Theorem: MeasureTheory.Measure.measure_toMeasurable_inter_of_sum
File: Mathlib/MeasureTheory/Measure/Typeclasses.lean

tactic: have A : ∃ t', t' ⊇ t ∧ MeasurableSet t' ∧ ∀ u, MeasurableSet u → μ (t' ∩ u) = μ (t ∩ u) := by
  let w n := toMeasurable (m n) t
  have T : t ⊆ ⋂ n, w n := subset_iInter (fun i ↦ subset_toMeasurable (m i) t)
  have M : MeasurableSet (⋂ n, w n) :=
    MeasurableSet.iInter (fun i ↦ measurableSet_toMeasurable (m i) t)
  refine ⟨⋂ n, w n, T, M, fun u hu ↦ ?_⟩
  refine le_antisymm ?_ (by gcongr)
  rw [hμ, sum_apply _ (M.inter hu)]
  apply le_trans _ (le_sum_apply _ _)
  apply ENNReal.tsum_le_tsum (fun i ↦ ?_)
  calc
  m i ((⋂ n, w n) ∩ u) ≤ m i (w i ∩ u) := by gcongr; apply iInter_subset
  _ = m i (t ∩ u) := measure_toMeasurable_inter hu (hv i)
state_before: α : Type u_1
β : Type u_2
δ : Type u_3
ι : Type u_4
m0 : MeasurableSpace α
inst✝ : MeasurableSpace β
μ ν ν₁ ν₂ : Measure α
s✝ t✝ s : Set α
hs : MeasurableSet s
t : Set α
m : ℕ → Measure α
hv : ∀ (n : ℕ), (m n) t ≠ ⊤
hμ : μ = sum m
⊢ μ (toMeasurable μ t ∩ s) = μ (t ∩ s)
state_after: α : Type u_1
β : Type u_2
δ : Type u_3
ι : Type u_4
m0 : MeasurableSpace α
inst✝ : MeasurableSpace β
μ ν ν₁ ν₂ : Measure α
s✝ t✝ 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 (toMeasurable μ t ∩ s) = μ (t ∩ s)

tactic: rw [toMeasurable]
state_before: α : Type u_1
β : Type u_2
δ : Type u_3
ι : Type u_4
m0 : MeasurableSpace α
inst✝ : MeasurableSpace β
μ ν ν₁ ν₂ : Measure α
s✝ t✝ 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 (toMeasurable μ t ∩ s) = μ (t ∩ s)
state_after: α : Type u_1
β : Type u_2
δ : Type u_3
ι : Type u_4
m0 : MeasurableSpace α
inst✝ : MeasurableSpace β
μ ν ν₁ ν₂ : Measure α
s✝ t✝ 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 =ᶠ[ae 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: split_ifs with ht
state_before: α : Type u_1
β : Type u_2
δ : Type u_3
ι : Type u_4
m0 : MeasurableSpace α
inst✝ : MeasurableSpace β
μ ν ν₁ ν₂ : Measure α
s✝ t✝ 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 =ᶠ[ae 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)
state_after: case pos
α : Type u_1
β : Type u_2
δ : Type u_3
ι : Type u_4
m0 : MeasurableSpace α
inst✝ : MeasurableSpace β
μ ν ν₁ ν₂ : Measure α
s✝ t✝ 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)
ht : ∃ t_1 ⊇ t, MeasurableSet t_1 ∧ t_1 =ᶠ[ae μ] t
⊢ μ (ht.choose ∩ s) = μ (t ∩ s)

case neg
α : Type u_1
β : Type u_2
δ : Type u_3
ι : Type u_4
m0 : MeasurableSpace α
inst✝ : MeasurableSpace β
μ ν ν₁ ν₂ : Measure α
s✝ t✝ 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)
ht : ¬∃ t_1 ⊇ t, MeasurableSet t_1 ∧ t_1 =ᶠ[ae μ] t
⊢ μ (A.choose ∩ s) = μ (t ∩ s)

tactic: let w n := toMeasurable (m n) t
state_before: α : Type u_1
β : Type u_2
δ : Type u_3
ι : Type u_4
m0 : MeasurableSpace α
inst✝ : MeasurableSpace β
μ ν ν₁ ν₂ : Measure α
s✝ t✝ s : Set α
hs : MeasurableSet s
t : Set α
m : ℕ → Measure α
hv : ∀ (n : ℕ), (m n) t ≠ ⊤
hμ : μ = sum m
⊢ ∃ t' ⊇ t, MeasurableSet t' ∧ ∀ (u : Set α), MeasurableSet u → μ (t' ∩ u) = μ (t ∩ u)
state_after: α : Type u_1
β : Type u_2
δ : Type u_3
ι : Type u_4
m0 : MeasurableSpace α
inst✝ : MeasurableSpace β
μ ν ν₁ ν₂ : Measure α
s✝ t✝ 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, MeasurableSet t' ∧ ∀ (u : Set α), MeasurableSet u → μ (t' ∩ u) = μ (t ∩ u)

tactic: have T : t ⊆ ⋂ n, w n := subset_iInter (fun i ↦ subset_toMeasurable (m i) t)
state_before: α : Type u_1
β : Type u_2
δ : Type u_3
ι : Type u_4
m0 : MeasurableSpace α
inst✝ : MeasurableSpace β
μ ν ν₁ ν₂ : Measure α
s✝ t✝ 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, MeasurableSet t' ∧ ∀ (u : Set α), MeasurableSet u → μ (t' ∩ u) = μ (t ∩ u)
state_after: α : Type u_1
β : Type u_2
δ : Type u_3
ι : Type u_4
m0 : MeasurableSpace α
inst✝ : MeasurableSpace β
μ ν ν₁ ν₂ : Measure α
s✝ t✝ 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
⊢ ∃ t' ⊇ t, MeasurableSet t' ∧ ∀ (u : Set α), MeasurableSet u → μ (t' ∩ u) = μ (t ∩ u)

tactic: have M : MeasurableSet (⋂ n, w n) :=
  MeasurableSet.iInter (fun i ↦ measurableSet_toMeasurable (m i) t)
state_before: α : Type u_1
β : Type u_2
δ : Type u_3
ι : Type u_4
m0 : MeasurableSpace α
inst✝ : MeasurableSpace β
μ ν ν₁ ν₂ : Measure α
s✝ t✝ 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
⊢ ∃ t' ⊇ t, MeasurableSet t' ∧ ∀ (u : Set α), MeasurableSet u → μ (t' ∩ u) = μ (t ∩ u)
state_after: α : Type u_1
β : Type u_2
δ : Type u_3
ι : Type u_4
m0 : MeasurableSpace α
inst✝ : MeasurableSpace β
μ ν ν₁ ν₂ : Measure α
s✝ t✝ 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 ↦ ?_⟩
state_before: α : Type u_1
β : Type u_2
δ : Type u_3
ι : Type u_4
m0 : MeasurableSpace α
inst✝ : MeasurableSpace β
μ ν ν₁ ν₂ : Measure α
s✝ t✝ 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)
state_after: α : Type u_1
β : Type u_2
δ : Type u_3
ι : Type u_4
m0 : MeasurableSpace α
inst✝ : MeasurableSpace β
μ ν ν₁ ν₂ : Measure α
s✝ t✝ 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)

tactic: refine le_antisymm ?_ (by gcongr)
state_before: α : Type u_1
β : Type u_2
δ : Type u_3
ι : Type u_4
m0 : MeasurableSpace α
inst✝ : MeasurableSpace β
μ ν ν₁ ν₂ : Measure α
s✝ t✝ 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)
state_after: α : Type u_1
β : Type u_2
δ : Type u_3
ι : Type u_4
m0 : MeasurableSpace α
inst✝ : MeasurableSpace β
μ ν ν₁ ν₂ : Measure α
s✝ t✝ 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)

tactic: rw [hμ, sum_apply _ (M.inter hu)]
state_before: α : Type u_1
β : Type u_2
δ : Type u_3
ι : Type u_4
m0 : MeasurableSpace α
inst✝ : MeasurableSpace β
μ ν ν₁ ν₂ : Measure α
s✝ t✝ 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)
state_after: α : Type u_1
β : Type u_2
δ : Type u_3
ι : Type u_4
m0 : MeasurableSpace α
inst✝ : MeasurableSpace β
μ ν ν₁ ν₂ : Measure α
s✝ t✝ 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: apply le_trans _ (le_sum_apply _ _)
state_before: α : Type u_1
β : Type u_2
δ : Type u_3
ι : Type u_4
m0 : MeasurableSpace α
inst✝ : MeasurableSpace β
μ ν ν₁ ν₂ : Measure α
s✝ t✝ 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)
state_after: α : Type u_1
β : Type u_2
δ : Type u_3
ι : Type u_4
m0 : MeasurableSpace α
inst✝ : MeasurableSpace β
μ ν ν₁ ν₂ : Measure α
s✝ t✝ 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) ≤ ∑' (i : ℕ), (m i) (t ∩ u)

tactic: apply ENNReal.tsum_le_tsum (fun i ↦ ?_)
state_before: α : Type u_1
β : Type u_2
δ : Type u_3
ι : Type u_4
m0 : MeasurableSpace α
inst✝ : MeasurableSpace β
μ ν ν₁ ν₂ : Measure α
s✝ t✝ 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) ≤ ∑' (i : ℕ), (m i) (t ∩ u)
state_after: α : Type u_1
β : Type u_2
δ : Type u_3
ι : Type u_4
m0 : MeasurableSpace α
inst✝ : MeasurableSpace β
μ ν ν₁ ν₂ : Measure α
s✝ t✝ 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) ≤ (m i) (t ∩ u)

tactic: calc
m i ((⋂ n, w n) ∩ u) ≤ m i (w i ∩ u) := by gcongr; apply iInter_subset
_ = m i (t ∩ u) := measure_toMeasurable_inter hu (hv i)
state_before: α : Type u_1
β : Type u_2
δ : Type u_3
ι : Type u_4
m0 : MeasurableSpace α
inst✝ : MeasurableSpace β
μ ν ν₁ ν₂ : Measure α
s✝ t✝ 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) ≤ (m i) (t ∩ u)
state_after: no goals

tactic: gcongr
state_before: α : Type u_1
β : Type u_2
δ : Type u_3
ι : Type u_4
m0 : MeasurableSpace α
inst✝ : MeasurableSpace β
μ ν ν₁ ν₂ : Measure α
s✝ t✝ 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 (t ∩ u) ≤ μ ((⋂ n, w n) ∩ u)
state_after: no goals

tactic: gcongr
state_before: α : Type u_1
β : Type u_2
δ : Type u_3
ι : Type u_4
m0 : MeasurableSpace α
inst✝ : MeasurableSpace β
μ ν ν₁ ν₂ : Measure α
s✝ t✝ 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) ≤ (m i) (w i ∩ u)
state_after: case h.H
α : Type u_1
β : Type u_2
δ : Type u_3
ι : Type u_4
m0 : MeasurableSpace α
inst✝ : MeasurableSpace β
μ ν ν₁ ν₂ : Measure α
s✝ t✝ 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
⊢ ⋂ n, w n ⊆ w i

tactic: apply iInter_subset
state_before: case h.H
α : Type u_1
β : Type u_2
δ : Type u_3
ι : Type u_4
m0 : MeasurableSpace α
inst✝ : MeasurableSpace β
μ ν ν₁ ν₂ : Measure α
s✝ t✝ 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
⊢ ⋂ n, w n ⊆ w i
state_after: no goals

tactic: apply measure_congr
state_before: case pos
α : Type u_1
β : Type u_2
δ : Type u_3
ι : Type u_4
m0 : MeasurableSpace α
inst✝ : MeasurableSpace β
μ ν ν₁ ν₂ : Measure α
s✝ t✝ 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)
ht : ∃ t_1 ⊇ t, MeasurableSet t_1 ∧ t_1 =ᶠ[ae μ] t
⊢ μ (ht.choose ∩ s) = μ (t ∩ s)
state_after: case pos.H
α : Type u_1
β : Type u_2
δ : Type u_3
ι : Type u_4
m0 : MeasurableSpace α
inst✝ : MeasurableSpace β
μ ν ν₁ ν₂ : Measure α
s✝ t✝ 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)
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 _)
state_before: case pos.H
α : Type u_1
β : Type u_2
δ : Type u_3
ι : Type u_4
m0 : MeasurableSpace α
inst✝ : MeasurableSpace β
μ ν ν₁ ν₂ : Measure α
s✝ t✝ 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)
ht : ∃ t_1 ⊇ t, MeasurableSet t_1 ∧ t_1 =ᶠ[ae μ] t
⊢ ht.choose ∩ s =ᶠ[ae μ] t ∩ s
state_after: no goals

tactic: exact A.choose_spec.2.2 s hs
state_before: case neg
α : Type u_1
β : Type u_2
δ : Type u_3
ι : Type u_4
m0 : MeasurableSpace α
inst✝ : MeasurableSpace β
μ ν ν₁ ν₂ : Measure α
s✝ t✝ 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)
ht : ¬∃ t_1 ⊇ t, MeasurableSet t_1 ∧ t_1 =ᶠ[ae μ] t
⊢ μ (A.choose ∩ s) = μ (t ∩ s)
state_after: no goals