/-- A measurable set of negative measure has a negative subset of negative measure. -/
theorem exists_subset_restrict_nonpos (hi : s i < 0) :
    ∃ j : Set α, MeasurableSet j ∧ j ⊆ i ∧ s ≤[j] 0 ∧ s j < 0 := by
  have hi₁ : MeasurableSet i := by_contradiction fun h => ne_of_lt hi <| s.not_measurable h
  by_cases h : s ≤[i] 0; · exact ⟨i, hi₁, Set.Subset.refl _, h, hi⟩
  by_cases hn : ∀ n : ℕ, ¬s ≤[i \ ⋃ l < n, restrictNonposSeq s i l] 0
  swap; · exact exists_subset_restrict_nonpos' hi₁ hi hn
  set A := i \ ⋃ l, restrictNonposSeq s i l with hA
  set bdd : ℕ → ℕ := fun n => findExistsOneDivLT s (i \ ⋃ k ≤ n, restrictNonposSeq s i k)
  have hn' : ∀ n : ℕ, ¬s ≤[i \ ⋃ l ≤ n, restrictNonposSeq s i l] 0 := by
    intro n
    convert hn (n + 1) using 5 <;>
      · ext l
        simp only [exists_prop, Set.mem_iUnion, and_congr_left_iff]
        exact fun _ => Nat.lt_succ_iff.symm
  have h₁ : s i = s A + ∑' l, s (restrictNonposSeq s i l) := by
    rw [hA, ← s.of_disjoint_iUnion_nat, add_comm, of_add_of_diff]
    exact MeasurableSet.iUnion fun _ => restrictNonposSeq_measurableSet _
    exacts [hi₁, Set.iUnion_subset fun _ => restrictNonposSeq_subset _, fun _ =>
      restrictNonposSeq_measurableSet _, restrictNonposSeq_disjoint]
  have h₂ : s A ≤ s i := by
    rw [h₁]
    apply le_add_of_nonneg_right
    exact tsum_nonneg fun n => le_of_lt (measure_of_restrictNonposSeq h _ (hn n))
  have h₃' : Summable fun n => (1 / (bdd n + 1) : ℝ) := by
    have : Summable fun l => s (restrictNonposSeq s i l) :=
      HasSum.summable
        (s.m_iUnion (fun _ => restrictNonposSeq_measurableSet _) restrictNonposSeq_disjoint)
    refine' .of_nonneg_of_le (fun n => _) (fun n => _)
        (this.comp_injective Nat.succ_injective)
    · exact le_of_lt Nat.one_div_pos_of_nat
    · exact le_of_lt (restrictNonposSeq_lt n (hn' n))
  have h₃ : Tendsto (fun n => (bdd n : ℝ) + 1) atTop atTop := by
    simp only [one_div] at h₃'
    exact Summable.tendsto_atTop_of_pos h₃' fun n => Nat.cast_add_one_pos (bdd n)
  have h₄ : Tendsto (fun n => (bdd n : ℝ)) atTop atTop := by
    convert atTop.tendsto_atTop_add_const_right (-1) h₃; simp
  have A_meas : MeasurableSet A :=
    hi₁.diff (MeasurableSet.iUnion fun _ => restrictNonposSeq_measurableSet _)
  refine' ⟨A, A_meas, Set.diff_subset _ _, _, h₂.trans_lt hi⟩
  by_contra hnn
  rw [restrict_le_restrict_iff _ _ A_meas] at hnn; push_neg at hnn
  obtain ⟨E, hE₁, hE₂, hE₃⟩ := hnn
  have : ∃ k, 1 ≤ bdd k ∧ 1 / (bdd k : ℝ) < s E := by
    rw [tendsto_atTop_atTop] at h₄
    obtain ⟨k, hk⟩ := h₄ (max (1 / s E + 1) 1)
    refine' ⟨k, _, _⟩
    · have hle := le_of_max_le_right (hk k le_rfl)
      norm_cast at hle
    · have : 1 / s E < bdd k := by
        linarith only [le_of_max_le_left (hk k le_rfl)]
      rw [one_div] at this ⊢
      

Proof state:
case intro.refine'_2
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ l, ⋃ (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ l, ⋃ (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
A : Set α := i \ ⋃ l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l
hA : A = i \ ⋃ l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l
bdd : ℕ → ℕ :=
  fun n =>
    MeasureTheory.SignedMeasure.findExistsOneDivLT s
      (i \ ⋃ k, ⋃ (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
hn' :
  ∀ (n : ℕ),
    ¬restrict s (i \ ⋃ l, ⋃ (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
        restrict 0 (i \ ⋃ l, ⋃ (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₁ : ↑s i = ↑s A + ∑' (l : ℕ), ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₂ : ↑s A ≤ ↑s i
h₃' : Summable fun n => 1 / (↑(bdd n) + 1)
h₃ : Tendsto (fun n => ↑(bdd n) + 1) atTop atTop
h₄ : ∀ (b : ℝ), ∃ i, ∀ (a : ℕ), i ≤ a → b ≤ ↑(bdd a)
A_meas : MeasurableSet A
E : Set α
hE₁ : MeasurableSet E
hE₂ : E ⊆ A
hE₃ : ↑0 E < ↑s E
k : ℕ
hk : ∀ (a : ℕ), k ≤ a → max (1 / ↑s E + 1) 1 ≤ ↑(bdd a)
this : (↑s E)⁻¹ < ↑(bdd k)
⊢ (↑(bdd k))⁻¹ < ↑s E

Next tactic:
rwa [inv_lt (lt_trans (inv_pos.2 hE₃) this) hE₃]