File: Mathlib/MeasureTheory/Measure/Haar/InnerProductSpace.lean

full_name: Orientation.measure_orthonormalBasis
code: /-- 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_orthonormalBasis (o : Orientation ℝ F (Fin n))
    (b : OrthonormalBasis ι ℝ F) : o.volumeForm.measure (parallelepiped b) = 1 
goal: o.volumeForm.measure (parallelepiped ⇑b) = 1

full_name: Orientation.measure_eq_volume
code: /-- In an oriented inner product space, the measure coming from the canonical volume form
associated to an orientation coincides with the volume. -/
theorem Orientation.measure_eq_volume (o : Orientation ℝ F (Fin n)) :
    o.volumeForm.measure = volume 
goal: o.volumeForm.measure = MeasureTheory.volume

full_name: OrthonormalBasis.volume_parallelepiped
code: /-- The volume measure in a finite-dimensional inner product space gives measure `1` to the
parallelepiped spanned by any orthonormal basis. -/
theorem OrthonormalBasis.volume_parallelepiped (b : OrthonormalBasis ι ℝ F) :
    volume (parallelepiped b) = 1 
goal: MeasureTheory.volume (parallelepiped ⇑b) = 1

full_name: OrthonormalBasis.addHaar_eq_volume
code: /-- The Haar measure defined by any orthonormal basis of a finite-dimensional inner product space
is equal to its volume measure. -/
theorem OrthonormalBasis.addHaar_eq_volume {ι F : Type*} [Fintype ι] [NormedAddCommGroup F]
    [InnerProductSpace ℝ F] [FiniteDimensional ℝ F] [MeasurableSpace F] [BorelSpace F]
    (b : OrthonormalBasis ι ℝ F) :
    b.toBasis.addHaar = volume 
goal: b.toBasis.addHaar = MeasureTheory.volume

full_name: OrthonormalBasis.measurableEquiv
code: /-- 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. -/
noncomputable def OrthonormalBasis.measurableEquiv (b : OrthonormalBasis ι ℝ F) :
    F ≃ᵐ EuclideanSpace ℝ ι 
goal: F ≃ᵐ EuclideanSpace ℝ ι

full_name: OrthonormalBasis.measurePreserving_measurableEquiv
code: /-- The measurable equivalence defined by an orthonormal basis is volume preserving. -/
theorem OrthonormalBasis.measurePreserving_measurableEquiv (b : OrthonormalBasis ι ℝ F) :
    MeasurePreserving b.measurableEquiv volume volume 
goal: MeasureTheory.MeasurePreserving (⇑b.measurableEquiv) MeasureTheory.volume MeasureTheory.volume

full_name: OrthonormalBasis.measurePreserving_repr
code: theorem OrthonormalBasis.measurePreserving_repr (b : OrthonormalBasis ι ℝ F) :
    MeasurePreserving b.repr volume volume 
goal: MeasureTheory.MeasurePreserving (⇑b.repr) MeasureTheory.volume MeasureTheory.volume

full_name: OrthonormalBasis.measurePreserving_repr_symm
code: theorem OrthonormalBasis.measurePreserving_repr_symm (b : OrthonormalBasis ι ℝ F) :
    MeasurePreserving b.repr.symm volume volume 
goal: MeasureTheory.MeasurePreserving (⇑b.repr.symm) MeasureTheory.volume MeasureTheory.volume

full_name: EuclideanSpace.volume_preserving_measurableEquiv
code: /-- The measure equivalence between `EuclideanSpace ℝ ι` and `ι → ℝ` is volume preserving. -/
theorem EuclideanSpace.volume_preserving_measurableEquiv :
    MeasurePreserving (EuclideanSpace.measurableEquiv ι) 
goal: MeasureTheory.MeasurePreserving (⇑(EuclideanSpace.measurableEquiv ι)) MeasureTheory.volume MeasureTheory.volume

full_name: PiLp.volume_preserving_equiv
code: /-- A copy of `EuclideanSpace.volume_preserving_measurableEquiv` for the canonical spelling of the
equivalence. -/
theorem PiLp.volume_preserving_equiv : MeasurePreserving (WithLp.equiv 2 (ι → ℝ)) 
goal: MeasureTheory.MeasurePreserving (⇑(WithLp.equiv 2 (ι → ℝ))) MeasureTheory.volume MeasureTheory.volume

full_name: PiLp.volume_preserving_equiv_symm
code: /-- The reverse direction of `PiLp.volume_preserving_measurableEquiv`, since
`MeasurePreserving.symm` only works for `MeasurableEquiv`s. -/
theorem PiLp.volume_preserving_equiv_symm : MeasurePreserving (WithLp.equiv 2 (ι → ℝ)).symm 
goal: MeasureTheory.MeasurePreserving (⇑(WithLp.equiv 2 (ι → ℝ)).symm) MeasureTheory.volume MeasureTheory.volume

full_name: volume_euclideanSpace_eq_dirac
code: lemma volume_euclideanSpace_eq_dirac [IsEmpty ι] :
    (volume : Measure (EuclideanSpace ℝ ι)) = Measure.dirac 0 
goal: MeasureTheory.volume = MeasureTheory.Measure.dirac 0

full_name: LinearIsometryEquiv.measurePreserving
code: /-- Every linear isometry on a real finite dimensional Hilbert space is measure-preserving. -/
theorem measurePreserving : MeasurePreserving f 
goal: MeasureTheory.MeasurePreserving (⇑f) MeasureTheory.volume MeasureTheory.volume

full_name: LinearIsometryEquiv.toMeasureEquiv
code: /-- Every linear isometry equivalence is a measurable equivalence. -/
def toMeasureEquiv : E ≃ᵐ F 
goal: E ≃ᵐ F

full_name: LinearIsometryEquiv.coe_toMeasureEquiv
code: @[simp] theorem coe_toMeasureEquiv : (f.toMeasureEquiv : E → F) = f 
goal: ⇑f.toMeasureEquiv = ⇑f

full_name: LinearIsometryEquiv.toMeasureEquiv_symm
code: theorem toMeasureEquiv_symm : f.toMeasureEquiv.symm = f.symm.toMeasureEquiv 
goal: f.toMeasureEquiv.symm = f.symm.toMeasureEquiv