Volume forms and measures on inner product spaces

A volume form induces a Lebesgue measure on general finite-dimensional real vector spaces. In this file, we discuss the specific situation of inner product spaces, where an orientation gives rise to a canonical volume form. We show that the measure coming from this volume form gives measure 1 to the parallelepiped spanned by any orthonormal basis, and that it coincides with the canonical volume from the MeasureSpace instance.

theorem Orientation.measure_orthonormalBasis {ι : Type u_1} {F : Type u_2} [Fintype ι] [NormedAddCommGroup F] [InnerProductSpace ℝ F] [FiniteDimensional ℝ F] [MeasurableSpace F] [BorelSpace F] {n : ℕ} [_i : Fact (FiniteDimensional.finrank ℝ F = n)] (o : Orientation ℝ F (Fin n)) (b : OrthonormalBasis ι ℝ F) : ↑↑(AlternatingMap.measure (Orientation.volumeForm o)) (parallelepiped ↑b) = 1

The volume form coming from an orientation in an inner product space gives measure 1 to the parallelepiped associated to any orthonormal basis. This is a rephrasing of abs_volumeForm_apply_of_orthonormal in terms of measures.

theorem Orientation.measure_eq_volume {F : Type u_2} [NormedAddCommGroup F] [InnerProductSpace ℝ F] [FiniteDimensional ℝ F] [MeasurableSpace F] [BorelSpace F] {n : ℕ} [_i : Fact (FiniteDimensional.finrank ℝ F = n)] (o : Orientation ℝ F (Fin n)) : AlternatingMap.measure (Orientation.volumeForm o) = MeasureTheory.volume

In an oriented inner product space, the measure coming from the canonical volume form associated to an orientation coincides with the volume.

theorem OrthonormalBasis.volume_parallelepiped {ι : Type u_1} {F : Type u_2} [Fintype ι] [NormedAddCommGroup F] [InnerProductSpace ℝ F] [FiniteDimensional ℝ F] [MeasurableSpace F] [BorelSpace F] (b : OrthonormalBasis ι ℝ F) : ↑↑MeasureTheory.volume (parallelepiped ↑b) = 1

The volume measure in a finite-dimensional inner product space gives measure 1 to the parallelepiped spanned by any orthonormal basis.

theorem OrthonormalBasis.addHaar_eq_volume {ι : Type u_3} {F : Type u_4} [Fintype ι] [NormedAddCommGroup F] [InnerProductSpace ℝ F] [FiniteDimensional ℝ F] [MeasurableSpace F] [BorelSpace F] (b : OrthonormalBasis ι ℝ F) : Basis.addHaar (OrthonormalBasis.toBasis b) = MeasureTheory.volume

The Haar measure defined by any orthonormal basis of a finite-dimensional inner product space is equal to its volume measure.

theorem EuclideanSpace.volume_preserving_measurableEquiv (ι : Type u_3) [Fintype ι] : MeasureTheory.MeasurePreserving ↑(EuclideanSpace.measurableEquiv ι)

The measure equivalence between EuclideanSpace ℝ ι and ι → ℝ is volume preserving.

theorem PiLp.volume_preserving_equiv (ι : Type u_3) [Fintype ι] : MeasureTheory.MeasurePreserving ↑(WithLp.equiv 2 (ι → ℝ))

A copy of EuclideanSpace.volume_preserving_measurableEquiv for the canonical spelling of the equivalence.

theorem PiLp.volume_preserving_equiv_symm (ι : Type u_3) [Fintype ι] : MeasureTheory.MeasurePreserving ↑(WithLp.equiv 2 (ι → ℝ)).symm

The reverse direction of PiLp.volume_preserving_measurableEquiv, since MeasurePreserving.symm only works for MeasurableEquivs.