measure_biUnion_eq_iSup — Mathlib

English

Continuity from above: for a countable directed downward family of sets, μ(⋂ s_i) = ⨅ μ(s_i).

Русский

Непрерывность сверху: для счётной направленной вниз семьи множеств μ(⋂ s_i) = ⨅ μ(s_i).

LaTeX

$$$ μ(\bigcap_i s_i) = \big\langle i \rangle_i μ(s_i) $$$

Lean4

theorem measure_biUnion_eq_iSup {s : ι → Set α} {t : Set ι} (ht : t.Countable) (hd : DirectedOn ((· ⊆ ·) on s) t) :
    μ (⋃ i ∈ t, s i) = ⨆ i ∈ t, μ (s i) := by
  haveI := ht.to_subtype
  rw [biUnion_eq_iUnion, hd.directed_val.measure_iUnion, ← iSup_subtype'']

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