Lean4
/-- Consider a real set `s`. If a property is true almost everywhere in `s ∩ (a, b)` for
all `a, b ∈ s`, then it is true almost everywhere in `s`. Formulated with `μ.restrict`.
See also `ae_of_mem_of_ae_of_mem_inter_Ioo`. -/
theorem ae_restrict_of_ae_restrict_inter_Ioo {μ : Measure ℝ} [NoAtoms μ] {s : Set ℝ} {p : ℝ → Prop}
(h : ∀ a b, a ∈ s → b ∈ s → a < b → ∀ᵐ x ∂μ.restrict (s ∩ Ioo a b), p x) : ∀ᵐ x ∂μ.restrict s, p x := by
/- By second-countability, we cover `s` by countably many intervals `(a, b)` (except maybe for
two endpoints, which don't matter since `μ` does not have any atom). -/
let T : s × s → Set ℝ := fun p => Ioo p.1 p.2
let u := ⋃ i : ↥s × ↥s, T i
have hfinite : (s \ u).Finite := s.finite_diff_iUnion_Ioo'
obtain ⟨A, A_count, hA⟩ : ∃ A : Set (↥s × ↥s), A.Countable ∧ ⋃ i ∈ A, T i = ⋃ i : ↥s × ↥s, T i :=
isOpen_iUnion_countable _ fun p => isOpen_Ioo
have : s ⊆ s \ u ∪ ⋃ p ∈ A, s ∩ T p := by
intro x hx
by_cases h'x : x ∈ ⋃ i : ↥s × ↥s, T i
· rw [← hA] at h'x
obtain ⟨p, pA, xp⟩ : ∃ p : ↥s × ↥s, p ∈ A ∧ x ∈ T p := by
simpa only [mem_iUnion, exists_prop, SetCoe.exists, exists_and_right] using h'x
right
exact mem_biUnion pA ⟨hx, xp⟩
· exact Or.inl ⟨hx, h'x⟩
apply ae_restrict_of_ae_restrict_of_subset this
rw [ae_restrict_union_iff, ae_restrict_biUnion_iff _ A_count]
constructor
· have : μ.restrict (s \ u) = 0 := by simp only [restrict_eq_zero, hfinite.measure_zero]
simp only [this, ae_zero, eventually_bot]
· rintro ⟨⟨a, as⟩, ⟨b, bs⟩⟩ -
dsimp [T]
rcases le_or_gt b a with (hba | hab)
· simp only [Ioo_eq_empty_of_le hba, inter_empty, restrict_empty, ae_zero, eventually_bot]
· exact h a b as bs hab
