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Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace

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.

Every linear isometry equivalence is a measurable equivalence.

Equations
  • f.toMeasureEquiv = { toEquiv := f, measurable_toFun := , measurable_invFun := }
Instances For
    theorem Orientation.measure_orthonormalBasis {ι : Type u_1} {F : Type u_3} [NormedAddCommGroup F] [InnerProductSpace F] [MeasurableSpace F] [BorelSpace F] [Fintype ι] [FiniteDimensional F] {n : } [_i : Fact (Module.finrank F = n)] (o : Orientation F (Fin n)) (b : OrthonormalBasis ι F) :
    o.volumeForm.measure (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_3} [NormedAddCommGroup F] [InnerProductSpace F] [MeasurableSpace F] [BorelSpace F] [FiniteDimensional F] {n : } [_i : Fact (Module.finrank F = n)] (o : Orientation F (Fin n)) :
    o.volumeForm.measure = MeasureTheory.volume

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

    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_4} {F : Type u_5} [Fintype ι] [NormedAddCommGroup F] [InnerProductSpace F] [FiniteDimensional F] [MeasurableSpace F] [BorelSpace F] (b : OrthonormalBasis ι F) :
    b.toBasis.addHaar = MeasureTheory.volume

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

    An orthonormal basis of a finite-dimensional inner product space defines a measurable equivalence between the space and the Euclidean space of the same dimension.

    Equations
    • b.measurableEquiv = b.repr.toHomeomorph.toMeasurableEquiv
    Instances For
      theorem OrthonormalBasis.measurePreserving_measurableEquiv {ι : Type u_1} {F : Type u_3} [NormedAddCommGroup F] [InnerProductSpace F] [MeasurableSpace F] [BorelSpace F] [Fintype ι] [FiniteDimensional F] (b : OrthonormalBasis ι F) :
      MeasureTheory.MeasurePreserving (⇑b.measurableEquiv) MeasureTheory.volume MeasureTheory.volume

      The measurable equivalence defined by an orthonormal basis is volume preserving.

      theorem OrthonormalBasis.measurePreserving_repr {ι : Type u_1} {F : Type u_3} [NormedAddCommGroup F] [InnerProductSpace F] [MeasurableSpace F] [BorelSpace F] [Fintype ι] [FiniteDimensional F] (b : OrthonormalBasis ι F) :
      MeasureTheory.MeasurePreserving (⇑b.repr) MeasureTheory.volume MeasureTheory.volume
      theorem OrthonormalBasis.measurePreserving_repr_symm {ι : Type u_1} {F : Type u_3} [NormedAddCommGroup F] [InnerProductSpace F] [MeasurableSpace F] [BorelSpace F] [Fintype ι] [FiniteDimensional F] (b : OrthonormalBasis ι F) :
      MeasureTheory.MeasurePreserving (⇑b.repr.symm) MeasureTheory.volume MeasureTheory.volume
      theorem EuclideanSpace.volume_preserving_measurableEquiv (ι : Type u_4) [Fintype ι] :
      MeasureTheory.MeasurePreserving (⇑(EuclideanSpace.measurableEquiv ι)) MeasureTheory.volume MeasureTheory.volume

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

      theorem PiLp.volume_preserving_equiv (ι : Type u_4) [Fintype ι] :
      MeasureTheory.MeasurePreserving (⇑(WithLp.equiv 2 (ι))) MeasureTheory.volume MeasureTheory.volume

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

      theorem PiLp.volume_preserving_equiv_symm (ι : Type u_4) [Fintype ι] :
      MeasureTheory.MeasurePreserving (⇑(WithLp.equiv 2 (ι)).symm) MeasureTheory.volume MeasureTheory.volume

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

      theorem volume_euclideanSpace_eq_dirac (ι : Type u_4) [Fintype ι] [IsEmpty ι] :
      MeasureTheory.volume = MeasureTheory.Measure.dirac 0

      Every linear isometry on a real finite dimensional Hilbert space is measure-preserving.