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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.

theorem Orientation.measure_orthonormalBasis {ι : Type u_1} {F : Type u_3} [Fintype ι] [NormedAddCommGroup F] [InnerProductSpace F] [FiniteDimensional F] [MeasurableSpace F] [BorelSpace F] {n : } [_i : Fact (FiniteDimensional.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] [FiniteDimensional F] [MeasurableSpace F] [BorelSpace F] {n : } [_i : Fact (FiniteDimensional.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} [Fintype ι] [NormedAddCommGroup F] [InnerProductSpace F] [FiniteDimensional F] [MeasurableSpace F] [BorelSpace 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} [Fintype ι] [NormedAddCommGroup F] [InnerProductSpace F] [FiniteDimensional F] [MeasurableSpace F] [BorelSpace 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} [Fintype ι] [NormedAddCommGroup F] [InnerProductSpace F] [FiniteDimensional F] [MeasurableSpace F] [BorelSpace 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.

    Every linear isometry equivalence is a measurable equivalence.

    Equations
    • f.toMeasureEquiv = { toEquiv := f, measurable_toFun := , measurable_invFun := }
    Instances For