restrict_biUnion_le — Mathlib

English

For a countable index family, ae of the restriction to the iUnion equals the sum of ae's of the restrictions: ae(μ.restrict (⋃ i, s_i)) = ⨆ i ae(μ.restrict (s_i)).

Русский

Для счётной цепи индексов ae-ограничения по iUnion равны sup-объединениям ae-ограничений по s_i.

LaTeX

$$$\\operatorname{ae}(\\mu\\restrict (\\bigcup_i s_i)) = \\bigvee_i \\operatorname{ae}(\\mu\\restrict (s_i)).$$$

Lean4

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

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