Theorems · Definition · measure theory

MeasureTheory.average

{α : Type u_1} →
  {E : Type u_2} →
    {m0 : MeasurableSpace α} → [inst : NormedAddCommGroup E] → [NormedSpace ℝ E] → MeasureTheory.Measure α → (α → E) → E

Average value of a function f w.r.t. a measure μ, denoted ⨍ x, f x ∂μ. It is equal to (μ.real univ)⁻¹ • ∫ x, f x ∂μ, so it takes value zero if f is not integrable or if μ is an infinite measure. If μ is a probability measure, then the average of any function is equal to its integral. For the average on a set, use ⨍ x in s, f x ∂μ, defined as ⨍ x, f x ∂(μ.restrict s). For the average w.r.t. the volume, one can omit ∂volume.

Defined in

Mathlib.MeasureTheory.Integral.Average

Cited by

87 results in Mathlib

Foundations

Depth 250 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpace