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

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
s t : Set α
inst✝ : IsFiniteMeasure μ
hs : MeasurableSet s
ht : MeasurableSet t
ε : ℝ≥0∞
h : μ s ≤ μ t + ε
⊢ μ univ - μ s + (μ t + ε) = μ univ - μ s + ε + μ t

Tactic:
by

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

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
s t : Set α
inst✝ : IsFiniteMeasure μ
hs : MeasurableSet s
ht : MeasurableSet t
ε : ℝ≥0∞
h : μ s ≤ μ t + ε
⊢ μ tᶜ ≤ μ sᶜ + ε

Tactic:
by

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

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
s t : Set α
inst✝ : IsFiniteMeasure μ
hs : MeasurableSet s
ht : MeasurableSet t
ε : ℝ≥0∞
h : μ s ≤ μ t + ε
⊢ μ tᶜ ≤ μ sᶜ + ε

Tactic:
by
  rw [measure_compl ht (by finiteness), measure_compl hs (by finiteness), tsub_le_iff_right]
  calc
    μ univ = μ univ - μ s + μ s := (tsub_add_cancel_of_le <| measure_mono s.subset_univ).symm
    _ ≤ μ univ - μ s + (μ t + ε) := by gcongr
    _ = _ := by rw [add_right_comm, add_assoc]

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
t : ℕ → Set α
hd : Directed (fun x1 x2 ↦ x1 ⊆ x2) t
T : ℕ → Set α := fun n ↦ toMeasurable μ (t n)
Td : ℕ → Set α := disjointed T
hm : ∀ (n : ℕ), MeasurableSet (Td n)
I : Finset ℕ
⊢ ∑ n ∈ I, μ (Td n) ≤ ⨆ n, μ (t n)

Tactic:
by

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
⊢ ∃ ν, IsFiniteMeasure ν ∧ μ ≪ ν ∧ ν ≪ μ

Tactic:
by

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

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
⊢ μ.cofinite = ⊥ ↔ IsFiniteMeasure μ

Tactic:
by

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

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
⊢ μ univ ≠ ∞

Tactic:
by

Goal state after:
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
⊢ 0 univ < ∞

Tactic:
by

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

Goal state before:
pp: α : Type u_2
m0 : MeasurableSpace α
μ : Measure α
s : Set α
h : MeasurableSet s
⊢ (μ.restrict s).toOuterMeasure = (OuterMeasure.restrict s) μ.toOuterMeasure

Tactic:
by

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✝ : MeasurableSpace β
μ✝ ν : Measure α
s t : Set α
a : α
μ : Measure α
h : SigmaFinite μ
n : ℕ
⊢ μ (toMeasurable μ (⋯.some.set n)) < ∞

Tactic:
by

Goal state after:
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
inst✝ : IsEmpty α
⊢ IsFiniteMeasure μ

Tactic:
by

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

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
inst✝ : SigmaFinite μ
⊢ ⋃ i, spanningSets μ i = univ

Tactic:
by

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

Goal state before:
pp: α : Type u_1
ι : Type u_5
m : MeasurableSpace α
μ : Measure α
inst✝ : Countable ι
s : ι → Set α
hd : Directed (fun x1 x2 ↦ x1 ⊆ x2) s
⊢ μ (⋃ i, s i) = ⨆ i, μ (s i)

Tactic:
by

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

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
inst✝ : SigmaFinite μ
⊢ Measurable (spanningSetsIndex μ)

Tactic:
by

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

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
inst✝ : SigmaFinite μ
n : ℕ
⊢ spanningSetsIndex μ ⁻¹' {n} = disjointed (spanningSets μ) n

Tactic:
by

Goal state after:
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
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
⊢ (μ + ν) univ < ∞

Tactic:
by

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

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
inst✝ : SigmaFinite μ
x : α
n : ℕ
⊢ spanningSetsIndex μ x = n ↔ x ∈ disjointed (spanningSets μ) n

Tactic:
by

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

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

Tactic:
by

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

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ ν : Measure α
inst✝ : SigmaFinite ν
⊢ (sum fun n ↦ μ.restrict (disjointed (spanningSets ν) n)) = μ

Tactic:
by

Goal state after:
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
⊢ ((μ univ)⁻¹ • μ) univ < ∞

Tactic:
by

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✝ : SigmaFinite μ
this : ∀ (n : ℕ), Fact (μ (disjointed (spanningSets μ) n) < ∞)
n : ℕ
⊢ IsFiniteMeasure ((fun n ↦ μ.restrict (disjointed (spanningSets μ) n)) n)

Tactic:
by

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

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

Tactic:
by

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✝ : SigmaFinite μ
⊢ SFinite μ

Tactic:
by

Goal state after:
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:
by

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

Goal state before:
pp: α : Type u_1
ι : Type u_5
m : MeasurableSpace α
μ : Measure α
inst✝² : Preorder ι
inst✝¹ : IsDirectedOrder ι
inst✝ : atTop.IsCountablyGenerated
s : ι → Set α
hs : Monotone s
⊢ μ (⋃ i, s i) = ⨆ i, μ (s i)

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝¹ : MeasurableSpace α
μ : Measure α
inst✝ : SigmaFinite μ
s : Set α
⊢ (∀ (n : ℕ), μ (s ∩ spanningSets μ n) = 0) ↔ μ s = 0

Tactic:
by

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

Goal state before:
pp: α : Type u_1
ι : Type u_5
m : MeasurableSpace α
μ : Measure α
inst✝² : Preorder ι
inst✝¹ : IsDirectedOrder ι
inst✝ : atTop.IsCountablyGenerated
f : ι → Set α
⊢ μ (⋃ i, f i) = ⨆ i, μ (accumulate f i)

Tactic:
by

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

Goal state before:
pp: α : Type u_1
β : Type u_2
mβ : MeasurableSpace β
m : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
f : α → β
⊢ IsFiniteMeasure (map f μ)

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝¹ : MeasurableSpace α
μ : Measure α
inst✝ : SigmaFinite μ
s : Set α
⊢ (∃ n, 0 < μ (s ∩ spanningSets μ n)) ↔ 0 < μ s

Tactic:
by

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

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
inst✝ : MeasurableSingletonClass α
s : Set α
hs : s.Infinite
h' : ∃ ε, ε ≠ 0 ∧ ∀ x ∈ s, ε ≤ μ {x}
ε : ℝ≥0∞
hne✝ : ε ≠ 0
hε : ∀ x ∈ s, ε ≤ μ {x}
this : Infinite ↑s
x y : ↑s
hne : x ≠ y
⊢ (Disjoint on fun x ↦ {↑x}) x y

Tactic:
by

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

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 : AEMeasurable f μ
inst✝ : IsFiniteMeasure (map f μ)
⊢ μ univ < ∞

Tactic:
by

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

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
inst✝ : MeasurableSingletonClass α
s : Set α
hs : s.Infinite
h' : ∃ ε, ε ≠ 0 ∧ ∀ x ∈ s, ε ≤ μ {x}
ε : ℝ≥0∞
hne : ε ≠ 0
hε : ∀ x ∈ s, ε ≤ μ {x}
this : Infinite ↑s
⊢ μ (⋃ x, {↑x}) = μ s

Tactic:
by

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

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
⊢ measureUnivNNReal μ = 0 ↔ μ = 0

Tactic:
by

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

Goal state before:
pp: α : Type u_1
ι : Type u_5
m : MeasurableSpace α
μ : Measure α
s : ι → Set α
t : Set ι
ht : t.Countable
hd : DirectedOn ((fun x1 x2 ↦ x1 ⊆ x2) on s) t
⊢ μ (⋃ i ∈ t, s i) = ⨆ i ∈ t, μ (s i)

Tactic:
by

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

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
hμ : μ ≠ 0
⊢ 0 < measureUnivNNReal μ

Tactic:
by

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

Goal state before:
pp: α : Type u_2
m0 : MeasurableSpace α
μ : Measure α
s : Set α
C : Set (Set α)
hC : IsCountablySpanning C
ht : ∀ t ∈ C, μ (s ∩ t) = 0
⊢ μ s = 0

Tactic:
by

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

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ ν : Measure α
inst✝ : IsFiniteMeasure μ
hμν : μ ≤ ν
s : Set α
h_univ : μ s + μ sᶜ = ν s + ν sᶜ
hs : MeasurableSet s
x✝ : s.Nonempty
h_lt : μ s < ν s
h_disj : Disjoint s sᶜ
⊢ μ sᶜ ≠ ∞

Tactic:
by

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

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ ν : Measure α
inst✝ : IsFiniteMeasure μ
hμν : μ ≤ ν
h_univ : μ univ = ν univ
⊢ μ = ν

Tactic:
by

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

Goal state before:
pp: α : Type u_1
ι : Type u_4
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)
Union_As_finite : μ (⋃ i, As i) ≠ ∞
posmeas : Set ι := {i | 0 < μ (As i)}
posmeas_def : posmeas = {i | 0 < μ (As i)}
as : ℕ → ℝ≥0∞
left✝ : StrictAnti as
as_mem : ∀ (n : ℕ), as n ∈ Ioo 0 1
as_lim : Tendsto as atTop (𝓝 0)
fairmeas : ℕ → Set ι := fun n ↦ {i | as n ≤ μ (As i)}
n : ℕ
⊢ fairmeas n = (fun i ↦ μ (As i)) ⁻¹' Ici (as n)

Tactic:
by

Goal state after:
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:
by

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

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
s : Set α
inst✝ : IsFiniteMeasure μ
hs : NullMeasurableSet s μ
h : μ s = μ univ
⊢ μ univ ≠ ∞

Tactic:
by

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

Goal state before:
pp: α : Type u_1
ι : Type u_4
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)
Union_As_finite : μ (⋃ i, As i) ≠ ∞
posmeas : Set ι := {i | 0 < μ (As i)}
posmeas_def : posmeas = {i | 0 < μ (As i)}
as : ℕ → ℝ≥0∞
left✝ : StrictAnti as
as_mem : ∀ (n : ℕ), as n ∈ Ioo 0 1
as_lim : Tendsto as atTop (𝓝 0)
fairmeas : ℕ → Set ι := fun n ↦ {i | as n ≤ μ (As i)}
⊢ posmeas = ⋃ n, fairmeas n

Tactic:
by

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

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
p : α → Prop
hp : NullMeasurableSet {a | p a} μ
⊢ (∀ᵐ (a : α) ∂μ, p a) ↔ μ {a | p a} = μ univ

Tactic:
by

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

Goal state before:
pp: α : Type u_2
m0 : MeasurableSpace α
μ : Measure α
s : Set α
C : Set (Set α)
hC : IsCountablySpanning C
hm : ∀ t ∈ C, MeasurableSet t
ht : ∀ t ∈ C, (μ.restrict t) s = 0
⊢ μ s = 0

Tactic:
by

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

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

Tactic:
by

Goal state after:
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:
by

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

Goal state before:
pp: α : Type u_1
ι : Type u_4
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)
Union_As_finite : μ (⋃ i, As i) ≠ ∞
⊢ {i | 0 < μ (As i)}.Countable

Tactic:
by

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

Goal state before:
pp: α : Type u_2
m0 : MeasurableSpace α
μ : Measure α
s : Set α
⊢ (μ.restrict s) univ = μ s

Tactic:
by

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

Goal state before:
pp: α : Type u_2
_m0 : MeasurableSpace α
R : Type u_7
inst✝¹ : SMul R ℝ≥0∞
inst✝ : IsScalarTower R ℝ≥0∞ ℝ≥0∞
c : R
μ : Measure α
s : Set α
⊢ (c • μ).restrict s = c • μ.restrict s

Tactic:
by

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

Goal state before:
pp: α : Type u_2
m0 : MeasurableSpace α
μ : Measure α
s t : Set α
hs : NullMeasurableSet s (μ.restrict t)
u : Set α
hu : MeasurableSet u
⊢ ((μ.restrict t).restrict s) u = (μ.restrict (s ∩ t)) u

Tactic:
by

Goal state after:
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.restrict t).restrict s = μ.restrict s

Tactic:
by

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

Goal state before:
pp: α : Type u_2
m0 : MeasurableSpace α
μ : Measure α
s t : Set α
ht : NullMeasurableSet t μ
u : Set α
hu : MeasurableSet u
⊢ ((μ.restrict t).restrict s) u = (μ.restrict (s ∩ t)) u

Tactic:
by

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

Goal state before:
pp: α : Type u_1
ι : Type u_5
m : MeasurableSpace α
μ : Measure α
inst✝ : Countable ι
s : ι → Set α
h : ∀ (i : ι), NullMeasurableSet (s i) μ
hd : Directed (fun x1 x2 ↦ x1 ⊇ x2) s
hfin : ∃ i, μ (s i) ≠ ∞
⊢ μ (⋂ i, s i) = ⨅ i, μ (s i)

Tactic:
by

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

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

Tactic:
by

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

Goal state before:
pp: α : Type u_2
m0 : MeasurableSpace α
μ : Measure α
s t : Set α
ht : MeasurableSet t
⊢ (μ.restrict s) t = 0 ↔ μ (t ∩ s) = 0

Tactic:
by

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

Goal state before:
pp: X : Type u_5
inst✝¹ : MeasurableSpace X
μ : Measure X
inst✝ : IsFiniteMeasure μ
Es : ℕ → Set X
Es_mble : ∀ (i : ℕ), NullMeasurableSet (Es i) μ
Es_disj : Pairwise fun n m ↦ Disjoint (Es n) (Es m)
decr : Antitone fun n ↦ ⋃ i, ⋃ (_ : i ≥ n), Es i
⊢ ⋂ n, ⋃ i, ⋃ (_ : i ≥ n), Es i = ∅

Tactic:
by

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

Goal state before:
pp: α : Type u_2
m0 : MeasurableSpace α
μ : Measure α
s t : Set α
hs : MeasurableSet s
⊢ (μ.restrict s) t = 0 ↔ μ (t ∩ s) = 0

Tactic:
by

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

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

Tactic:
by

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

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

Tactic:
by

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

Goal state before:
pp: X : Type u_5
inst✝¹ : MeasurableSpace X
μ : Measure X
inst✝ : IsFiniteMeasure μ
Es : ℕ → Set X
Es_mble : ∀ (i : ℕ), NullMeasurableSet (Es i) μ
Es_disj : Pairwise fun n m ↦ Disjoint (Es n) (Es m)
decr : Antitone fun n ↦ ⋃ i, ⋃ (_ : i ≥ n), Es i
nothing : ⋂ n, ⋃ i, ⋃ (_ : i ≥ n), Es i = ∅
n : ℕ
⊢ NullMeasurableSet (⋃ i, ⋃ (_ : i ≥ n), Es i) μ

Tactic:
by

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

Goal state before:
pp: α : Type u_2
m0 : MeasurableSpace α
μ : Measure α
t s : Set α
ht : NullMeasurableSet t μ
⊢ μ.restrict (s ∩ t) + μ.restrict (s \ t) = μ.restrict s

Tactic:
by

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

Goal state before:
pp: α : Type u_1
ι : Type u_5
m : MeasurableSpace α
μ : Measure α
inst✝² : Preorder ι
inst✝¹ : IsCodirectedOrder ι
inst✝ : atBot.IsCountablyGenerated
s : ι → Set α
hs : Monotone s
hsm : ∀ (i : ι), NullMeasurableSet (s i) μ
hfin : ∃ i, μ (s i) ≠ ∞
this : Nonempty ι
x : ℕ → ι
hxm : Antitone x
hx : Tendsto x atTop atBot
⊢ ⨅ n, μ (s (x n)) = μ (⋂ n, s (x n))

Tactic:
by

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

Goal state before:
pp: X : Type u_5
inst✝¹ : MeasurableSpace X
μ : Measure X
inst✝ : IsFiniteMeasure μ
Es : ℕ → Set X
Es_mble : ∀ (i : ℕ), NullMeasurableSet (Es i) μ
Es_disj : Pairwise fun n m ↦ Disjoint (Es n) (Es m)
⊢ Tendsto (⇑μ ∘ fun n ↦ ⋃ i, ⋃ (_ : i ≥ n), Es i) atTop (𝓝 0)

Tactic:
by

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

Goal state before:
pp: α : Type u_1
ι : Type u_4
x✝ : MeasurableSpace α
μ : Measure α
inst✝ : SFinite μ
As : ι → Set α
As_mble : ∀ (i : ι), NullMeasurableSet (As i) (sum (sfiniteSeq μ))
As_disj : Pairwise (AEDisjoint (sum (sfiniteSeq μ)) on As)
⊢ {i | 0 < (sum (sfiniteSeq μ)) (As i)} ⊆ ⋃ n, {i | 0 < (sfiniteSeq μ n) (As i)}

Tactic:
by

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

Goal state before:
pp: α : Type u_1
ι : Type u_5
m : MeasurableSpace α
μ : Measure α
inst✝² : Preorder ι
inst✝¹ : IsCodirectedOrder ι
inst✝ : atBot.IsCountablyGenerated
s : ι → Set α
hs : Monotone s
hsm : ∀ (i : ι), NullMeasurableSet (s i) μ
hfin : ∃ i, μ (s i) ≠ ∞
this : Nonempty ι
x : ℕ → ι
hxm : Antitone x
hx : Tendsto x atTop atBot
⊢ μ (⋂ n, s (x n)) ≤ μ (⋂ i, s i)

Tactic:
by

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

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

Tactic:
by

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

Goal state before:
pp: α : Type u_1
ι : Type u_5
m : MeasurableSpace α
μ : Measure α
inst✝² : Preorder ι
inst✝¹ : IsCodirectedOrder ι
inst✝ : atBot.IsCountablyGenerated
s : ι → Set α
hs : Monotone s
hsm : ∀ (i : ι), NullMeasurableSet (s i) μ
hfin : ∃ i, μ (s i) ≠ ∞
⊢ μ (⋂ i, s i) = ⨅ i, μ (s i)

Tactic:
by

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

Goal state before:
pp: α : Type u_2
m0 : MeasurableSpace α
μ : Measure α
s : Set α
hs : MeasurableSet s
t : Set α
⊢ μ.restrict (s ∪ t) + μ.restrict (s ∩ t) = μ.restrict s + μ.restrict t

Tactic:
by

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

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

Tactic:
by

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

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
s t : Set α
hs : NullMeasurableSet s μ
ht : NullMeasurableSet t μ
hs' : μ s ≠ ∞
ht' : μ t ≠ ∞
hst : μ (s \ t) ≠ ∞
hts : μ (t \ s) ≠ ∞
this : (μ s).toReal - (μ t).toReal = (μ (s \ t)).toReal - (μ (t \ s)).toReal
⊢ |(μ s).toReal - (μ t).toReal| ≤ (μ (s ∆ t)).toReal

Tactic:
rw [this, measure_symmDiff_eq hs ht, ENNReal.toReal_add hst hts]

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
s t : Set α
hs : NullMeasurableSet s μ
ht : NullMeasurableSet t μ
hs' : μ s ≠ ∞
ht' : μ t ≠ ∞
hst : μ (s \ t) ≠ ∞
hts : μ (t \ s) ≠ ∞
this : (μ s).toReal - (μ t).toReal = (μ (s \ t)).toReal - (μ (t \ s)).toReal
⊢ |(μ (s \ t)).toReal - (μ (t \ s)).toReal| ≤ (μ (s \ t)).toReal + (μ (t \ s)).toReal
Module: Mathlib.MeasureTheory.Measure.Restrict

Goal state before:
pp: α : Type u_2
m0 : MeasurableSpace α
μ : Measure α
s t : Set α
h : Disjoint s t
hs : MeasurableSet s
⊢ μ.restrict (s ∪ t) = μ.restrict s + μ.restrict t

Tactic:
by

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

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

Tactic:
by

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

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

Tactic:
by

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

Goal state before:
pp: α : Type u_1
ι : Type u_5
m : MeasurableSpace α
μ : Measure α
inst✝² : Preorder ι
inst✝¹ : IsCodirectedOrder ι
inst✝ : atBot.IsCountablyGenerated
s : ι → Set α
hs : ∀ᵐ (ω : α) ∂μ, Monotone fun x ↦ ω ∈ s x
hsm : ∀ (i : ι), NullMeasurableSet (s i) μ
i✝ : ι
hi : μ (s i✝) ≠ ∞
this : Nonempty ι
t : ι → Set α := fun i ↦ s i ∩ {ω | Monotone fun x ↦ ω ∈ s x}
i : ι
⊢ s i =ᵐ[μ] t i

Tactic:
by

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

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
s t : Set α
hs : NullMeasurableSet s μ
ht : NullMeasurableSet t μ
hs' : μ s ≠ ∞
ht' : μ t ≠ ∞
hst : μ (s \ t) ≠ ∞
hts : μ (t \ s) ≠ ∞
this : (μ s).toReal - (μ t).toReal = (μ (s \ t)).toReal - (μ (t \ s)).toReal
⊢ |(μ s).toReal - (μ t).toReal| ≤ (μ (s ∆ t)).toReal

Tactic:
by

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

Goal state before:
pp: α : Type u_2
m0 : MeasurableSpace α
μ : Measure α
s s' t : Set α
ht : MeasurableSet t
⊢ (μ.restrict (s ∪ s')) t ≤ (μ.restrict s + μ.restrict s') t

Tactic:
by

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

Goal state before:
pp: α : Type u_2
ι : Type u_6
m0 : MeasurableSpace α
μ : Measure α
inst✝ : Countable ι
s : ι → Set α
hd : Pairwise (AEDisjoint μ on s)
hm : ∀ (i : ι), NullMeasurableSet (s i) μ
t : Set α
ht : MeasurableSet t
⊢ (μ.restrict (⋃ i, s i)) t = ∑' (i : ι), (μ.restrict (s i)) t

Tactic:
by

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

Goal state before:
pp: α : Type u_1
ι : Type u_4
x✝ : MeasurableSpace α
μ : Measure α
inst✝ : SFinite μ
As : ι → Set α
As_mble : ∀ (i : ι), NullMeasurableSet (As i) μ
As_disj : Pairwise (AEDisjoint μ on As)
⊢ {i | 0 < μ (As i)}.Countable

Tactic:
by

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

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
s t : Set α
hs : NullMeasurableSet s μ
ht : NullMeasurableSet t μ
hs' : μ s ≠ ∞
ht' : μ t ≠ ∞
hst : μ (s \ t) ≠ ∞
hts : μ (t \ s) ≠ ∞
⊢ μ (s ∪ t) ≠ ∞

Tactic:
by

Goal state after: