sum_measure_le_measure_univ — Mathlib

English

For any family s, the union over i of toMeasurable μ(s_i) has the same measure as the union over i of s_i.

Русский

Для любой семейства множеств μ(⋃i toMeasurable μ(s_i)) равно μ(⋃i s_i).

LaTeX

$$$ μ(⋃ i, toMeasurable μ (s i)) = μ(⋃ i, s i) $$$

Lean4

theorem sum_measure_le_measure_univ {s : Finset ι} {t : ι → Set α} (h : ∀ i ∈ s, NullMeasurableSet (t i) μ)
    (H : Set.Pairwise s (AEDisjoint μ on t)) : (∑ i ∈ s, μ (t i)) ≤ μ (univ : Set α) :=
  by
  rw [← measure_biUnion_finset₀ H h]
  exact measure_mono (subset_univ _)

Похожие утверждения