laverage_union — Mathlib

English

If s is measurable and f=g a.e. on s, then laverage over μ restricted to s is the same for f and g.

Русский

Если s измеримо и f=g почти пожалуй на s, то laverage по μ|s одинакова для f и g.

LaTeX

$$$hs: MeasurableSet s\\Rightarrow (f=g\\text{ a.e. on } s) \\Rightarrow laverage_{μ|s}(f) = laverage_{μ|s}(g)$$$

Lean4

theorem laverage_union (hd : AEDisjoint μ s t) (ht : NullMeasurableSet t μ) :
    ⨍⁻ x in s ∪ t, f x ∂μ = μ s / (μ s + μ t) * ⨍⁻ x in s, f x ∂μ + μ t / (μ s + μ t) * ⨍⁻ x in t, f x ∂μ := by
  rw [restrict_union₀ hd ht, laverage_add_measure, restrict_apply_univ, restrict_apply_univ]

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