Theorems · Theorem · measure theory

MeasureTheory.integral_fun_norm_addHaar

∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] [inst_2 : MeasurableSpace E] {F : Type u_2}
  [inst_3 : NormedAddCommGroup F] [inst_4 : NormedSpace ℝ F] [Nontrivial E] (μ : MeasureTheory.Measure E)
  [FiniteDimensional ℝ E] [BorelSpace E] [μ.IsAddHaarMeasure] (f : ℝ → F),
  ∫ (x : E), f ‖x‖ ∂μ =
    Module.finrank ℝ E • μ.real (Metric.ball 0 1) • ∫ (y : ℝ) in Set.Ioi 0, y ^ (Module.finrank ℝ E - 1) • f y

Defined in

Mathlib.MeasureTheory.Constructions.HaarToSphere

Cited by

1 results in Mathlib

Foundations

Depth 276 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpaceMeasurableSpaceNormedAddCommGroupNormedSpaceNontrivialFiniteDimensionalBorelSpaceMeasureTheory.Measure.IsAddHaarMeasure