of_nonneg_disjoint_union_eq_zero — Mathlib

English

If s is a signed measure and A, B are disjoint measurable, with s(A) ≥ 0, s(B) ≥ 0 and s(A ∪ B) = 0, then s(A) = 0.

Русский

Если s — знак-мера и A, B — дисjoint измеримые, при s(A) ≥ 0, s(B) ≥ 0 и s(A ∪ B) = 0, тогда s(A) = 0.

LaTeX

$$s A = 0$$

Lean4

theorem of_nonneg_disjoint_union_eq_zero {s : SignedMeasure α} {A B : Set α} (h : Disjoint A B) (hA₁ : MeasurableSet A)
    (hB₁ : MeasurableSet B) (hA₂ : 0 ≤ s A) (hB₂ : 0 ≤ s B) (hAB : s (A ∪ B) = 0) : s A = 0 :=
  by
  rw [of_union h hA₁ hB₁] at hAB
  linarith

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