-- Module: Mathlib.MeasureTheory.VectorMeasure.AddContent
Goal:
α : Type u_1
hα : MeasurableSpace α
E : Type u_2
inst : NormedAddCommGroup E
inst_1 : CompleteSpace E
μ : Measure α
inst_2 : IsFiniteMeasure μ
C : Set (Set α)
m : AddContent E C
hC : IsSetRing C
hCmeas : ∀ s ∈ C, MeasurableSet s
hm : ∀ s ∈ C, enorm.{u_2} (E := E) ((m : (a : Set α) → E) s : E) ≤ (μ : (a : Set α) → ℝ≥0∞) s
h'C : ∀ (t : Set α) (ε : ℝ≥0∞), MeasurableSet t → 0 < ε → ∃ s ∈ C, (μ : (a : Set α) → ℝ≥0∞) (s ∆ t) < ε
C' : Set (MeasuredSets μ) := {s : MeasuredSets μ | ∃ c ∈ C, (↑s : Set α) = c}
C'C : ∀ (s : MeasuredSets μ) (hs : s ∈ C'), (↑s : Set α) ∈ C
C'_dense : Dense C'
m₀ : (↑C' : Type u_1) → E := fun (x : (↑C' : Type u_1)) ↦ (m : (a : Set α) → E) (↑(↑x : MeasuredSets μ) : Set α)
lip : LipschitzWith 1 m₀
m₁ : MeasuredSets μ → E := C'_dense.extend (s := C') m₀
m₁_lip : LipschitzWith 1 m₁
hBound : ∀ (s : MeasuredSets μ), ‖m₁ s‖ₑ ≤ (μ : (a : Set α) → ℝ≥0∞) (↑s : Set α)
⊢ ∀ (s t : MeasuredSets μ) (h : Disjoint (↑s : Set α) (↑t : Set α)),
    m₁ (Subtype.mk (p := fun (s : Set α) ↦ MeasurableSet s) ((↑s : Set α) ∪ (↑t : Set α)) ⋯) = m₁ s + m₁ t

Tactic:
have hAddit (s t : MeasuredSets μ) (h : Disjoint (s : Set α) t) :
      m₁ ⟨s ∪ t, s.2.union t.2⟩ = m₁ s + m₁ t := by
    suffices ∀ ε > 0, ‖m₁ (⟨s ∪ t, s.2.union t.2⟩) - m₁ s - m₁ t‖ₑ < ε by
      rw [← sub_eq_zero, ← enorm_eq_zero, sub_add_eq_sub_sub]
      exact eq_bot_iff.2 (le_of_forall_gt this)
    intro ε εpos
    obtain ⟨δ, δpos, hδ⟩ : ∃ δ, 0 < δ ∧ 8 * δ = ε :=
      ⟨ε / 8, (ENNReal.div_pos εpos.ne' (by simp)), ENNReal.mul_div_cancel (by simp) (by simp)⟩
    -- approximate `s` and `t` up to `δ` by sets `s'` and `t'` in `C`.
    obtain ⟨s', s'C, hs'⟩ : ∃ s' ∈ C, μ (s' ∆ s) < δ := h'C _ _ s.2 δpos
    obtain ⟨t', t'C, ht'⟩ : ∃ t' ∈ C, μ (t' ∆ t) < δ := h'C _ _ t.2 δpos
    have It : ‖m t' - m₁ t‖ₑ < δ := by
      have : m₁ ⟨t', hCmeas _ t'C⟩ = m t' :=
        C'_dense.extend_eq lip.continuous ⟨⟨t', hCmeas _ t'C⟩, ⟨t', t'C, rfl⟩⟩
      rw [← this, ← edist_eq_enorm_sub]
      apply (m₁_lip _ _).trans_lt
      simp only [ENNReal.coe_one, MeasuredSets.edist_def, one_mul]
      exact ht'
    -- `s'` and `t'` have no reason to be disjoint, but their intersection has small measure
    have hμ' : μ (s' ∩ t') < 2 * δ := calc
      μ (s' ∩ t')
      _ ≤ μ (s ∩ t ∪ (s' ∆ s) ∪ (t' ∆ t)) := measure_mono (by grind)
      _ = μ ((s' ∆ s) ∪ (t' ∆ t)) := by simp [Set.disjoint_iff_inter_eq_empty.mp h]
      _ ≤ μ (s' ∆ s) + μ (t' ∆ t) := measure_union_le _ _
      _ < δ + δ := by gcongr
      _ = 2 * δ := by ring
    -- Therefore, the set `s'' := s' \ t'` still approximates well the original set `s`, it belongs
    -- to `C`, and moreover `s''` and `t'` are disjoint.
    let s'' := s' \ t'
    have s''C : s'' ∈ C := hC.diff_mem s'C t'C
    have hs'' : μ (s'' ∆ s) < 3 * δ := calc
      μ (s'' ∆ s)
      _ ≤ μ (s'' ∆ s') + μ (s' ∆ s) := measure_symmDiff_le _ _ _
      _ < 2 * δ + δ := by gcongr; simp [s'', symmDiff, hμ']
      _ = 3 * δ := by ring
    have Is : ‖m s'' - m₁ s‖ₑ < 3 * δ := by
      have : m₁ ⟨s'', hCmeas _ s''C⟩ = m s'' :=
        C'_dense.extend_eq lip.continuous ⟨⟨s'', hCmeas _ s''C⟩, ⟨s'', s''C, rfl⟩⟩
      rw [← this, ← edist_eq_enorm_sub]
      apply (m₁_lip _ _).trans_lt
      simp only [ENNReal.coe_one, MeasuredSets.edist_def, one_mul]
      exact hs''
    -- `s'' ∪ t'` also approximates well `s ∪ t`.
    have Ist : ‖m (s'' ∪ t') - m₁ ⟨s ∪ t, s.2.union t.2⟩‖ₑ < 4 * δ := by
      have s''t'C : s'' ∪ t' ∈ C := hC.union_mem s''C t'C
      have : m₁ ⟨s'' ∪ t', hCmeas _ s''t'C⟩ = m (s'' ∪ t') :=
        C'_dense.extend_eq lip.continuous ⟨⟨s'' ∪ t', hCmeas _ s''t'C⟩, ⟨s'' ∪ t', s''t'C, rfl⟩⟩
      rw [← this, ← edist_eq_enorm_sub]
      apply (m₁_lip _ _).trans_lt
      simp only [ENNReal.coe_one, MeasuredSets.edist_def, one_mul]
      change μ ((s'' ∪ t') ∆ (s ∪ t)) < 4 * δ
      calc μ ((s'' ∪ t') ∆ (s ∪ t))
      _ ≤ μ (s'' ∆ s ∪ t' ∆ t) := measure_mono (Set.union_symmDiff_union_subset ..)
      _ ≤ μ (s'' ∆ s) + μ (t' ∆ t) := measure_union_le _ _
      _ < 3 * δ + δ := by gcongr
      _ = 4 * δ := by ring
    -- conclusion: to estimate `m₁ (s ∪ t) - m₁ s - m₁ t`, replace it up to a small error by
    -- `m₁ (s'' ∪ t') - m₁ s'' - m₁ t'`, which is zero as `m₁` is additive on `C` and these
    -- two sets are disjoint
    calc ‖m₁ (⟨s ∪ t, s.2.union t.2⟩) - m₁ s - m₁ t‖ₑ
    _ = ‖(m (s'' ∪ t') - m s'' - m t') + (m₁ ⟨s ∪ t, s.2.union t.2⟩ - m (s'' ∪ t'))
          + (m s'' - m₁ s) + (m t' - m₁ t)‖ₑ := by abel_nf
    _ ≤ ‖m (s'' ∪ t') - m s'' - m t'‖ₑ + ‖m₁ ⟨s ∪ t, s.2.union t.2⟩ - m (s'' ∪ t')‖ₑ
          + ‖m s'' - m₁ s‖ₑ + ‖m t' - m₁ t‖ₑ := enorm_add₄_le
    _ = ‖m₁ ⟨s ∪ t, s.2.union t.2⟩ - m (s'' ∪ t')‖ₑ + ‖m s'' - m₁ s‖ₑ + ‖m t' - m₁ t‖ₑ := by
      rw [addContent_union hC s''C t'C Set.disjoint_sdiff_left]
      simp
    _ < 4 * δ + 3 * δ + δ := by
      gcongr
      rwa [enorm_sub_rev]
    _ = 8 * δ := by ring
    _ = ε := hδ