restrict_iUnion_le — Mathlib

English

Let s_i be a family of sets and T a Finset. If the family is pairwise disjoint on T and each s_i is measurable, then μ.restrict (⋃ i ∈ T, s_i) = sum_{i ∈ T} μ.restrict (s_i).

Русский

Пусть s_i образуют дисjoint-структуру на Finset T и все s_i измеримы. Тогда ограничение объединения равно сумме ограничений по каждому.

LaTeX

$$$\\mu\\restrict (\\bigcup_{i\\in T} s_i) = \\sum_{i\\in T} (\\mu\\restrict (s_i))$.$$

Lean4

theorem restrict_iUnion_le [Countable ι] {s : ι → Set α} : μ.restrict (⋃ i, s i) ≤ sum fun i => μ.restrict (s i) :=
  le_iff.2 fun t ht ↦ by simpa [ht, inter_iUnion] using measure_iUnion_le (t ∩ s ·)

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