Theorems · Theorem · measure theory

MeasureTheory.Measure.integral_comp_smul_of_nonneg

∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] [inst_2 : MeasurableSpace E] [BorelSpace E]
  [FiniteDimensional ℝ E] (μ : MeasureTheory.Measure E) [μ.IsAddHaarMeasure] {F : Type u_2}
  [inst_6 : NormedAddCommGroup F] [inst_7 : NormedSpace ℝ F] (f : E → F) (R : ℝ) {hR : 0 ≤ R},
  ∫ (x : E), f (R • x) ∂μ = (R ^ Module.finrank ℝ E)⁻¹ • ∫ (x : E), f x ∂μ

The integral of f (R • x) with respect to an additive Haar measure is a multiple of the integral of f. The formula we give works even when f is not integrable or R = 0 thanks to the convention that a non-integrable function has integral zero.

Defined in

Mathlib.MeasureTheory.Measure.Haar.NormedSpace

Cited by

0 results in Mathlib

Foundations

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

Assumes

NormedAddCommGroupNormedSpaceMeasurableSpaceBorelSpaceFiniteDimensionalMeasureTheory.Measure.IsAddHaarMeasureNormedAddCommGroupNormedSpace