Theorems · Theorem · measure theory

MeasureTheory.average_union

∀ {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E]
  {μ : MeasureTheory.Measure α} {f : α → E} {s t : Set α},
  MeasureTheory.AEDisjoint μ s t →
    MeasureTheory.NullMeasurableSet t μ →
      μ s ≠ ⊤ →
        μ t ≠ ⊤ →
          MeasureTheory.IntegrableOn f s μ →
            MeasureTheory.IntegrableOn f t μ →
              ⨍ (x : α) in s ∪ t, f x ∂μ =
                (μ.real s / (μ.real s + μ.real t)) • ⨍ (x : α) in s, f x ∂μ +
                  (μ.real t / (μ.real s + μ.real t)) • ⨍ (x : α) in t, f x ∂μ

Defined in

Mathlib.MeasureTheory.Integral.Average

Cited by

2 results in Mathlib

Foundations

Depth 256 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpace