mathlib3 documentation

measure_theory.measure.haar.of_basis

Additive Haar measure constructed from a basis #

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Given a basis of a finite-dimensional real vector space, we define the corresponding Lebesgue measure, which gives measure 1 to the parallelepiped spanned by the basis.

Main definitions #

In particular, we declare a measure_space instance on any finite-dimensional inner product space, by using the Lebesgue measure associated to some orthonormal basis (which is in fact independent of the basis).

def parallelepiped {ι : Type u_1} {E : Type u_3} [fintype ι] [add_comm_group E] [module E] (v : ι E) :
set E

The closed parallelepiped spanned by a finite family of vectors.

Equations
theorem mem_parallelepiped_iff {ι : Type u_1} {E : Type u_3} [fintype ι] [add_comm_group E] [module E] (v : ι E) (x : E) :
x parallelepiped v (t : ι ) (ht : t set.Icc 0 1), x = finset.univ.sum (λ (i : ι), t i v i)
theorem image_parallelepiped {ι : Type u_1} {E : Type u_3} {F : Type u_4} [fintype ι] [add_comm_group E] [module E] [add_comm_group F] [module F] (f : E →ₗ[] F) (v : ι E) :
@[simp]
theorem parallelepiped_comp_equiv {ι : Type u_1} {ι' : Type u_2} {E : Type u_3} [fintype ι] [fintype ι'] [add_comm_group E] [module E] (v : ι E) (e : ι' ι) :

Reindexing a family of vectors does not change their parallelepiped.

theorem parallelepiped_eq_sum_segment {ι : Type u_1} {E : Type u_3} [fintype ι] [add_comm_group E] [module E] (v : ι E) :
parallelepiped v = finset.univ.sum (λ (i : ι), segment 0 (v i))
theorem convex_parallelepiped {ι : Type u_1} {E : Type u_3} [fintype ι] [add_comm_group E] [module E] (v : ι E) :
theorem parallelepiped_eq_convex_hull {ι : Type u_1} {E : Type u_3} [fintype ι] [add_comm_group E] [module E] (v : ι E) :
parallelepiped v = (convex_hull ) (finset.univ.sum (λ (i : ι), {0, v i}))

A parallelepiped is the convex hull of its vertices

theorem parallelepiped_single {ι : Type u_1} [fintype ι] [decidable_eq ι] (a : ι ) :
parallelepiped (λ (i : ι), pi.single i (a i)) = set.uIcc 0 a

The axis aligned parallelepiped over ι → ℝ is a cuboid.

The parallelepiped spanned by a basis, as a compact set with nonempty interior.

Equations
@[simp]
@[simp]
theorem basis.parallelepiped_reindex {ι : Type u_1} {ι' : Type u_2} {E : Type u_3} [fintype ι] [fintype ι'] [normed_add_comm_group E] [normed_space E] (b : basis ι E) (e : ι ι') :
@[irreducible]
noncomputable def basis.add_haar {ι : Type u_1} {E : Type u_3} [fintype ι] [normed_add_comm_group E] [normed_space E] [measurable_space E] [borel_space E] (b : basis ι E) :

The Lebesgue measure associated to a basis, giving measure 1 to the parallelepiped spanned by the basis.

Equations
Instances for basis.add_haar
@[protected, instance]

A finite dimensional inner product space has a canonical measure, the Lebesgue measure giving volume 1 to the parallelepiped spanned by any orthonormal basis. We define the measure using some arbitrary choice of orthonormal basis. The fact that it works with any orthonormal basis is proved in orthonormal_basis.volume_parallelepiped.

Equations

Miscellaneous instances for euclidean_space #

In combination with measure_space_of_inner_product_space, these put a measure_space structure on euclidean_space.

@[protected, instance]