pi.add_apply, toMeasure_apply, add_toOuterMeasure,
      FunLike.coe_add, hs]

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

Goal state before:
pp: 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), MeasurableSet s → (map f μ).toOuterMeasure s = ((OuterMeasure.map f) μ.toOuterMeasure) s

Tactic:
intro s

Goal state after:
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
⊢ MeasurableSet s → (map f μ).toOuterMeasure s = ((OuterMeasure.map f) μ.toOuterMeasure) s
Module: Mathlib.MeasureTheory.Measure.AbsolutelyContinuous

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
h : ∀ ⦃s : Set α⦄, MeasurableSet s → ν s = 0 → μ s = 0
⊢ μ ≪ ν

Tactic:
intro s

Goal state after:
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
h : ∀ ⦃s : Set α⦄, MeasurableSet s → ν s = 0 → μ s = 0
s : Set α
⊢ ν s = 0 → μ s = 0
Module: Mathlib.MeasureTheory.Measure.AbsolutelyContinuous

Goal state before:
pp: α : Type u_1
R : Type u_5
mα : MeasurableSpace α
μ ν : Measure α
inst✝¹ : SMul R ℝ≥0∞
inst✝ : IsScalarTower R ℝ≥0∞ ℝ≥0∞
h : μ ≪ ν
c : R
⊢ c • μ ≪ c • ν

Tactic:
intro s

Goal state after:
α : Type u_1
R : Type u_5
mα : MeasurableSpace α
μ ν : Measure α
inst✝¹ : SMul R ℝ≥0∞
inst✝ : IsScalarTower R ℝ≥0∞ ℝ≥0∞
h : μ ≪ ν
c : R
s : Set α
⊢ (c • ν) s = 0 → (c • μ) s = 0
Module: Mathlib.MeasureTheory.Measure.AbsolutelyContinuous

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ₁ μ₂ ν ν' : Measure α
h1 : μ₁ ≪ ν
h2 : μ₂ ≪ ν'
⊢ μ₁ + μ₂ ≪ ν + ν'

Tactic:
intro s

Goal state after:
α : Type u_1
mα : MeasurableSpace α
μ₁ μ₂ ν ν' : Measure α
h1 : μ₁ ≪ ν
h2 : μ₂ ≪ ν'
s : Set α
⊢ (ν + ν') s = 0 → (μ₁ + μ₂) s = 0
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), ν s = 0 → μ₁ s = 0 ∧ μ₂ s = 0

Tactic:
intro s

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

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
h1 : μ ≪ ν
ν' : Measure α
⊢ μ ≪ ν + ν'

Tactic:
intro s

Goal state after:
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
h1 : μ ≪ ν
ν' : Measure α
s : Set α
⊢ (ν + ν') s = 0 → μ s = 0
Module: Mathlib.MeasureTheory.OuterMeasure.BorelCantelli

Goal state before:
pp: α : Type u_1
ι : Type u_2
F : Type u_3
inst✝² : FunLike F (Set α) ℝ≥0∞
inst✝¹ : OuterMeasureClass F α
inst✝ : Countable ι
μ : F
s : ι → Set α
hs : ∑' (i : ι), μ (s i) ≠ ∞
⊢ μ (limsup s cofinite) = 0

Tactic:
refine bot_unique <| ge_of_tendsto' (ENNReal.tendsto_tsum_compl_atTop_zero hs) fun t ↦ ?_
  calc
    μ (limsup s cofinite) ≤ μ (⋃ i : {i // i ∉ t}, s i) := by
      gcongr
      rw [hasBasis_cofinite.limsup_eq_iInf_iSup, iUnion_subtype]
      exact iInter₂_subset _ t.finite_toSet
    _ ≤ ∑' i : {i // i ∉ t}, μ (s i) := measure_iUnion_le _

Goal state after:
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

Tactic:
have : f ⁻¹' t =ᵐ[μ] t := (h.preimage_ae_eq ht.symm).trans (hs'.trans ht)

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
this : f ⁻¹' t =ᵐ[μ] t
⊢ limsup (fun x ↦ f^[x] ⁻¹' t) atTop =ᵐ[μ] 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.map (⇑e.symm) μ = Measure.comap (⇑e) μ

Tactic:
← comap_symm,

Goal state after:
α : Type u_1
β : Type u_2
ma : MeasurableSpace α
mb : MeasurableSpace β
μ : Measure α
e : β ≃ᵐ α
⊢ Measure.comap (⇑e.symm.symm) μ = Measure.comap (⇑e) μ
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:
intro s

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)
s : Finset ι
⊢ ∑ a ∈ s, μ (As a) ≤ μ (⋃ i, As i)
Module: Mathlib.MeasureTheory.Measure.Typeclasses.Finite

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
hμ : IsFiniteMeasure μ
f : ℕ → Set α
hf₁ : ∀ (i : ℕ), MeasurableSet (f i)
hf₂ : Pairwise (Disjoint on f)
⊢ Summable fun x ↦ μ.real (f x)

Tactic:
apply ENNReal.summable_toReal

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
hμ : IsFiniteMeasure μ
f : ℕ → Set α
hf₁ : ∀ (i : ℕ), MeasurableSet (f i)
hf₂ : Pairwise (Disjoint on f)
⊢ ∑' (x : ℕ), μ (f x) ≠ ∞
Module: Mathlib.MeasureTheory.Measure.Typeclasses.SFinite

Goal state before:
pp: α : Type u_4
β : Type u_5
x✝ : MeasurableSpace α
μ : Measure α
inst✝² : SFinite μ
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSingletonClass β
g : α → β
g_mble : NullMeasurable g μ
⊢ {t | 0 < μ {a | g a = t}}.Countable

Tactic:
have level_sets_disjoint : Pairwise (Disjoint on fun t : β => { a : α | g a = t }) :=
    fun s t hst => Disjoint.preimage g (disjoint_singleton.mpr hst)

Goal state after:
α : Type u_4
β : Type u_5
x✝ : MeasurableSpace α
μ : Measure α
inst✝² : SFinite μ
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSingletonClass β
g : α → β
g_mble : NullMeasurable g μ
level_sets_disjoint : Pairwise (Disjoint on fun t ↦ {a | g a = t})
⊢ {t | 0 < μ {a | g a = t}}.Countable
Module: Mathlib.MeasureTheory.Measure.Typeclasses.SFinite

Goal state before:
pp: α : Type u_4
β : Type u_5
x✝ : MeasurableSpace α
μ : Measure α
inst✝² : SFinite μ
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSingletonClass β
g : α → β
g_mble : NullMeasurable g μ
level_sets_disjoint : Pairwise (Disjoint on fun t ↦ {a | g a = t})
⊢ {t | 0 < μ {a | g a = t}}.Countable

Tactic:
exact Measure.countable_meas_pos_of_disjoint_iUnion₀
    (fun b => g_mble (‹MeasurableSingletonClass β›.measurableSet_singleton b))
    ((fun _ _ h ↦ Disjoint.aedisjoint (level_sets_disjoint h)))

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

Goal state before:
pp: case neg
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
⊢ Tendsto (⇑μ ∘ s) (𝓝[>] a) (𝓝 (μ (⋂ r, ⋂ (_ : r > a), s r)))

Tactic:
Order.not_isPredPrelimit_iff

Goal state after:
case neg
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 : ∃ b, a ⋖ b
⊢ Tendsto (⇑μ ∘ s) (𝓝[>] a) (𝓝 (μ (⋂ r, ⋂ (_ : r > a), s r)))
Module: Mathlib.MeasureTheory.Measure.Restrict

Goal state before:
pp: α : Type u_2
δ : Type u_4
m0 : MeasurableSpace α
μ : Measure α
f g : α → δ
s : Set α
hfg : f =ᵐ[μ] g
⊢ μ.restrict s ≪ μ

Tactic:
exact absolutelyContinuous_restrict

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

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
m : Set (Measure α)
s : Set α
hs : MeasurableSet s
t : Set α
μ : Measure α
hμ : μ ∈ m
u : Set α
htu : t ⊆ u
_hu : MeasurableSet u
⊢ ∀ {s t : Set α}, s ⊆ t → OuterMeasure.sInfGen (toOuterMeasure '' m) s ≤ μ t

Tactic:
intro s

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
m : Set (Measure α)
s✝ : Set α
hs : MeasurableSet s✝
t : Set α
μ : Measure α
hμ : μ ∈ m
u : Set α
htu : t ⊆ u
_hu : MeasurableSet u
s : Set α
⊢ ∀ {t : Set α}, s ⊆ t → OuterMeasure.sInfGen (toOuterMeasure '' m) s ≤ μ t
Module: Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : Countable α
⊢ inst✝¹ ≤ generateFrom (⋃ y, {measurableAtom y})

Tactic:
intro s

Goal state after:
α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : Countable α
s : Set α
⊢ MeasurableSet s → MeasurableSet s
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
⊢ μ + ν₁ = μ + ν₂ ↔ ν₁ = ν₂

Tactic:
refine ⟨fun h ↦ ?_, fun h ↦ by rw [h]⟩

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ ν₁ ν₂ : Measure α
inst✝ : SigmaFinite μ
h : μ + ν₁ = μ + ν₂
⊢ ν₁ = ν₂
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
μ : Measure α
inst✝ : SigmaFinite μ
s : Set α
⊢ SigmaFinite (μ.restrict s)

Tactic:
by
  refine ⟨⟨⟨spanningSets μ, fun _ => trivial, fun i => ?_, iUnion_spanningSets μ⟩⟩⟩
  rw [Measure.restrict_apply (measurableSet_spanningSets μ i)]
  exact (measure_mono inter_subset_left).trans_lt (measure_spanningSets_lt_top μ i)

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

Goal state before:
pp: case refine_3
α : Type u_1
mα : MeasurableSpace α
inst✝ : CountablyGenerated α
x : α
A : ℕ → Set α := natGeneratingSequence α
hA : ∀ (n : ℕ), MeasurableSet (A n)
hA_gen : generateFrom (range A) = mα
⊢ ∀ (i : Set α), x ∈ i → MeasurableSet i → (countablyGeneratedAtom α fun x_1 ↦ x ∈ A x_1) ⊆ i

Tactic:
intro s

Goal state after:
case refine_3
α : Type u_1
mα : MeasurableSpace α
inst✝ : CountablyGenerated α
x : α
A : ℕ → Set α := natGeneratingSequence α
hA : ∀ (n : ℕ), MeasurableSet (A n)
hA_gen : generateFrom (range A) = mα
s : Set α
⊢ x ∈ s → MeasurableSet s → (countablyGeneratedAtom α fun x_1 ↦ x ∈ A x_1) ⊆ s
Module: Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated

Goal state before:
pp: case refine_1.iUnion.refine_1
α : Type u_1
t : ℕ → Set α
n : ℕ
s : Set α
f : ℕ → Set α
hs✝ : ∀ (n_1 : ℕ), MeasurableSet (f n_1)
S : ℕ → Finset (Set α)
hS_subset : ∀ (n_1 : ℕ), ↑(S n_1) ⊆ memPartition t n
hS_eq : ∀ (n : ℕ), f n = ⋃₀ ↑(S n)
this : Fintype ↑(⋃ n, ↑(S n))
⊢ ↑(⋃ n, ↑(S n)).toFinset ⊆ memPartition t n

Tactic:
simp only [coe_toFinset, iUnion_subset_iff]
        exact hS_subset

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

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

Tactic:
sum_eq_tsum_indicator

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

Goal state before:
pp: case refine_2.mpr
ι : Type u_1
α : Type u_2
β : Type u_3
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
f : α → β
inst✝ : Countable ι
μ : ι → Measure α
h : ∀ (i : ι), AEMeasurable f (μ i)
a✝ : Nontrivial β
inhabited_h : Inhabited β
s : ι → Set α := fun i ↦ toMeasurable (μ i) {x | f x ≠ mk f ⋯ x}
hsμ : ∀ (i : ι), (μ i) (s i) = 0
hsm : MeasurableSet (⋂ i, s i)
hs : ∀ (i : ι), ∀ x ∉ s i, f x = mk f ⋯ x
g : α → β := (⋂ i, s i).piecewise (const α default) f
t : Set β
ht : MeasurableSet t
x : α
i : ι
hi : x ∉ s i
⊢ f x ∈ t → ∃ i, mk f ⋯ x ∈ t ∧ x ∉ s i

Tactic:
rw [hs _ _ hi]

Goal state after:
case refine_2.mpr
ι : Type u_1
α : Type u_2
β : Type u_3
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
f : α → β
inst✝ : Countable ι
μ : ι → Measure α
h : ∀ (i : ι), AEMeasurable f (μ i)
a✝ : Nontrivial β
inhabited_h : Inhabited β
s : ι → Set α := fun i ↦ toMeasurable (μ i) {x | f x ≠ mk f ⋯ x}
hsμ : ∀ (i : ι), (μ i) (s i) = 0
hsm : MeasurableSet (⋂ i, s i)
hs : ∀ (i : ι), ∀ x ∉ s i, f x = mk f ⋯ x
g : α → β := (⋂ i, s i).piecewise (const α default) f
t : Set β
ht : MeasurableSet t
x : α
i : ι
hi : x ∉ s i
⊢ mk f ⋯ x ∈ t → ∃ i, mk f ⋯ x ∈ t ∧ x ∉ s i
Module: Mathlib.MeasureTheory.Measure.MeasureSpace

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

Tactic:
rw [mem_ofPred_eq] at hk₁ hk₂

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

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
⊢ μ.cofinite ≤ ae μ

Tactic:
intro s

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
s : Set α
⊢ s ∈ ae μ → s ∈ μ.cofinite
Module: Mathlib.MeasureTheory.Group.Arithmetic

Goal state before:
pp: α✝ : Type u_1
M✝ : Type u_2
X : Type u_3
α : Type u_4
β : Type u_5
inst✝⁷ : MeasurableSpace X
inst✝⁶ : SMul M✝ X
m : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure α
f : α → M✝
g : α → X
inst✝⁵ : MeasurableSpace M✝
inst✝⁴ : MeasurableSMul M✝ X
M : Type u_6
inst✝³ : SubNegMonoid M
inst✝² : MeasurableSpace M
inst✝¹ : MeasurableAdd₂ M
inst✝ : MeasurableNeg M
⊢ Measurable (Function.uncurry fun x1 x2 ↦ x1 • x2)

Tactic:
by
    suffices Measurable fun p : M × ℤ => p.2 • p.1 by apply this.comp measurable_swap
    refine measurable_from_prod_countable_left fun n => ?_
    cases n with
    | ofNat n =>
      simp only [Int.ofNat_eq_natCast, natCast_zsmul]
      exact measurable_const_smul _
    | negSucc n =>
      simp only [negSucc_zsmul]
      exact (measurable_const_smul _).neg

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Basic

Goal state before:
pp: γ : Type u_3
δ : Type u_5
inst✝³ : TopologicalSpace γ
inst✝² : MeasurableSpace γ
inst✝¹ : BorelSpace γ
inst✝ : MeasurableSpace δ
f : δ → γ
hf : ∀ (s : Set γ), IsClosed[inst✝³] s → MeasurableSet (f ⁻¹' s)
⊢ ∀ (s : Set γ), IsOpen[inst✝³] s → MeasurableSet (f ⁻¹' s)

Tactic:
intro s

Goal state after:
γ : Type u_3
δ : Type u_5
inst✝³ : TopologicalSpace γ
inst✝² : MeasurableSpace γ
inst✝¹ : BorelSpace γ
inst✝ : MeasurableSpace δ
f : δ → γ
hf : ∀ (s : Set γ), IsClosed[inst✝³] s → MeasurableSet (f ⁻¹' s)
s : Set γ
⊢ IsOpen[inst✝³] s → MeasurableSet (f ⁻¹' s)