Theorems · Theorem · measure theory

MeasureTheory.average_mem_openSegment_compl_self

∀ {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] {f : α → E} {s : Set α},
  MeasureTheory.NullMeasurableSet s μ →
    μ s ≠ 0 →
      μ sᶜ ≠ 0 →
        MeasureTheory.Integrable f μ →
          ⨍ (x : α), f x ∂μ ∈ openSegment ℝ (⨍ (x : α) in s, f x ∂μ) (⨍ (x : α) in sᶜ, f x ∂μ)

Defined in

Mathlib.MeasureTheory.Integral.Average

Cited by

4 results in Mathlib

Foundations

Depth 258 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpaceMeasureTheory.IsFiniteMeasure