Theorems · Theorem · measure theory

MeasureTheory.average_add_measure

∀ {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E]
  {μ : MeasureTheory.Measure α} [MeasureTheory.IsFiniteMeasure μ] {ν : MeasureTheory.Measure α}
  [MeasureTheory.IsFiniteMeasure ν] {f : α → E},
  MeasureTheory.Integrable f μ →
    MeasureTheory.Integrable f ν →
      ⨍ (x : α), f x ∂(μ + ν) =
        (μ.real Set.univ / (μ.real Set.univ + ν.real Set.univ)) • ⨍ (x : α), f x ∂μ +
          (ν.real Set.univ / (μ.real Set.univ + ν.real Set.univ)) • ⨍ (x : α), f x ∂ν

Defined in

Mathlib.MeasureTheory.Integral.Average

Cited by

1 results in Mathlib

Foundations

Depth 255 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpaceMeasureTheory.IsFiniteMeasureMeasureTheory.IsFiniteMeasure