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

Goal state before:
pp: α : Type u_3
ι : Type u_4
π : ι → Set (Set α)
hpi : ∀ (x : ι), IsPiSystem (π x)
S : Set ι
t1 : Set α
p1 : Finset ι
hp1S : ↑p1 ⊆ S
f1 : ι → Set α
hf1m : ∀ x ∈ p1, f1 x ∈ π x
ht1_eq : t1 = ⋂ x ∈ p1, f1 x
t2 : Set α
p2 : Finset ι
hp2S : ↑p2 ⊆ S
f2 : ι → Set α
hf2m : ∀ x ∈ p2, f2 x ∈ π x
ht2_eq : t2 = ⋂ x ∈ p2, f2 x
h_nonempty : (t1 ∩ t2).Nonempty
⊢ t1 ∩ t2 ∈ piiUnionInter π S

Tactic:
simp_rw [piiUnionInter, Set.mem_ofPred_eq]

Goal state after:
α : Type u_3
ι : Type u_4
π : ι → Set (Set α)
hpi : ∀ (x : ι), IsPiSystem (π x)
S : Set ι
t1 : Set α
p1 : Finset ι
hp1S : ↑p1 ⊆ S
f1 : ι → Set α
hf1m : ∀ x ∈ p1, f1 x ∈ π x
ht1_eq : t1 = ⋂ x ∈ p1, f1 x
t2 : Set α
p2 : Finset ι
hp2S : ↑p2 ⊆ S
f2 : ι → Set α
hf2m : ∀ x ∈ p2, f2 x ∈ π x
ht2_eq : t2 = ⋂ x ∈ p2, f2 x
h_nonempty : (t1 ∩ t2).Nonempty
⊢ ∃ t, ∃ (_ : ↑t ⊆ S), ∃ f, ∃ (_ : ∀ x ∈ t, f x ∈ π x), t1 ∩ t2 = ⋂ x ∈ t, f x
Module: Mathlib.MeasureTheory.Measure.MeasureSpaceDef

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
s : Set α
hs : MeasurableSet s
⊢ μ s = extend (fun t _ht ↦ μ t) s

Tactic:
extend_eq

Goal state after:
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
s : Set α
hs : MeasurableSet s
⊢ μ s = μ s

case h
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
s : Set α
hs : MeasurableSet s
⊢ MeasurableSet s
Module: Mathlib.MeasureTheory.Measure.MeasureSpaceDef

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
s : Set α
⊢ MeasurableSet (toMeasurable μ s)

Tactic:
by
  rw [toMeasurable_def]; split_ifs with hs h's
  exacts [hs.choose_spec.2.1, h's.choose_spec.2.1,
          (exists_measurable_superset μ s).choose_spec.2.1]

Goal state after:
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:
set ms := ⨅ (t : Set α) (_ : s ⊆ t) (_ : MeasurableSet t), m t

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
⊢ ∃ t, s ⊆ t ∧ MeasurableSet t ∧ m t = ms
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:
← hf.2.1

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.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)
⊢ ∀ ⦃s : Set α⦄, MeasurableSet s → MeasurableSet (f '' s)

Tactic:
← hf.2.1

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)
⊢ ∀ ⦃s : Set α⦄, MeasurableSet s → MeasurableSet (f '' s)
Module: Mathlib.MeasureTheory.Measure.NullMeasurable

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

Tactic:
rw [union_comm, measure_union₀ hs (AEDisjoint.symm hd), add_comm]

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
R : Type u_4
inst✝¹ : SMul R ℝ≥0∞
inst✝ : IsScalarTower R ℝ≥0∞ ℝ≥0∞
c : R
μ : Measure α
f : α → β
this : ∀ (c : ℝ≥0∞), map f (c • μ) = c • map f μ
⊢ map f (c • μ) = c • map f μ

Tactic:
by simpa using this (c • 1)

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

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

Tactic:
rw [← Set.biUnion_preimage_singleton, measure_biUnion hs (pairwiseDisjoint_fiber f s) hf]

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
s₁ s₂ : Set α
h : s₂ ⊆ s₁
h₂ : NullMeasurableSet s₂ μ
h_fin : μ s₂ ≠ ∞
⊢ μ (s₁ \ s₂) = μ s₁ - μ s₂

Tactic:
measure_sdiff' _ h₂ h_fin,

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
s₁ s₂ : Set α
h : s₂ ⊆ s₁
h₂ : NullMeasurableSet s₂ μ
h_fin : μ s₂ ≠ ∞
⊢ μ (s₁ ∪ s₂) - μ s₂ = μ s₁ - μ s₂
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✝² : MeasurableSpace β
μ ν : Measure α
s t : Set α
a : α
inst✝¹ : Countable ι
m : ι → Measure α
inst✝ : ∀ (n : ι), SFinite (m n)
⊢ SFinite (sum m)

Tactic:
change SFinite (Measure.sum (fun i ↦ m i))

Goal state after:
α : Type u_1
β : Type u_2
ι : Type u_3
m0 : MeasurableSpace α
inst✝² : MeasurableSpace β
μ ν : Measure α
s t : Set α
a : α
inst✝¹ : Countable ι
m : ι → Measure α
inst✝ : ∀ (n : ι), SFinite (m n)
⊢ SFinite (sum fun i ↦ m i)
Module: Mathlib.MeasureTheory.Measure.Typeclasses.Finite

Goal state before:
pp: α : Type u_1
β : Type u_2
δ : Type u_3
ι : Type u_4
m0 : MeasurableSpace α
mβ : MeasurableSpace β
μ ν ν₁ ν₂ : Measure α
s t : Set α
R : Type u_5
inst✝⁴ : SMul R ℝ≥0
inst✝³ : SMul R ℝ≥0∞
inst✝² : IsScalarTower R ℝ≥0 ℝ≥0∞
inst✝¹ : IsScalarTower R ℝ≥0∞ ℝ≥0∞
inst✝ : IsFiniteMeasure μ
r : R
⊢ IsFiniteMeasure (r • μ)

Tactic:
rw [← smul_one_smul ℝ≥0 r μ]

Goal state after:
α : Type u_1
β : Type u_2
δ : Type u_3
ι : Type u_4
m0 : MeasurableSpace α
mβ : MeasurableSpace β
μ ν ν₁ ν₂ : Measure α
s t : Set α
R : Type u_5
inst✝⁴ : SMul R ℝ≥0
inst✝³ : SMul R ℝ≥0∞
inst✝² : IsScalarTower R ℝ≥0 ℝ≥0∞
inst✝¹ : IsScalarTower R ℝ≥0∞ ℝ≥0∞
inst✝ : IsFiniteMeasure μ
r : R
⊢ IsFiniteMeasure ((r • 1) • μ)
Module: Mathlib.MeasureTheory.Measure.Restrict

Goal state before:
pp: α : Type u_2
m0 : MeasurableSpace α
μ : Measure α
s : Set α
⊢ μ.restrict s = 0 ↔ μ s = 0

Tactic:
rw [← measure_univ_eq_zero, restrict_apply_univ]

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

Goal state before:
pp: α : Type u_2
m0 : MeasurableSpace α
μ ν : Measure α
C : Set (Set α)
B : ℕ → Set α
hA : m0 = generateFrom C
hC : IsPiSystem C
h1B : ⋃ i, B i = univ
h2B : ∀ (i : ℕ), B i ∈ C
hμB : ∀ (i : ℕ), μ (B i) ≠ ∞
h_eq : ∀ s ∈ C, μ s = ν s
⊢ μ = ν

Tactic:
refine ext_of_generateFrom_of_cover_subset hA hC ?_ (countable_range B) h1B ?_ h_eq

Goal state after:
case refine_1
α : Type u_2
m0 : MeasurableSpace α
μ ν : Measure α
C : Set (Set α)
B : ℕ → Set α
hA : m0 = generateFrom C
hC : IsPiSystem C
h1B : ⋃ i, B i = univ
h2B : ∀ (i : ℕ), B i ∈ C
hμB : ∀ (i : ℕ), μ (B i) ≠ ∞
h_eq : ∀ s ∈ C, μ s = ν s
⊢ range B ⊆ C

case refine_2
α : Type u_2
m0 : MeasurableSpace α
μ ν : Measure α
C : Set (Set α)
B : ℕ → Set α
hA : m0 = generateFrom C
hC : IsPiSystem C
h1B : ⋃ i, B i = univ
h2B : ∀ (i : ℕ), B i ∈ C
hμB : ∀ (i : ℕ), μ (B i) ≠ ∞
h_eq : ∀ s ∈ C, μ s = ν s
⊢ ∀ s ∈ range B, μ s ≠ ∞
Module: Mathlib.MeasureTheory.Measure.MeasureSpace

Goal state before:
pp: α : Type u_1
ι : Type u_5
m : MeasurableSpace α
I : Finset ι
μ : ι → Measure α
s : Set α
⊢ (∑ i ∈ I, μ i) s = ∑ i ∈ I, (μ i) s

Tactic:
coe_finsetSum,

Goal state after:
α : Type u_1
ι : Type u_5
m : MeasurableSpace α
I : Finset ι
μ : ι → Measure α
s : Set α
⊢ (∑ i ∈ I, ⇑(μ i)) s = ∑ i ∈ I, (μ i) s
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⦄, (∀ (i : ℕ), MeasurableSet (f i)) → Pairwise (Disjoint on f) → ⊤ (⋃ i, f i) = ∑' (i : ℕ), ⊤ (f i)

Tactic:
by
          intro f _ _
          refine (measure_iUnion_le _).antisymm ?_
          if hne : (⋃ i, f i).Nonempty then
            rw [OuterMeasure.top_apply hne]
            exact le_top
          else
            simp_all [Set.not_nonempty_iff_eq_empty]

Goal state after:
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 (Subtype.val '' ⋃ i, f i) μ

Tactic:
image_iUnion

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: 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 (f '' s))

Tactic:
rw [comap_restrict hf, preimage_image_eq _ hf.injective]

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
⊢ (map f μ) s = (comap g μ) s

Tactic:
rw [map_apply hf hs, hg.comap_apply, ← measure_sdiff_null hμg]

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.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:
by
  rw [← generateFrom_natGeneratingSequence α] at hs
  refine generateFrom_induction (range (natGeneratingSequence α)) _ ?_ ?_ ?_ ?_ s hs
  · simp only [mem_range, forall_exists_index, forall_apply_eq_imp_iff]
    refine fun n _ ↦ ⟨fun p ↦ p n, ?_⟩
    ext x
    simp only [mem_iUnion, mem_ite_empty_right]
    refine ⟨fun hx ↦ ?_, ?_⟩
    · exact ⟨(x ∈ natGeneratingSequence α ·), by simp [countablyGeneratedAtom]; grind⟩
    · rintro ⟨p, hpn, hpx⟩
      simp only [countablyGeneratedAtom, mem_iInter] at hpx
      specialize hpx n
      simpa [hpn] using hpx
  · exact ⟨fun _ ↦ False, by simp⟩
  · simp only [forall_exists_index]
    intro t ht q htq
    refine ⟨fun p ↦ ¬ q p, ?_⟩
    ext x
    simp only [htq, compl_iUnion, mem_iInter, mem_compl_iff, mem_ite_empty_right, not_and,
      mem_iUnion]
    refine ⟨fun h ↦ ?_, ?_⟩
    · refine ⟨fun n ↦ x ∈ natGeneratingSequence α n, ?_,
        mem_countablyGeneratedAtom_natGeneratingSequence x⟩
      contrapose! h
      refine ⟨fun n ↦ x ∈ natGeneratingSequence α n, h,
        mem_countablyGeneratedAtom_natGeneratingSequence x⟩
    · rintro ⟨p, hpq, hpx⟩ p' hqp' hx_mem
      have hpp' : p ≠ p' := by grind
      have h_disj : Disjoint (countablyGeneratedAtom α p) (countablyGeneratedAtom α p') :=
        disjoint_countablyGeneratedAtom hpp'
      grind
  · intro t ht h
    choose q hq using h
    refine ⟨fun p ↦ ⨆ n, q n p, ?_⟩
    ext
    simp [hq]
    grind

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

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝ : MeasurableSpace α
h : CountablySeparated α
s : Set α
⊢ CountablySeparated ↑s

Tactic:
rw [CountablySeparated.subtype_iff]

Goal state after:
α : Type u_1
β : Type u_2
inst✝ : MeasurableSpace α
h : CountablySeparated α
s : Set α
⊢ HasCountableSeparatingOn α MeasurableSet s
Module: Mathlib.MeasureTheory.Measure.Restrict

Goal state before:
pp: case refine_2
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)
i : ι
hi : i ∈ F
x : α
hx : x ∈ s i
⊢ x ∈ ⋂ i ∈ {x_1 ∈ F | x ∈ s x_1}, s i ∧ x ∈ ⋂ i ∈ F \ {x_1 ∈ F | x ∈ s x_1}, (s i)ᶜ

Tactic:
mem_iInter,

Goal state after:
case refine_2
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)
i : ι
hi : i ∈ F
x : α
hx : x ∈ s i
⊢ (∀ i ∈ {x_1 ∈ F | x ∈ s x_1}, x ∈ s i) ∧ ∀ i ∈ F \ {x_1 ∈ F | x ∈ s x_1}, x ∈ (s i)ᶜ
Module: Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated

Goal state before:
pp: α : Type u_1
t : ℕ → Set α
n : ℕ
⊢ t n = ⋃ u ∈ memPartition t n, u ∩ t n

Tactic:
simp_rw [← iUnion_inter, ← sUnion_eq_biUnion, sUnion_memPartition, univ_inter]

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

Goal state before:
pp: case refine_2
α : Type u_2
β : Type u_3
m0 : MeasurableSpace α
inst✝ : MeasurableSpace β
f : α → β
μ : Measure α
H : AEMeasurable f μ
t : Set β
ht : ∀ᵐ (x : α) ∂μ, f x ∈ t
h₀ : t.Nonempty
s : Set α := toMeasurable μ {x | f x = mk f H x ∧ f x ∈ t}ᶜ
g : α → β := s.piecewise (fun x ↦ h₀.some) (mk f H)
⊢ range g ⊆ t

Tactic:
rintro _ ⟨x, rfl⟩

Goal state after:
case refine_2
α : Type u_2
β : Type u_3
m0 : MeasurableSpace α
inst✝ : MeasurableSpace β
f : α → β
μ : Measure α
H : AEMeasurable f μ
t : Set β
ht : ∀ᵐ (x : α) ∂μ, f x ∈ t
h₀ : t.Nonempty
s : Set α := toMeasurable μ {x | f x = mk f H x ∧ f x ∈ t}ᶜ
g : α → β := s.piecewise (fun x ↦ h₀.some) (mk f H)
x : α
⊢ g x ∈ t
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Basic

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : OpensMeasurableSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : OpensMeasurableSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : TopologicalSpace γ₂
inst✝² : MeasurableSpace γ₂
inst✝¹ : BorelSpace γ₂
inst✝ : MeasurableSpace δ
s : Set α
h : HasCountableSeparatingOn α IsOpen[inst✝¹²] s
⊢ CountablySeparated ↑s

Tactic:
rw [CountablySeparated.subtype_iff]

Goal state after:
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : OpensMeasurableSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : OpensMeasurableSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : TopologicalSpace γ₂
inst✝² : MeasurableSpace γ₂
inst✝¹ : BorelSpace γ₂
inst✝ : MeasurableSpace δ
s : Set α
h : HasCountableSeparatingOn α IsOpen[inst✝¹²] s
⊢ HasCountableSeparatingOn α MeasurableSet s
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Order

Goal state before:
pp: case mpr
α : Type u_5
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
hd : Dense s
hbot : ∀ (x : α), IsBot x → x ∈ s
hIoo : ∀ (x y : α), x < y → Ioo x y = ∅ → x ∈ s
S : Set (Set α) := {S | ∃ l ∈ s, ∃ u ∈ s, l ≤ u ∧ Icc l u = S}
this : MeasurableSpace α := MeasurableSpace.generateFrom S
a : α
t : Set α
hts : t ⊆ s
hc : t.Countable
htd : Dense t
htb : ∀ (x : α), IsBot x → x ∈ s → x ∈ t
ha : ∀ b < a, (Ioo b a).Nonempty
y l u : α
hua : u < a
hyu : y ≤ u
⊢ y < a

Tactic:
exact hyu.trans_lt hua

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

Goal state before:
pp: case refine_1
α : Type u_5
inst✝⁵ : TopologicalSpace α
m : MeasurableSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : BorelSpace α
μ ν : Measure α
inst✝ : IsFiniteMeasure μ
h : ∀ (a : α), μ (Iic a) = ν (Iic a)
⊢ μ univ = ν univ

Tactic:
rcases exists_countable_dense_bot_top α with ⟨s, hsc, hsd, -, hst⟩

Goal state after:
case refine_1
α : Type u_5
inst✝⁵ : TopologicalSpace α
m : MeasurableSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : BorelSpace α
μ ν : Measure α
inst✝ : IsFiniteMeasure μ
h : ∀ (a : α), μ (Iic a) = ν (Iic a)
s : Set α
hsc : s.Countable
hsd : Dense s
hst : ∀ (x : α), IsTop x → x ∈ s
⊢ μ univ = ν univ
Module: Mathlib.Topology.MetricSpace.Thickening

Goal state before:
pp: α : Type u
inst✝ : PseudoEMetricSpace α
δ : ℝ
s : Set α
x : α
r : ℝ≥0∞
h : infEDist x (cthickening δ s) + ENNReal.ofReal δ < r
⊢ infEDist x s < r

Tactic:
simp_rw [← lt_tsub_iff_right, infEDist_lt_iff, mem_cthickening_iff] at h

Goal state after:
α : Type u
inst✝ : PseudoEMetricSpace α
δ : ℝ
s : Set α
x : α
r : ℝ≥0∞
h : ∃ y, infEDist y s ≤ ENNReal.ofReal δ ∧ edist x y < r - ENNReal.ofReal δ
⊢ infEDist x s < r
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Order

Goal state before:
pp: case convert_4
α : Type u_1
δ : Type u_4
inst✝⁴ : TopologicalSpace α
mα : MeasurableSpace α
inst✝³ : BorelSpace α
mδ : MeasurableSpace δ
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
f : δ → α
hf : ∀ (x : α), MeasurableSet (f ⁻¹' Iio x)
⊢ ∀ t ∈ range Iio, MeasurableSet (f ⁻¹' t)

Tactic:
rintro _ ⟨x, rfl⟩

Goal state after:
case convert_4
α : Type u_1
δ : Type u_4
inst✝⁴ : TopologicalSpace α
mα : MeasurableSpace α
inst✝³ : BorelSpace α
mδ : MeasurableSpace δ
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
f : δ → α
hf : ∀ (x : α), MeasurableSet (f ⁻¹' Iio x)
x : α
⊢ MeasurableSet (f ⁻¹' Iio x)
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Order

Goal state before:
pp: case convert_4
α : Type u_1
δ : Type u_4
inst✝⁴ : TopologicalSpace α
mα : MeasurableSpace α
inst✝³ : BorelSpace α
mδ : MeasurableSpace δ
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
f : δ → α
hf : ∀ (x : α), MeasurableSet (f ⁻¹' Iio x)
⊢ ∀ t ∈ range Iio, MeasurableSet (f ⁻¹' t)

Tactic:
· rintro _ ⟨x, rfl⟩; exact hf x

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

Goal state before:
pp: case convert_4
α : Type u_1
δ : Type u_4
inst✝⁴ : TopologicalSpace α
mα : MeasurableSpace α
inst✝³ : BorelSpace α
mδ : MeasurableSpace δ
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
f : δ → α
hf : ∀ (x : α), MeasurableSet (f ⁻¹' Ioi x)
⊢ ∀ t ∈ range Ioi, MeasurableSet (f ⁻¹' t)

Tactic:
rintro _ ⟨x, rfl⟩

Goal state after:
case convert_4
α : Type u_1
δ : Type u_4
inst✝⁴ : TopologicalSpace α
mα : MeasurableSpace α
inst✝³ : BorelSpace α
mδ : MeasurableSpace δ
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
f : δ → α
hf : ∀ (x : α), MeasurableSet (f ⁻¹' Ioi x)
x : α
⊢ MeasurableSet (f ⁻¹' Ioi x)
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Order

Goal state before:
pp: case convert_4
α : Type u_1
δ : Type u_4
inst✝⁴ : TopologicalSpace α
mα : MeasurableSpace α
inst✝³ : BorelSpace α
mδ : MeasurableSpace δ
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
f : δ → α
hf : ∀ (x : α), MeasurableSet (f ⁻¹' Ioi x)
⊢ ∀ t ∈ range Ioi, MeasurableSet (f ⁻¹' t)

Tactic:
· rintro _ ⟨x, rfl⟩; exact hf x

Goal state after:
Module: Mathlib.MeasureTheory.Function.SimpleFunc

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
c : ℝ≥0∞
⊢ (const α c).lintegral μ = c * μ univ

Tactic:
rw [lintegral]

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
c : ℝ≥0∞
⊢ ∑ x ∈ (const α c).range, x * μ (⇑(const α c) ⁻¹' {x}) = c * μ univ
Module: Mathlib.MeasureTheory.Function.SimpleFunc

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
c : ℝ≥0∞
s : Set α
hs : MeasurableSet s
⊢ ((const α c).restrict s).lintegral μ = c * μ s

Tactic:
rw [restrict_lintegral_eq_lintegral_restrict _ hs, const_lintegral_restrict]

Goal state after:
Module: Mathlib.MeasureTheory.Function.SimpleFunc

Goal state before:
pp: α : Type u_5
γ : Type u_6
inst✝¹ : MeasurableSpace α
inst✝ : AddZeroClass γ
motive : (α →ₛ γ) → Prop
const :
  ∀ (c : γ) {s : Set α} (hs : MeasurableSet s), motive (piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0))
add : ∀ ⦃f g : α →ₛ γ⦄, Disjoint (support ⇑f) (support ⇑g) → motive f → motive g → motive (f + g)
x : γ
s : Finset γ
hxs : x ∉ s
ih : ∀ (f : α →ₛ γ), range ⇑f \ {0} = ↑s → motive f
f : α →ₛ γ
h : range ⇑f \ {0} = ↑(insert x s)
mx : MeasurableSet (⇑f ⁻¹' {x})
g : α →ₛ γ := piecewise (⇑f ⁻¹' {x}) mx 0 f
⊢ ⇑0 '' ⇑f ⁻¹' {x} ⊆ {0}

pp: α : Type u_5
γ : Type u_6
inst✝¹ : MeasurableSpace α
inst✝ : AddZeroClass γ
motive : (α →ₛ γ) → Prop
const :
  ∀ (c : γ) {s : Set α} (hs : MeasurableSet s), motive (piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0))
add : ∀ ⦃f g : α →ₛ γ⦄, Disjoint (support ⇑f) (support ⇑g) → motive f → motive g → motive (f + g)
x : γ
s : Finset γ
hxs : x ∉ s
ih : ∀ (f : α →ₛ γ), range ⇑f \ {0} = ↑s → motive f
f : α →ₛ γ
h : range ⇑f \ {0} = ↑(insert x s)
mx : MeasurableSet (⇑f ⁻¹' {x})
g : α →ₛ γ := piecewise (⇑f ⁻¹' {x}) mx 0 f
⊢ x ∉ ↑s

Tactic:
· rw [Set.image_subset_iff]
        convert! Set.subset_univ _
        exact preimage_const_of_mem (mem_singleton _)

Goal state after:
α : Type u_5
γ : Type u_6
inst✝¹ : MeasurableSpace α
inst✝ : AddZeroClass γ
motive : (α →ₛ γ) → Prop
const :
  ∀ (c : γ) {s : Set α} (hs : MeasurableSet s), motive (piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0))
add : ∀ ⦃f g : α →ₛ γ⦄, Disjoint (support ⇑f) (support ⇑g) → motive f → motive g → motive (f + g)
x : γ
s : Finset γ
hxs : x ∉ s
ih : ∀ (f : α →ₛ γ), range ⇑f \ {0} = ↑s → motive f
f : α →ₛ γ
h : range ⇑f \ {0} = ↑(insert x s)
mx : MeasurableSet (⇑f ⁻¹' {x})
g : α →ₛ γ := piecewise (⇑f ⁻¹' {x}) mx 0 f
⊢ x ∉ ↑s
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Real

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
ι : Type u_5
f : ι → α → ℝ≥0∞
g : α → ℝ≥0∞
μ : Measure α
u : Filter ι
inst✝¹ : u.NeBot
inst✝ : u.IsCountablyGenerated
hf : ∀ (i : ι), AEMeasurable (f i) μ
hlim : ∀ᵐ (a : α) ∂μ, Tendsto (fun i ↦ f i a) u (𝓝 (g a))
⊢ AEMeasurable g μ

Tactic:
rcases u.exists_seq_tendsto with ⟨v, hv⟩
  have h'f : ∀ n, AEMeasurable (f (v n)) μ := fun n ↦ hf (v n)
  set p : α → (ℕ → ℝ≥0∞) → Prop := fun x f' ↦ Tendsto f' atTop (𝓝 (g x))
  have hp : ∀ᵐ x ∂μ, p x fun n ↦ f (v n) x := by
    filter_upwards [hlim] with x hx using hx.comp hv
  classical
  set aeSeqLim := fun x ↦ ite (x ∈ aeSeqSet h'f p) (g x) (⟨f (v 0) x⟩ : Nonempty ℝ≥0∞).some
  refine ⟨aeSeqLim, measurable_of_tendsto' atTop (aeSeq.measurable h'f p)
    (tendsto_pi_nhds.mpr fun x ↦ ?_), ?_⟩
  · unfold aeSeqLim
    simp_rw [aeSeq]
    split_ifs with hx
    · simp_rw [aeSeq.mk_eq_fun_of_mem_aeSeqSet h'f hx]
      exact aeSeq.fun_prop_of_mem_aeSeqSet h'f hx
    · exact tendsto_const_nhds
  · exact (ite_ae_eq_of_measure_compl_zero g (fun x ↦ (⟨f (v 0) x⟩ : Nonempty ℝ≥0∞).some)
      (aeSeqSet h'f p) (aeSeq.measure_compl_aeSeqSet_eq_zero h'f hp)).symm

Goal state after: