lintegral_biUnion₀ — Mathlib

English

If t is countable and the sets s i are null-measurable with respect to μ, then the lintegral over the biUnion equals the sum over i of the integrals over s_i.

Русский

Если t счётно, а множества s_i нуль - measurables по μ, тогда линеграл по объединению по i в t равен сумме интегралов по s_i.

LaTeX

$$$$ \\int^- a in \\bigcup_{i \\in t} s_i, f(a) \\, d\\mu = \\sum_{i \\in t} \\int^- a in s_i, f(a) \\, d\\mu. $$$$

Lean4

theorem lintegral_biUnion₀ {t : Set β} {s : β → Set α} (ht : t.Countable) (hm : ∀ i ∈ t, NullMeasurableSet (s i) μ)
    (hd : t.Pairwise (AEDisjoint μ on s)) (f : α → ℝ≥0∞) :
    ∫⁻ a in ⋃ i ∈ t, s i, f a ∂μ = ∑' i : t, ∫⁻ a in s i, f a ∂μ :=
  by
  haveI := ht.toEncodable
  rw [biUnion_eq_iUnion, lintegral_iUnion₀ (SetCoe.forall'.1 hm) (hd.subtype _ _)]

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