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

Goal state before:
pp: α : Type u_2
β : Type u_3
m0 : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s : Set α
ν : Measure β
f : α → β
hf : QuasiMeasurePreserving f μ ν
t : Set β
hmaps : MapsTo f s t
u : Set β
hum : MeasurableSet u
hu : ν (u ∩ t) = 0
⊢ f ⁻¹' u ∩ s ⊆ f ⁻¹' u ∩ f ⁻¹' t

Tactic:
]

Goal state after:
α : Type u_2
β : Type u_3
m0 : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s : Set α
ν : Measure β
f : α → β
hf : QuasiMeasurePreserving f μ ν
t : Set β
hmaps : MapsTo f s t
u : Set β
hum : MeasurableSet u
hu : ν (u ∩ t) = 0
⊢ f ⁻¹' u ∩ s ⊆ f ⁻¹' u ∩ f ⁻¹' t
Module: Mathlib.MeasureTheory.Measure.MeasureSpace

Goal state before:
pp: α : Type u_8
ι : Type u_9
x✝ : MeasurableSpace α
μ : Measure α
inst✝¹ : SemilatticeSup ι
inst✝ : Countable ι
f : ι → Set α
hm : ∀ (i : ι), NullMeasurableSet (f i) μ
ε : ℝ≥0∞
hε : 0 < ε
hfin : ∃ i, μ (f i) ≠ ∞
hfem : ⋂ n, f n = ∅
F : ι → ℝ≥0∞ := fun m ↦ μ (⋂ n, ⋂ (_ : n ≤ m), f n)
hFAnti : Antitone F
hzero : μ (⋂ n, f n) = 0
⊢ Tendsto F atTop (𝓝 (μ (⋂ n, f n)))

Tactic:
]

Goal state after:
α : Type u_8
ι : Type u_9
x✝ : MeasurableSpace α
μ : Measure α
inst✝¹ : SemilatticeSup ι
inst✝ : Countable ι
f : ι → Set α
hm : ∀ (i : ι), NullMeasurableSet (f i) μ
ε : ℝ≥0∞
hε : 0 < ε
hfin : ∃ i, μ (f i) ≠ ∞
hfem : ⋂ n, f n = ∅
F : ι → ℝ≥0∞ := fun m ↦ μ (⋂ n, ⋂ (_ : n ≤ m), f n)
hFAnti : Antitone F
hzero : μ (⋂ n, f n) = 0
⊢ Tendsto F atTop (𝓝 (μ (⋂ n, f n)))
Module: Mathlib.MeasureTheory.Measure.Typeclasses.SFinite

Goal state before:
pp: case h₂
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)
D : Set (Set α)
D_mem : D ∈ {D | D ⊆ C ∧ D.Countable}
hD : μ (⋃₀ D) = m
s : Set α
hs : s ∈ C
⊢ μ (⋃₀ (D ∪ {s})) ≤ m

Tactic:
]

Goal state after:
case h₂
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)
D : Set (Set α)
D_mem : D ∈ {D | D ⊆ C ∧ D.Countable}
hD : μ (⋃₀ D) = m
s : Set α
hs : s ∈ C
⊢ μ (⋃₀ (D ∪ {s})) ≤ m
Module: Mathlib.MeasureTheory.Measure.Restrict

Goal state before:
pp: α : Type u_2
ι : Type u_6
m0 : MeasurableSpace α
μ ν : Measure α
inst✝ : Countable ι
s : ι → Set α
hs : ⋃ i, s i = univ
⊢ μ = ν ↔ μ = ν

Tactic:
]

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

Goal state before:
pp: 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
⊢ (comap Subtype.val (𝓝[>] a)).IsCountablyGenerated

Tactic:
]

Goal state after:
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
⊢ (comap Subtype.val (𝓝[>] a)).IsCountablyGenerated
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 ↔ μ = ν

Tactic:
]

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

Goal state before:
pp: α : Type u_2
m0 : MeasurableSpace α
μ ν : Measure α
S : Set (Set α)
hc : S.Countable
hs : ⋃₀ S = univ
⊢ ⋃₀ S = univ

Tactic:
]

Goal state after:
α : Type u_2
m0 : MeasurableSpace α
μ ν : Measure α
S : Set (Set α)
hc : S.Countable
hs : ⋃₀ S = univ
⊢ ⋃₀ S = univ
Module: Mathlib.MeasureTheory.Measure.MeasureSpace

Goal state before:
pp: case pos.hs
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
⊢ ∀ (a_1 : ι) (b : a_1 ∈ Ioi a), NullMeasurableSet (s ↑⟨a_1, b⟩) μ

Tactic:
]

Goal state after:
case pos.hs
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
⊢ ∀ (a_1 : ι) (b : a_1 ∈ Ioi a), NullMeasurableSet (s ↑⟨a_1, b⟩) μ
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
⊢ s ≤ᵐ[sum (sfiniteSeq μ)] ⋃₀ ⋃ n, D n

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
⊢ s ≤ᵐ[sum (sfiniteSeq μ)] ⋃₀ ⋃ n, D n
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
m : ℕ → Measure α
hv : ∀ (n : ℕ), (m n) t ≠ ∞
hμ : μ = sum m
w : ℕ → Set α := fun n ↦ toMeasurable (m n) t
T : t ⊆ ⋂ n, w n
M : MeasurableSet (⋂ n, w n)
⊢ ∃ t' ⊇ t, MeasurableSet t' ∧ ∀ (u : Set α), MeasurableSet u → μ (t' ∩ u) = μ (t ∩ u)

Tactic:
refine ⟨⋂ n, w n, T, M, fun u hu ↦ ?_⟩

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
s : Set α
hs : MeasurableSet s
t : Set α
m : ℕ → Measure α
hv : ∀ (n : ℕ), (m n) t ≠ ∞
hμ : μ = sum m
w : ℕ → Set α := fun n ↦ toMeasurable (m n) t
T : t ⊆ ⋂ n, w n
M : MeasurableSet (⋂ n, w n)
u : Set α
hu : MeasurableSet u
⊢ μ ((⋂ n, w n) ∩ u) = μ (t ∩ u)
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 : ∃ b, a ⋖ b
⊢ Tendsto (⇑μ ∘ s) (𝓝[>] a) (𝓝 (μ (⋂ r, ⋂ (_ : r > a), s r)))

Tactic:
]

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.Typeclasses.Finite

Goal state before:
pp: case inr
α : Type u_1
β : Type u_2
δ : Type u_3
ι : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : SecondCountableTopology α
m : MeasurableSpace α
μ : Measure α
inst✝ : IsLocallyFiniteMeasure μ
h✝ : Nonempty α
inhabited_h : Inhabited α
S : Set (Set α) := {s | IsOpen[inst✝²] s ∧ μ s < ∞}
T : Set (Set α)
T_count : T.Countable
TS : T ⊆ S
hT : ⋃₀ T = univ
⊢ Nonempty (μ.FiniteSpanningSetsIn {K | IsOpen[inst✝²] K})

Tactic:
]

Goal state after:
case inr
α : Type u_1
β : Type u_2
δ : Type u_3
ι : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : SecondCountableTopology α
m : MeasurableSpace α
μ : Measure α
inst✝ : IsLocallyFiniteMeasure μ
h✝ : Nonempty α
inhabited_h : Inhabited α
S : Set (Set α) := {s | IsOpen[inst✝²] s ∧ μ s < ∞}
T : Set (Set α)
T_count : T.Countable
TS : T ⊆ S
hT : ⋃₀ T = univ
⊢ Nonempty (μ.FiniteSpanningSetsIn {K | IsOpen[inst✝²] K})
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
m : ℕ → Measure α
hv : ∀ (n : ℕ), (m n) t ≠ ∞
hμ : μ = sum m
w : ℕ → Set α := fun n ↦ toMeasurable (m n) t
T : t ⊆ ⋂ n, w n
M : MeasurableSet (⋂ n, w n)
u : Set α
hu : MeasurableSet u
⊢ ∑' (i : ℕ), (m i) ((⋂ n, w n) ∩ u) ≤ (sum m) (t ∩ u)

Tactic:
]

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
s : Set α
hs : MeasurableSet s
t : Set α
m : ℕ → Measure α
hv : ∀ (n : ℕ), (m n) t ≠ ∞
hμ : μ = sum m
w : ℕ → Set α := fun n ↦ toMeasurable (m n) t
T : t ⊆ ⋂ n, w n
M : MeasurableSet (⋂ n, w n)
u : Set α
hu : MeasurableSet u
⊢ ∑' (i : ℕ), (m i) ((⋂ n, w n) ∩ u) ≤ (sum m) (t ∩ u)
Module: Mathlib.MeasureTheory.Measure.Restrict

Goal state before:
pp: case compl
α : Type u_2
m0 : MeasurableSpace α
μ ν : Measure α
S T : Set (Set α)
h_gen : m0 = generateFrom S
hc : T.Countable
h_inter : IsPiSystem S
hU : ⋃₀ T = univ
htop : ∀ t ∈ T, μ t ≠ ∞
ST_eq : ∀ t ∈ T, ∀ s ∈ S, μ (s ∩ t) = ν (s ∩ t)
T_eq : ∀ t ∈ T, μ t = ν t
t : Set α
ht : t ∈ T
u✝ u : Set α
hu : MeasurableSet u
ihu : μ (t ∩ u) = ν (t ∩ u)
this : μ t = ν t
⊢ μ (t ∩ uᶜ) = ν (t ∩ uᶜ)

Tactic:
]

Goal state after:
case compl
α : Type u_2
m0 : MeasurableSpace α
μ ν : Measure α
S T : Set (Set α)
h_gen : m0 = generateFrom S
hc : T.Countable
h_inter : IsPiSystem S
hU : ⋃₀ T = univ
htop : ∀ t ∈ T, μ t ≠ ∞
ST_eq : ∀ t ∈ T, ∀ s ∈ S, μ (s ∩ t) = ν (s ∩ t)
T_eq : ∀ t ∈ T, μ t = ν t
t : Set α
ht : t ∈ T
u✝ u : Set α
hu : MeasurableSet u
ihu : μ (t ∩ u) = ν (t ∩ u)
this : μ t = ν t
⊢ μ (t ∩ uᶜ) = ν (t ∩ uᶜ)
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
inst✝² : TopologicalSpace α
inst✝¹ : SecondCountableTopology α
m : MeasurableSpace α
μ : Measure α
inst✝ : IsLocallyFiniteMeasure μ
h✝ : Nonempty α
inhabited_h : Inhabited α
S : Set (Set α) := {s | IsOpen[inst✝²] s ∧ μ s < ∞}
T : Set (Set α)
T_count : T.Countable
TS : T ⊆ S
hT : ∅ = univ
h'T : ¬T.Nonempty
⊢ False

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
δ : Type u_3
ι : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : SecondCountableTopology α
m : MeasurableSpace α
μ : Measure α
inst✝ : IsLocallyFiniteMeasure μ
h✝ : Nonempty α
inhabited_h : Inhabited α
S : Set (Set α) := {s | IsOpen[inst✝²] s ∧ μ s < ∞}
T : Set (Set α)
T_count : T.Countable
TS : T ⊆ S
hT : ∅ = univ
h'T : ¬T.Nonempty
⊢ False
Module: Mathlib.MeasureTheory.Measure.MeasureSpace

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
inst✝ : Countable α
μ ν : Measure α
h : ∀ (x : α), μ (measurableAtom x) = ν (measurableAtom x)
s : Set α
hs : MeasurableSet s
h1 : s = ⋃₀ (measurableAtom '' s)
⊢ μ s = ν s

Tactic:
]

Goal state after:
α : Type u_1
m : MeasurableSpace α
inst✝ : Countable α
μ ν : Measure α
h : ∀ (x : α), μ (measurableAtom x) = ν (measurableAtom x)
s : Set α
hs : MeasurableSet s
h1 : s = ⋃₀ (measurableAtom '' s)
⊢ μ s = ν s
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
m : ℕ → Measure α
hv : ∀ (n : ℕ), (m n) t ≠ ∞
hμ : μ = sum m
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
m : ℕ → Measure α
hv : ∀ (n : ℕ), (m n) t ≠ ∞
hμ : μ = sum m
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
m : MeasurableSpace α
inst✝ : Countable α
μ ν : Measure α
h : ∀ (x : α), μ (measurableAtom x) = ν (measurableAtom x)
s : Set α
hs : MeasurableSet s
h1 : s = ⋃₀ (measurableAtom '' s)
⊢ μ (⋃₀ (measurableAtom '' s)) = ν (⋃₀ (measurableAtom '' s))

Tactic:
]

Goal state after:
α : Type u_1
m : MeasurableSpace α
inst✝ : Countable α
μ ν : Measure α
h : ∀ (x : α), μ (measurableAtom x) = ν (measurableAtom x)
s : Set α
hs : MeasurableSet s
h1 : s = ⋃₀ (measurableAtom '' s)
⊢ μ (⋃₀ (measurableAtom '' s)) = ν (⋃₀ (measurableAtom '' s))
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
inst✝² : TopologicalSpace α
inst✝¹ : SecondCountableTopology α
m : MeasurableSpace α
μ : Measure α
inst✝ : IsLocallyFiniteMeasure μ
h✝ : Nonempty α
inhabited_h : Inhabited α
S : Set (Set α) := {s | IsOpen[inst✝²] s ∧ μ s < ∞}
T : Set (Set α)
T_count : T.Countable
TS : T ⊆ S
hT : ⋃₀ T = univ
T_ne : T.Nonempty
f : ℕ → Set α
hf : T = range f
n : ℕ
⊢ f n ∈ range f

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
δ : Type u_3
ι : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : SecondCountableTopology α
m : MeasurableSpace α
μ : Measure α
inst✝ : IsLocallyFiniteMeasure μ
h✝ : Nonempty α
inhabited_h : Inhabited α
S : Set (Set α) := {s | IsOpen[inst✝²] s ∧ μ s < ∞}
T : Set (Set α)
T_count : T.Countable
TS : T ⊆ S
hT : ⋃₀ T = univ
T_ne : T.Nonempty
f : ℕ → Set α
hf : T = range f
n : ℕ
⊢ f n ∈ range f
Module: Mathlib.MeasureTheory.Measure.Restrict

Goal state before:
pp: case compl
α : Type u_2
m0 : MeasurableSpace α
μ ν : Measure α
S T : Set (Set α)
h_gen : m0 = generateFrom S
hc : T.Countable
h_inter : IsPiSystem S
hU : ⋃₀ T = univ
htop : ∀ t ∈ T, μ t ≠ ∞
ST_eq : ∀ t ∈ T, ∀ s ∈ S, μ (s ∩ t) = ν (s ∩ t)
T_eq : ∀ t ∈ T, μ t = ν t
t : Set α
ht : t ∈ T
u✝ u : Set α
hu : MeasurableSet u
ihu : μ (t ∩ u) = ν (t ∩ u)
this : μ (t \ u) = ν (t \ u)
⊢ μ (t ∩ uᶜ) = ν (t ∩ uᶜ)

pp: case compl
α : Type u_2
m0 : MeasurableSpace α
μ ν : Measure α
S T : Set (Set α)
h_gen : m0 = generateFrom S
hc : T.Countable
h_inter : IsPiSystem S
hU : ⋃₀ T = univ
htop : ∀ t ∈ T, μ t ≠ ∞
ST_eq : ∀ t ∈ T, ∀ s ∈ S, μ (s ∩ t) = ν (s ∩ t)
T_eq : ∀ t ∈ T, μ t = ν t
t : Set α
ht : t ∈ T
u✝ u : Set α
hu : MeasurableSet u
ihu : μ (t ∩ u) = ν (t ∩ u)
this : μ (t ∩ u) + μ (t \ u) = μ (t ∩ u) + ν (t \ u)
⊢ μ (t ∩ u) ≠ ∞

Tactic:
]

Goal state after:
case compl
α : Type u_2
m0 : MeasurableSpace α
μ ν : Measure α
S T : Set (Set α)
h_gen : m0 = generateFrom S
hc : T.Countable
h_inter : IsPiSystem S
hU : ⋃₀ T = univ
htop : ∀ t ∈ T, μ t ≠ ∞
ST_eq : ∀ t ∈ T, ∀ s ∈ S, μ (s ∩ t) = ν (s ∩ t)
T_eq : ∀ t ∈ T, μ t = ν t
t : Set α
ht : t ∈ T
u✝ u : Set α
hu : MeasurableSet u
ihu : μ (t ∩ u) = ν (t ∩ u)
this : μ (t \ u) = ν (t \ u)
⊢ μ (t ∩ uᶜ) = ν (t ∩ uᶜ)

case compl
α : Type u_2
m0 : MeasurableSpace α
μ ν : Measure α
S T : Set (Set α)
h_gen : m0 = generateFrom S
hc : T.Countable
h_inter : IsPiSystem S
hU : ⋃₀ T = univ
htop : ∀ t ∈ T, μ t ≠ ∞
ST_eq : ∀ t ∈ T, ∀ s ∈ S, μ (s ∩ t) = ν (s ∩ t)
T_eq : ∀ t ∈ T, μ t = ν t
t : Set α
ht : t ∈ T
u✝ u : Set α
hu : MeasurableSet u
ihu : μ (t ∩ u) = ν (t ∩ u)
this : μ (t ∩ u) + μ (t \ u) = μ (t ∩ u) + ν (t \ u)
⊢ μ (t ∩ u) ≠ ∞
Module: Mathlib.MeasureTheory.Measure.MeasureSpace

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
inst✝ : Countable α
μ ν : Measure α
h : ∀ (x : α), μ (measurableAtom x) = ν (measurableAtom x)
s : Set α
hs : MeasurableSet s
h1 : s = ⋃₀ (measurableAtom '' s)
h_count : (measurableAtom '' s).Countable
t t' : Set α
y : α
hys : y ∈ s
hy : measurableAtom y = t
y' : α
h_eq : measurableAtom y ≠ measurableAtom y'
hy's : y' ∈ s
hy' : measurableAtom y' = t'
⊢ Disjoint (measurableAtom y) (measurableAtom y')

Tactic:
]

Goal state after:
α : Type u_1
m : MeasurableSpace α
inst✝ : Countable α
μ ν : Measure α
h : ∀ (x : α), μ (measurableAtom x) = ν (measurableAtom x)
s : Set α
hs : MeasurableSet s
h1 : s = ⋃₀ (measurableAtom '' s)
h_count : (measurableAtom '' s).Countable
t t' : Set α
y : α
hys : y ∈ s
hy : measurableAtom y = t
y' : α
h_eq : measurableAtom y ≠ measurableAtom y'
hy's : y' ∈ s
hy' : measurableAtom y' = t'
⊢ Disjoint (measurableAtom y) (measurableAtom y')
Module: Mathlib.MeasureTheory.Measure.MeasureSpace

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
inst✝ : Countable α
μ ν : Measure α
h : ∀ (x : α), μ (measurableAtom x) = ν (measurableAtom x)
s : Set α
hs : MeasurableSet s
h1 : s = ⋃₀ (measurableAtom '' s)
h_count : (measurableAtom '' s).Countable
h_disj : (measurableAtom '' s).Pairwise Disjoint
t : Set α
x : α
hxs : x ∈ s
hx : measurableAtom x = t
⊢ MeasurableSet (measurableAtom x)

Tactic:
]

Goal state after:
α : Type u_1
m : MeasurableSpace α
inst✝ : Countable α
μ ν : Measure α
h : ∀ (x : α), μ (measurableAtom x) = ν (measurableAtom x)
s : Set α
hs : MeasurableSet s
h1 : s = ⋃₀ (measurableAtom '' s)
h_count : (measurableAtom '' s).Countable
h_disj : (measurableAtom '' s).Pairwise Disjoint
t : Set α
x : α
hxs : x ∈ s
hx : measurableAtom x = t
⊢ MeasurableSet (measurableAtom x)
Module: Mathlib.MeasureTheory.Measure.MeasureSpace

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
inst✝ : Countable α
μ ν : Measure α
h : ∀ (x : α), μ (measurableAtom x) = ν (measurableAtom x)
s : Set α
hs : MeasurableSet s
h1 : s = ⋃₀ (measurableAtom '' s)
h_count : (measurableAtom '' s).Countable
h_disj : (measurableAtom '' s).Pairwise Disjoint
h_meas : ∀ t ∈ measurableAtom '' s, MeasurableSet t
⊢ ∑' (s_1 : ↑(measurableAtom '' s)), μ ↑s_1 = ∑' (s_1 : ↑(measurableAtom '' s)), ν ↑s_1

Tactic:
]

Goal state after:
α : Type u_1
m : MeasurableSpace α
inst✝ : Countable α
μ ν : Measure α
h : ∀ (x : α), μ (measurableAtom x) = ν (measurableAtom x)
s : Set α
hs : MeasurableSet s
h1 : s = ⋃₀ (measurableAtom '' s)
h_count : (measurableAtom '' s).Countable
h_disj : (measurableAtom '' s).Pairwise Disjoint
h_meas : ∀ t ∈ measurableAtom '' s, MeasurableSet t
⊢ ∑' (s_1 : ↑(measurableAtom '' s)), μ ↑s_1 = ∑' (s_1 : ↑(measurableAtom '' s)), ν ↑s_1
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)
⊢ t ⊆ ⋃ i, t ∩ w i

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) ≠ ∞
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)
⊢ t ⊆ ⋃ i, t ∩ w i
Module: Mathlib.MeasureTheory.Measure.MeasureSpace

Goal state before:
pp: case e_f
α : Type u_1
m : MeasurableSpace α
inst✝ : Countable α
μ ν : Measure α
h : ∀ (x : α), μ (measurableAtom x) = ν (measurableAtom x)
s : Set α
hs : MeasurableSet s
h1 : s = ⋃₀ (measurableAtom '' s)
h_count : (measurableAtom '' s).Countable
h_disj : (measurableAtom '' s).Pairwise Disjoint
h_meas : ∀ t ∈ measurableAtom '' s, MeasurableSet t
s' : ↑(measurableAtom '' s)
x : α
hxs : x ∈ s
hx : measurableAtom x = ↑s'
⊢ μ (measurableAtom x) = ν (measurableAtom x)

Tactic:
]

Goal state after:
case e_f
α : Type u_1
m : MeasurableSpace α
inst✝ : Countable α
μ ν : Measure α
h : ∀ (x : α), μ (measurableAtom x) = ν (measurableAtom x)
s : Set α
hs : MeasurableSet s
h1 : s = ⋃₀ (measurableAtom '' s)
h_count : (measurableAtom '' s).Countable
h_disj : (measurableAtom '' s).Pairwise Disjoint
h_meas : ∀ t ∈ measurableAtom '' s, MeasurableSet t
s' : ↑(measurableAtom '' s)
x : α
hxs : x ∈ s
hx : measurableAtom x = ↑s'
⊢ μ (measurableAtom x) = ν (measurableAtom x)
Module: Mathlib.MeasureTheory.Measure.MeasureSpace

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
s t u : Set α
hs : MeasurableSet s
h : μ t = μ u
htu : t ⊆ u
ht_ne_top : μ u ≠ ∞
⊢ μ (t ∩ s) = μ (u ∩ s)

Tactic:
]

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
s t u : Set α
hs : MeasurableSet s
h : μ t = μ u
htu : t ⊆ u
ht_ne_top : μ u ≠ ∞
⊢ μ (t ∩ s) = μ (u ∩ s)
Module: Mathlib.MeasureTheory.Measure.Trim

Goal state before:
pp: α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
⊢ μ.trim = μ.trim

Tactic:
]

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

Goal state before:
pp: α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
s : Set α
hm : m ≤ m0
hs : MeasurableSet s
⊢ μ s = μ s

Tactic:
]

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

Goal state before:
pp: α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
s : Set α
hs : MeasurableSet s
⊢ μ s = μ s

Tactic:
]

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

Goal state before:
pp: α : Type u_1
β : Type u_2
m0 : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure α
f : α → β
hf : Measurable f
s : Set β
hs : MeasurableSet s
⊢ μ (f ⁻¹' s) = μ (f ⁻¹' s)

pp: case hs
α : Type u_1
β : Type u_2
m0 : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure α
f : α → β
hf : Measurable f
s : Set β
hs : MeasurableSet s
⊢ MeasurableSet (f ⁻¹' s)

pp: α : Type u_1
β : Type u_2
m0 : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure α
f : α → β
hf : Measurable f
s : Set β
hs : MeasurableSet s
⊢ Measurable f

Tactic:
]

Goal state after:
case hs
α : Type u_1
β : Type u_2
m0 : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure α
f : α → β
hf : Measurable f
s : Set β
hs : MeasurableSet s
⊢ MeasurableSet (f ⁻¹' s)

α : Type u_1
β : Type u_2
m0 : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure α
f : α → β
hf : Measurable f
s : Set β
hs : MeasurableSet s
⊢ Measurable f
Module: Mathlib.MeasureTheory.Measure.Trim

Goal state before:
pp: α : Type u_1
β : Type u_2
m0 : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure α
f : α → β
hf : Measurable f
s : Set β
hs : MeasurableSet s
⊢ (μ.trim ⋯) (f ⁻¹' s) = (μ.trim ⋯) (f ⁻¹' s)

Tactic:
]

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

Goal state before:
pp: α : Type u_1
β : Type u_2
m0 : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure α
f : α → β
hf : Measurable f
p : β → Prop
hp : MeasurableSet {x | p x}
⊢ (∀ᵐ (x : α) ∂μ.trim ⋯, p (f x)) ↔ ∀ᵐ (x : α) ∂μ.trim ⋯, p (f x)

Tactic:
]

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

Goal state before:
pp: α : Type u_1
inst✝ : IsEmpty α
x✝ : MeasurableSpace α
μ : Measure α
s : Set α
⊢ 0 = 0

Tactic:
]

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

Goal state before:
pp: α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
s : Set α
hm : m ≤ m0
h : (μ.trim hm) s = 0
⊢ (μ.trim hm) s ≤ (μ.trim hm) s

Tactic:
]

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

Goal state before:
pp: α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
s : Set α
hm : m ≤ m0
hs : (μ.trim hm) s = 0
⊢ (μ.trim hm) s = 0

Tactic:
]

Goal state after:
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
s : Set α
hm : m ≤ m0
hs : (μ.trim hm) s = 0
⊢ (μ.trim hm) s = 0
Module: Mathlib.MeasureTheory.Measure.Trim

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
m₁ m₂ : MeasurableSpace α
hm₁₂ : m₁ ≤ m₂
hm₂ : m₂ ≤ m0
t : Set α
ht : MeasurableSet t
⊢ μ t = μ t

Tactic:
]

Goal state after:
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✝¹ : MeasurableSpace γ
μ✝ μ₁ μ₂ μ₃ ν✝ ν' ν₁ ν₂ : Measure α
s✝ s' t : Set α
inst✝ : IsEmpty α
m : MeasurableSpace α
μ ν : Measure α
s : Set α
a✝ : MeasurableSet s
⊢ 0 = 0

Tactic:
]

Goal state after: