f_iUnion — Mathlib

English

If s i are Carathéodory-measurable for all i and the family is pairwise disjoint, then ⋃ i, s i is Carathéodory-measurable.

Русский

Если каждый s_i карatheodory-измерим и множества взаимно не перекрываются, то ⋃ i, s_i карatheodory-измеримо.

LaTeX

$$m.IsCaratheodory (⋃ i, s i)$$

Lean4

theorem f_iUnion {s : ℕ → Set α} (h : ∀ i, IsCaratheodory m (s i)) (hd : Pairwise (Disjoint on s)) :
    m (⋃ i, s i) = ∑' i, m (s i) := by
  refine le_antisymm (measure_iUnion_le s) ?_
  rw [ENNReal.tsum_eq_iSup_nat]
  refine iSup_le fun n => ?_
  have := @isCaratheodory_sum _ m _ h hd univ n
  simp only [inter_comm, inter_univ, univ_inter] at this; simp only [this]
  exact m.mono (iUnion₂_subset fun i _ => subset_iUnion _ i)

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