-- Mathlib/MeasureTheory/Measure/Typeclasses.lean
theorem measure_toMeasurable_inter_of_cover {s : Set α} (hs : MeasurableSet s) {t : Set α}
    {v : ℕ → Set α} (hv : t ⊆ ⋃ n, v n) (h'v : ∀ n, μ (t ∩ v n) ≠ ∞) :
    μ (toMeasurable μ t ∩ s) = μ (t ∩ s)

Proof state:
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:
intro x hx

Result:
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)
x : α
hx : x ∈ t
⊢ x ∈ ⋃ i, t ∩ w i

Proof state:
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)
⊢ ∃ t' ⊇ t, MeasurableSet t' ∧ ∀ (u : Set α), MeasurableSet u → μ (t' ∩ u) = μ (t ∩ u)

Tactic:
have hw : ∀ n, μ (w n) < ∞ := by
  intro n
  simp_rw [w, measure_toMeasurable]
  exact (h'v n).lt_top

Result:
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' ⊇ t, MeasurableSet t' ∧ ∀ (u : Set α), MeasurableSet u → μ (t' ∩ u) = μ (t ∩ u)

Proof state:
case pos.H
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) ≠ ⊤
A : ∃ t' ⊇ t, MeasurableSet t' ∧ ∀ (u : Set α), MeasurableSet u → μ (t' ∩ u) = μ (t ∩ u)
ht : ∃ t_1 ⊇ t, MeasurableSet t_1 ∧ t_1 =ᶠ[ae μ] t
⊢ ht.choose ∩ s =ᶠ[ae μ] t ∩ s

Tactic:
exact ae_eq_set_inter ht.choose_spec.2.2 (ae_eq_refl _)

Result:
no goals

Proof state:
case e_f
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)
tt' : t ⊆ t'
u : Set α
hu : MeasurableSet u
⊢ (fun n => μ (toMeasurable μ (t ∩ disjointed w n) ∩ u)) = fun n => μ (t ∩ disjointed w n ∩ u)

Tactic:
ext1 n

Result:
case e_f.h
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)
tt' : t ⊆ t'
u : Set α
hu : MeasurableSet u
n : ℕ
⊢ μ (toMeasurable μ (t ∩ disjointed w n) ∩ u) = μ (t ∩ disjointed w n ∩ u)

Proof state:
case e_f
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)
tt' : t ⊆ t'
u : Set α
hu : MeasurableSet u
⊢ (fun n => μ (t ∩ disjointed w n ∩ u)) = fun n => (μ.restrict (t ∩ u)) (disjointed w n)

Tactic:
ext1 n

Result:
case e_f.h
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)
tt' : t ⊆ t'
u : Set α
hu : MeasurableSet u
n : ℕ
⊢ μ (t ∩ disjointed w n ∩ u) = (μ.restrict (t ∩ u)) (disjointed w n)

Proof state:
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' ⊇ t, MeasurableSet t' ∧ ∀ (u : Set α), MeasurableSet u → μ (t' ∩ u) = μ (t ∩ u)

Tactic:
set t' := ⋃ n, toMeasurable μ (t ∩ disjointed w n) with ht'

Result:
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' ⊇ t, MeasurableSet t' ∧ ∀ (u : Set α), MeasurableSet u → μ (t' ∩ u) = μ (t ∩ u)

Proof state:
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)
x : α
hx : x ∈ t
⊢ x ∈ ⋃ i, t ∩ w i

Tactic:
rcases mem_iUnion.1 (hv hx) with ⟨n, hn⟩

Result:
case intro
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)
x : α
hx : x ∈ t
n : ℕ
hn : x ∈ v n
⊢ x ∈ ⋃ i, t ∩ w i

Proof state:
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)
tt' : t ⊆ t'
u : Set α
hu : MeasurableSet u
⊢ μ (t ∩ u) ≤ μ (t' ∩ u)

Tactic:
gcongr

Result:
no goals

Proof state:
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' ⊇ t, MeasurableSet t' ∧ ∀ (u : Set α), MeasurableSet u → μ (t' ∩ u) = μ (t ∩ u)

Tactic:
have tt' : t ⊆ t' :=
  calc
    t ⊆ ⋃ n, t ∩ disjointed w n := by
      rw [← inter_iUnion, iUnion_disjointed, inter_iUnion]
      intro x hx
      rcases mem_iUnion.1 (hv hx) with ⟨n, hn⟩
      refine mem_iUnion.2 ⟨n, ?_⟩
      have : x ∈ t ∩ v n := ⟨hx, hn⟩
      exact ⟨hx, subset_toMeasurable μ _ this⟩
    _ ⊆ ⋃ n, toMeasurable μ (t ∩ disjointed w n) :=
      iUnion_mono fun n => subset_toMeasurable _ _

Result:
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)
tt' : t ⊆ t'
⊢ ∃ t' ⊇ t, MeasurableSet t' ∧ ∀ (u : Set α), MeasurableSet u → μ (t' ∩ u) = μ (t ∩ u)