Theorems · Theorem · measure theory

MeasureTheory.Measure.addHaar_preimage_linearEquiv

∀ {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 : E ≃ₗ[ℝ] E) (s : Set E),
  μ (⇑f ⁻¹' s) = ENNReal.ofReal |LinearMap.det ↑f.symm| * μ s

The preimage of a set s under a linear equiv f has measure equal to μ s times the absolute value of the inverse of the determinant of f.

Defined in

Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar

Cited by

3 results in Mathlib

Foundations

Depth 256 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpaceMeasurableSpaceBorelSpaceFiniteDimensionalMeasureTheory.Measure.IsAddHaarMeasure