restrict_iUnion_apply_eq_iSup — Mathlib

English

If a countable index set ι and s_i are pairwise disjoint, then the restriction to the union equals the iSup of the restrictions.

Русский

Если множество индексов счётно и s_i попарно дисjoint, ограничение на объединение равно iSup ограничений.

LaTeX

$$$[Countable\, ι] \ (Pairwise\, Disjoint\, on\, s) \Rightarrow \mu.restrict (\bigcup i, s i) = \bigl(\bigsqcup_i \mu.restrict (s i)\bigr)$$$

Lean4

theorem restrict_iUnion_apply_eq_iSup [Countable ι] {s : ι → Set α} (hd : Directed (· ⊆ ·) s) {t : Set α}
    (ht : MeasurableSet t) : μ.restrict (⋃ i, s i) t = ⨆ i, μ.restrict (s i) t :=
  by
  simp only [restrict_apply ht, inter_iUnion]
  rw [Directed.measure_iUnion]
  exacts [hd.mono_comp _ fun s₁ s₂ => inter_subset_inter_right _]

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