l3lab/ntp-mathlib huggingface dataset:
/-- If a measure `μ` is the sum of a countable family `mₙ`, and a set `t` has finite measure for
each `mₙ`, then its measurable superset `toMeasurable μ t` (which has the same measure as `t`)
satisfies, for any measurable set `s`, the equality `μ (toMeasurable μ t ∩ s) = μ (t ∩ s)`. -/
theorem measure_toMeasurable_inter_of_sum {s : Set α} (hs : MeasurableSet s) {t : Set α}
    {m : ℕ → Measure α} (hv : ∀ n, m n t ≠ ∞) (hμ : μ = sum m) :
    μ (toMeasurable μ t ∩ s) = μ (t ∩ s) := by
  -- we show that there is a measurable superset of `t` satisfying the conclusion for any
  -- measurable set `s`. It is built for each measure `mₙ` using `toMeasurable`
  -- (which is well behaved for finite measure sets thanks to `measure_toMeasurable_inter`), and
  -- then taking the intersection over `n`.
  have A : ∃ t', t' ⊇ t ∧ MeasurableSet t' ∧ ∀ u, MeasurableSet u → μ (t' ∩ u) = μ (t ∩ u) := by
    let w n := toMeasurable (m n) t
    

Proof state:
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)

Next tactic:
have T : t ⊆ ⋂ n, w n := subset_iInter (fun i ↦ subset_toMeasurable (m i) t)
/-- If a measure `μ` is the sum of a countable family `mₙ`, and a set `t` has finite measure for
each `mₙ`, then its measurable superset `toMeasurable μ t` (which has the same measure as `t`)
satisfies, for any measurable set `s`, the equality `μ (toMeasurable μ t ∩ s) = μ (t ∩ s)`. -/
theorem measure_toMeasurable_inter_of_sum {s : Set α} (hs : MeasurableSet s) {t : Set α}
    {m : ℕ → Measure α} (hv : ∀ n, m n t ≠ ∞) (hμ : μ = sum m) :
    μ (toMeasurable μ t ∩ s) = μ (t ∩ s) := by
  -- we show that there is a measurable superset of `t` satisfying the conclusion for any
  -- measurable set `s`. It is built for each measure `mₙ` using `toMeasurable`
  -- (which is well behaved for finite measure sets thanks to `measure_toMeasurable_inter`), and
  -- then taking the intersection over `n`.
  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)
    

Proof state:
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)

Next tactic:
have M : MeasurableSet (⋂ n, w n) :=
      MeasurableSet.iInter (fun i ↦ measurableSet_toMeasurable (m i) t)
/-- If a measure `μ` is the sum of a countable family `mₙ`, and a set `t` has finite measure for
each `mₙ`, then its measurable superset `toMeasurable μ t` (which has the same measure as `t`)
satisfies, for any measurable set `s`, the equality `μ (toMeasurable μ t ∩ s) = μ (t ∩ s)`. -/
theorem measure_toMeasurable_inter_of_sum {s : Set α} (hs : MeasurableSet s) {t : Set α}
    {m : ℕ → Measure α} (hv : ∀ n, m n t ≠ ∞) (hμ : μ = sum m) :
    μ (toMeasurable μ t ∩ s) = μ (t ∩ s) := by
  -- we show that there is a measurable superset of `t` satisfying the conclusion for any
  -- measurable set `s`. It is built for each measure `mₙ` using `toMeasurable`
  -- (which is well behaved for finite measure sets thanks to `measure_toMeasurable_inter`), and
  -- then taking the intersection over `n`.
  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)
    

Proof state:
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)

Next tactic:
refine ⟨⋂ n, w n, T, M, fun u hu ↦ ?_⟩
/-- If a measure `μ` is the sum of a countable family `mₙ`, and a set `t` has finite measure for
each `mₙ`, then its measurable superset `toMeasurable μ t` (which has the same measure as `t`)
satisfies, for any measurable set `s`, the equality `μ (toMeasurable μ t ∩ s) = μ (t ∩ s)`. -/
theorem measure_toMeasurable_inter_of_sum {s : Set α} (hs : MeasurableSet s) {t : Set α}
    {m : ℕ → Measure α} (hv : ∀ n, m n t ≠ ∞) (hμ : μ = sum m) :
    μ (toMeasurable μ t ∩ s) = μ (t ∩ s) := by
  -- we show that there is a measurable superset of `t` satisfying the conclusion for any
  -- measurable set `s`. It is built for each measure `mₙ` using `toMeasurable`
  -- (which is well behaved for finite measure sets thanks to `measure_toMeasurable_inter`), and
  -- then taking the intersection over `n`.
  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 ↦ ?_⟩
    

Proof state:
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)

Next tactic:
refine le_antisymm ?_ (measure_mono (inter_subset_inter_left _ T))
/-- If a measure `μ` is the sum of a countable family `mₙ`, and a set `t` has finite measure for
each `mₙ`, then its measurable superset `toMeasurable μ t` (which has the same measure as `t`)
satisfies, for any measurable set `s`, the equality `μ (toMeasurable μ t ∩ s) = μ (t ∩ s)`. -/
theorem measure_toMeasurable_inter_of_sum {s : Set α} (hs : MeasurableSet s) {t : Set α}
    {m : ℕ → Measure α} (hv : ∀ n, m n t ≠ ∞) (hμ : μ = sum m) :
    μ (toMeasurable μ t ∩ s) = μ (t ∩ s) := by
  -- we show that there is a measurable superset of `t` satisfying the conclusion for any
  -- measurable set `s`. It is built for each measure `mₙ` using `toMeasurable`
  -- (which is well behaved for finite measure sets thanks to `measure_toMeasurable_inter`), and
  -- then taking the intersection over `n`.
  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 ?_ (measure_mono (inter_subset_inter_left _ T))
    

Proof state:
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)

Next tactic:
rw [hμ, sum_apply _ (M.inter hu)]
/-- If a measure `μ` is the sum of a countable family `mₙ`, and a set `t` has finite measure for
each `mₙ`, then its measurable superset `toMeasurable μ t` (which has the same measure as `t`)
satisfies, for any measurable set `s`, the equality `μ (toMeasurable μ t ∩ s) = μ (t ∩ s)`. -/
theorem measure_toMeasurable_inter_of_sum {s : Set α} (hs : MeasurableSet s) {t : Set α}
    {m : ℕ → Measure α} (hv : ∀ n, m n t ≠ ∞) (hμ : μ = sum m) :
    μ (toMeasurable μ t ∩ s) = μ (t ∩ s) := by
  -- we show that there is a measurable superset of `t` satisfying the conclusion for any
  -- measurable set `s`. It is built for each measure `mₙ` using `toMeasurable`
  -- (which is well behaved for finite measure sets thanks to `measure_toMeasurable_inter`), and
  -- then taking the intersection over `n`.
  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 ?_ (measure_mono (inter_subset_inter_left _ T))
    rw [hμ, sum_apply _ (M.inter hu)]
    

Proof state:
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)

Next tactic:
apply le_trans _ (le_sum_apply _ _)
/-- If a measure `μ` is the sum of a countable family `mₙ`, and a set `t` has finite measure for
each `mₙ`, then its measurable superset `toMeasurable μ t` (which has the same measure as `t`)
satisfies, for any measurable set `s`, the equality `μ (toMeasurable μ t ∩ s) = μ (t ∩ s)`. -/
theorem measure_toMeasurable_inter_of_sum {s : Set α} (hs : MeasurableSet s) {t : Set α}
    {m : ℕ → Measure α} (hv : ∀ n, m n t ≠ ∞) (hμ : μ = sum m) :
    μ (toMeasurable μ t ∩ s) = μ (t ∩ s) := by
  -- we show that there is a measurable superset of `t` satisfying the conclusion for any
  -- measurable set `s`. It is built for each measure `mₙ` using `toMeasurable`
  -- (which is well behaved for finite measure sets thanks to `measure_toMeasurable_inter`), and
  -- then taking the intersection over `n`.
  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 ?_ (measure_mono (inter_subset_inter_left _ T))
    rw [hμ, sum_apply _ (M.inter hu)]
    apply le_trans _ (le_sum_apply _ _)
    

Proof state:
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)

Next tactic:
apply ENNReal.tsum_le_tsum (fun i ↦ ?_)
/-- If a measure `μ` is the sum of a countable family `mₙ`, and a set `t` has finite measure for
each `mₙ`, then its measurable superset `toMeasurable μ t` (which has the same measure as `t`)
satisfies, for any measurable set `s`, the equality `μ (toMeasurable μ t ∩ s) = μ (t ∩ s)`. -/
theorem measure_toMeasurable_inter_of_sum {s : Set α} (hs : MeasurableSet s) {t : Set α}
    {m : ℕ → Measure α} (hv : ∀ n, m n t ≠ ∞) (hμ : μ = sum m) :
    μ (toMeasurable μ t ∩ s) = μ (t ∩ s) := by
  -- we show that there is a measurable superset of `t` satisfying the conclusion for any
  -- measurable set `s`. It is built for each measure `mₙ` using `toMeasurable`
  -- (which is well behaved for finite measure sets thanks to `measure_toMeasurable_inter`), and
  -- then taking the intersection over `n`.
  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 ?_ (measure_mono (inter_subset_inter_left _ T))
    rw [hμ, sum_apply _ (M.inter hu)]
    apply le_trans _ (le_sum_apply _ _)
    apply ENNReal.tsum_le_tsum (fun i ↦ ?_)
    

Proof state:
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)

Next tactic:
calc
    m i ((⋂ n, w n) ∩ u) ≤ m i (w i ∩ u) :=
      measure_mono (inter_subset_inter_left _ (iInter_subset _ _))
    _ = m i (t ∩ u) := measure_toMeasurable_inter hu (hv i)
/-- If a measure `μ` is the sum of a countable family `mₙ`, and a set `t` has finite measure for
each `mₙ`, then its measurable superset `toMeasurable μ t` (which has the same measure as `t`)
satisfies, for any measurable set `s`, the equality `μ (toMeasurable μ t ∩ s) = μ (t ∩ s)`. -/
theorem measure_toMeasurable_inter_of_sum {s : Set α} (hs : MeasurableSet s) {t : Set α}
    {m : ℕ → Measure α} (hv : ∀ n, m n t ≠ ∞) (hμ : μ = sum m) :
    μ (toMeasurable μ t ∩ s) = μ (t ∩ s) := by
  -- we show that there is a measurable superset of `t` satisfying the conclusion for any
  -- measurable set `s`. It is built for each measure `mₙ` using `toMeasurable`
  -- (which is well behaved for finite measure sets thanks to `measure_toMeasurable_inter`), and
  -- then taking the intersection over `n`.
  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 ?_ (measure_mono (inter_subset_inter_left _ T))
    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) :=
      measure_mono (inter_subset_inter_left _ (iInter_subset _ _))
    _ = m i (t ∩ u) := measure_toMeasurable_inter hu (hv i)
  -- thanks to the definition of `toMeasurable`, the previous property will also be shared
  -- by `toMeasurable μ t`, which is enough to conclude the proof.
  

Proof state:
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)

Next tactic:
rw [toMeasurable]
/-- If a measure `μ` is the sum of a countable family `mₙ`, and a set `t` has finite measure for
each `mₙ`, then its measurable superset `toMeasurable μ t` (which has the same measure as `t`)
satisfies, for any measurable set `s`, the equality `μ (toMeasurable μ t ∩ s) = μ (t ∩ s)`. -/
theorem measure_toMeasurable_inter_of_sum {s : Set α} (hs : MeasurableSet s) {t : Set α}
    {m : ℕ → Measure α} (hv : ∀ n, m n t ≠ ∞) (hμ : μ = sum m) :
    μ (toMeasurable μ t ∩ s) = μ (t ∩ s) := by
  -- we show that there is a measurable superset of `t` satisfying the conclusion for any
  -- measurable set `s`. It is built for each measure `mₙ` using `toMeasurable`
  -- (which is well behaved for finite measure sets thanks to `measure_toMeasurable_inter`), and
  -- then taking the intersection over `n`.
  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 ?_ (measure_mono (inter_subset_inter_left _ T))
    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) :=
      measure_mono (inter_subset_inter_left _ (iInter_subset _ _))
    _ = m i (t ∩ u) := measure_toMeasurable_inter hu (hv i)
  -- thanks to the definition of `toMeasurable`, the previous property will also be shared
  -- by `toMeasurable μ t`, which is enough to conclude the proof.
  rw [toMeasurable]
  

Proof state:
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 Exists.choose h
        else
          if h' : ∃ t_1 ⊇ t, MeasurableSet t_1 ∧ ∀ (u : Set α), MeasurableSet u → ↑↑μ (t_1 ∩ u) = ↑↑μ (t ∩ u) then
            Exists.choose h'
          else Exists.choose (_ : ∃ t_1, t ⊆ t_1 ∧ MeasurableSet t_1 ∧ ↑↑μ t_1 = ↑↑μ t)) ∩
        s) =
    ↑↑μ (t ∩ s)

Next tactic:
split_ifs with ht
/-- If a measure `μ` is the sum of a countable family `mₙ`, and a set `t` has finite measure for
each `mₙ`, then its measurable superset `toMeasurable μ t` (which has the same measure as `t`)
satisfies, for any measurable set `s`, the equality `μ (toMeasurable μ t ∩ s) = μ (t ∩ s)`. -/
theorem measure_toMeasurable_inter_of_sum {s : Set α} (hs : MeasurableSet s) {t : Set α}
    {m : ℕ → Measure α} (hv : ∀ n, m n t ≠ ∞) (hμ : μ = sum m) :
    μ (toMeasurable μ t ∩ s) = μ (t ∩ s) := by
  -- we show that there is a measurable superset of `t` satisfying the conclusion for any
  -- measurable set `s`. It is built for each measure `mₙ` using `toMeasurable`
  -- (which is well behaved for finite measure sets thanks to `measure_toMeasurable_inter`), and
  -- then taking the intersection over `n`.
  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 ?_ (measure_mono (inter_subset_inter_left _ T))
    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) :=
      measure_mono (inter_subset_inter_left _ (iInter_subset _ _))
    _ = m i (t ∩ u) := measure_toMeasurable_inter hu (hv i)
  -- thanks to the definition of `toMeasurable`, the previous property will also be shared
  -- by `toMeasurable μ t`, which is enough to conclude the proof.
  rw [toMeasurable]
  split_ifs with ht
  · 

Proof state:
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 (Exists.choose ht ∩ s) = ↑↑μ (t ∩ s)

Next tactic:
apply measure_congr
/-- If a measure `μ` is the sum of a countable family `mₙ`, and a set `t` has finite measure for
each `mₙ`, then its measurable superset `toMeasurable μ t` (which has the same measure as `t`)
satisfies, for any measurable set `s`, the equality `μ (toMeasurable μ t ∩ s) = μ (t ∩ s)`. -/
theorem measure_toMeasurable_inter_of_sum {s : Set α} (hs : MeasurableSet s) {t : Set α}
    {m : ℕ → Measure α} (hv : ∀ n, m n t ≠ ∞) (hμ : μ = sum m) :
    μ (toMeasurable μ t ∩ s) = μ (t ∩ s) := by
  -- we show that there is a measurable superset of `t` satisfying the conclusion for any
  -- measurable set `s`. It is built for each measure `mₙ` using `toMeasurable`
  -- (which is well behaved for finite measure sets thanks to `measure_toMeasurable_inter`), and
  -- then taking the intersection over `n`.
  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 ?_ (measure_mono (inter_subset_inter_left _ T))
    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) :=
      measure_mono (inter_subset_inter_left _ (iInter_subset _ _))
    _ = m i (t ∩ u) := measure_toMeasurable_inter hu (hv i)
  -- thanks to the definition of `toMeasurable`, the previous property will also be shared
  -- by `toMeasurable μ t`, which is enough to conclude the proof.
  rw [toMeasurable]
  split_ifs with ht
  · apply measure_congr
    

Proof state:
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
⊢ Exists.choose ht ∩ s =ᶠ[ae μ] t ∩ s

Next tactic:
exact ae_eq_set_inter ht.choose_spec.2.2 (ae_eq_refl _)