of_union — Mathlib

English

If A and B are disjoint measurable sets, then the measure of their union equals the sum of their measures.

Русский

Если A и B — дисjoint измеримые множества, то мера их объединения равна сумме мер.

LaTeX

$$v(A ∪ B) = v(A) + v(B)$$

Lean4

theorem of_union {A B : Set α} (h : Disjoint A B) (hA : MeasurableSet A) (hB : MeasurableSet B) :
    v (A ∪ B) = v A + v B :=
  by
  rw [Set.union_eq_iUnion, of_disjoint_iUnion, tsum_fintype, Fintype.sum_bool, cond, cond]
  exacts [fun b => Bool.casesOn b hB hA, pairwise_disjoint_on_bool.2 h]

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