exists_disjoint_diff — Mathlib

English

If s and t are μ-almost everywhere disjoint, there exists a measurable null set u such that (s \\ u) is disjoint from t.

Русский

Если s и t являются μ-почти что дисjoint, существует измеримое нуль-множество u такое, что (s \ u) дисjoint с t.

LaTeX

$$$\\exists u\\, \\big( \\text{MeasurableSet } u \\land μ(u)=0 \\land Disjoint (s \\ u) t \\big)$$$

Lean4

/-- If `s` and `t` are `μ`-a.e. disjoint, then `s \ u` and `t` are disjoint for some measurable null
set `u`. -/
theorem exists_disjoint_diff (h : AEDisjoint μ s t) : ∃ u, MeasurableSet u ∧ μ u = 0 ∧ Disjoint (s \ u) t :=
  ⟨toMeasurable μ (s ∩ t), measurableSet_toMeasurable _ _, (measure_toMeasurable _).trans h,
    disjoint_sdiff_self_left.mono_left (b := s \ t) fun x hx => by
      simpa using ⟨hx.1, fun hxt => hx.2 <| subset_toMeasurable _ _ ⟨hx.1, hxt⟩⟩⟩

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