/-- Caratheodory theorem. If `m` is a metric outer measure, then every Borel measurable set `t` is
Caratheodory measurable: for any (not necessarily measurable) set `s` we have
`μ (s ∩ t) + μ (s \ t) = μ s`. -/
theorem borel_le_caratheodory (hm : IsMetric μ) : borel X ≤ μ.caratheodory := by
  rw [borel_eq_generateFrom_isClosed]
  refine' MeasurableSpace.generateFrom_le fun t ht => μ.isCaratheodory_iff_le.2 fun s => _
  set S : ℕ → Set X := fun n => {x ∈ s | (↑n)⁻¹ ≤ infEdist x t}
  have n0 : ∀ {n : ℕ}, (n⁻¹ : ℝ≥0∞) ≠ 0 := fun {n} => ENNReal.inv_ne_zero.2 (ENNReal.nat_ne_top _)
  have Ssep : ∀ n, IsMetricSeparated (S n) t := fun n =>
    ⟨n⁻¹, n0, fun x hx y hy => hx.2.trans <| infEdist_le_edist_of_mem hy⟩
  have Ssep' : ∀ n, IsMetricSeparated (S n) (s ∩ t) := fun n =>
    (Ssep n).mono Subset.rfl (inter_subset_right _ _)
  have S_sub : ∀ n, S n ⊆ s \ t := fun n =>
    subset_inter (inter_subset_left _ _) (Ssep n).subset_compl_right
  have hSs : ∀ n, μ (s ∩ t) + μ (S n) ≤ μ s := fun n =>
    calc
      μ (s ∩ t) + μ (S n) = μ (s ∩ t ∪ S n) := Eq.symm <| hm _ _ <| (Ssep' n).symm
      _ ≤ μ (s ∩ t ∪ s \ t) := μ.mono <| union_subset_union_right _ <| S_sub n
      _ = μ s := by rw [inter_union_diff]
  have iUnion_S : ⋃ n, S n = s \ t := by
    refine' Subset.antisymm (iUnion_subset S_sub) _
    rintro x ⟨hxs, hxt⟩
    rw [mem_iff_infEdist_zero_of_closed ht] at hxt
    rcases ENNReal.exists_inv_nat_lt hxt with ⟨n, hn⟩
    exact mem_iUnion.2 ⟨n, hxs, hn.le⟩
  /- Now we have `∀ n, μ (s ∩ t) + μ (S n) ≤ μ s` and we need to prove
    `μ (s ∩ t) + μ (⋃ n, S n) ≤ μ s`. We can't pass to the limit because
    `μ` is only an outer measure. -/
  by_cases htop : μ (s \ t) = ∞
  · rw [htop, add_top, ← htop]
    exact μ.mono (diff_subset _ _)
  suffices : μ (⋃ n, S n) ≤ ⨆ n, μ (S n)
  calc
    μ (s ∩ t) + μ (s \ t) = μ (s ∩ t) + μ (⋃ n, S n) := by rw [iUnion_S]
    _ ≤ μ (s ∩ t) + ⨆ n, μ (S n) := (add_le_add le_rfl this)
    _ = ⨆ n, μ (s ∩ t) + μ (S n) := ENNReal.add_iSup
    _ ≤ μ s := iSup_le hSs
  /- It suffices to show that `∑' k, μ (S (k + 1) \ S k) ≠ ∞`. Indeed, if we have this,
    then for all `N` we have `μ (⋃ n, S n) ≤ μ (S N) + ∑' k, m (S (N + k + 1) \ S (N + k))`
    and the second term tends to zero, see `OuterMeasure.iUnion_nat_of_monotone_of_tsum_ne_top`
    for details. -/
  have : ∀ n, S n ⊆ S (n + 1) := fun n x hx =>
    ⟨hx.1, le_trans (ENNReal.inv_le_inv.2 <| Nat.cast_le.2 n.le_succ) hx.2⟩
  classical -- Porting note: Added this to get the next tactic to work
  refine' (μ.iUnion_nat_of_monotone_of_tsum_ne_top this _).le; clear this
  /- While the sets `S (k + 1) \ S k` are not pairwise metric separated, the sets in each
    subsequence `S (2 * k + 1) \ S (2 * k)` and `S (2 * k + 2) \ S (2 * k)` are metric separated,
    so `m` is additive on each of those sequences. -/
  rw [← tsum_even_add_odd ENNReal.summable ENNReal.summable, ENNReal.add_ne_top]
  suffices : ∀ a, (∑' k : ℕ, μ (S (2 * k + 1 + a) \ S (2 * k + a))) ≠ ∞
  exact ⟨by simpa using this 0, by simpa using this 1⟩
  refine' fun r => ne_top_of_le_ne_top htop _
  rw [← iUnion_S, ENNReal.tsum_eq_iSup_nat, iSup_le_iff]
  intro n
  rw [← hm.finset_iUnion_of_pairwise_separated]
  · exact μ.mono (iUnion_subset fun i => iUnion_subset fun _ x hx => mem_iUnion.2 ⟨_, hx.1⟩)
  suffices : ∀ i j, i < j → IsMetricSeparated (S (2 * i + 1 + r)) (s \ S (2 * j + r))
  exact fun i _ j _ hij =>
    hij.lt_or_lt.elim
      (fun h => (this i j h).mono (inter_subset_left _ _) fun x hx => by exact ⟨hx.1.1, hx.2⟩)
      fun h => (this j i h).symm.mono (fun x hx => by exact ⟨hx.1.1, hx.2⟩) (inter_subset_left _ _)
  intro i j hj
  have A : ((↑(2 * j + r))⁻¹ : ℝ≥0∞) < (↑(2 * i + 1 + r))⁻¹ := by
    rw [ENNReal.inv_lt_inv, Nat.cast_lt]; linarith
  refine' ⟨(↑(2 * i + 1 + r))⁻¹ - (↑(2 * j + r))⁻¹, by simpa using A, fun x hx y hy => _⟩
  have : infEdist y t < (↑(2 * j + r))⁻¹ := not_le.1 fun hle => hy.2 ⟨hy.1, hle⟩
  rcases infEdist_lt_iff.mp this with ⟨z, hzt, hyz⟩
  have hxz : (↑(2 * i + 1 + r))⁻¹ ≤ edist x z := le_infEdist.1 hx.2 _ hzt
  apply ENNReal.le_of_add_le_add_right hyz.ne_top
  

Proof state:
case this.intro.intro
ι : Type u_1
X : Type u_2
Y : Type u_3
inst✝¹ : EMetricSpace X
inst✝ : EMetricSpace Y
μ : OuterMeasure X
hm : IsMetric μ
t : Set X
ht : t ∈ {s | IsClosed s}
s : Set X
S : ℕ → Set X := fun n => {x | x ∈ s ∧ (↑n)⁻¹ ≤ infEdist x t}
n0 : ∀ {n : ℕ}, (↑n)⁻¹ ≠ 0
Ssep : ∀ (n : ℕ), IsMetricSeparated (S n) t
Ssep' : ∀ (n : ℕ), IsMetricSeparated (S n) (s ∩ t)
S_sub : ∀ (n : ℕ), S n ⊆ s \ t
hSs : ∀ (n : ℕ), ↑μ (s ∩ t) + ↑μ (S n) ≤ ↑μ s
iUnion_S : ⋃ n, S n = s \ t
htop : ¬↑μ (s \ t) = ⊤
r n i j : ℕ
hj : i < j
A : (↑(2 * j + r))⁻¹ < (↑(2 * i + 1 + r))⁻¹
x : X
hx : x ∈ S (2 * i + 1 + r)
y : X
hy : y ∈ s \ S (2 * j + r)
this : infEdist y t < (↑(2 * j + r))⁻¹
z : X
hzt : z ∈ t
hyz : edist y z < (↑(2 * j + r))⁻¹
hxz : (↑(2 * i + 1 + r))⁻¹ ≤ edist x z
⊢ (↑(2 * i + 1 + r))⁻¹ - (↑(2 * j + r))⁻¹ + edist y z ≤ edist x y + edist y z

Next tactic:
refine' le_trans _ (edist_triangle _ _ _)