Haar measure #
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In this file we prove the existence and uniqueness (up to scalar multiples) of Haar measure for a locally compact Hausdorff topological group.
For the construction, we follow the write-up by Jonathan Gleason, Existence and Uniqueness of Haar Measure. This is essentially the same argument as in https://en.wikipedia.org/wiki/Haar_measure#A_construction_using_compact_subsets.
We construct the Haar measure first on compact sets. For this we define (K : U) as the (smallest)
number of left-translates of U that are needed to cover K (index in the formalization).
Then we define a function h on compact sets as lim_U (K : U) / (K₀ : U),
where U becomes a smaller and smaller open neighborhood of 1, and K₀ is a fixed compact set
with nonempty interior. This function is chaar in the formalization, and we define the limit
formally using Tychonoff's theorem.
This function h forms a content, which we can extend to an outer measure and then a measure
(haar_measure).
We normalize the Haar measure so that the measure of K₀ is 1.
We show that for second countable spaces any left invariant Borel measure is a scalar multiple of
the Haar measure.
Note that μ need not coincide with h on compact sets, according to
[halmos1950measure, ch. X, §53 p.233]. However, we know that h(K) lies between μ(Kᵒ) and μ(K),
where ᵒ denotes the interior.
Main Declarations #
haar_measure: the Haar measure on a locally compact Hausdorff group. This is a left invariant regular measure. It takes as argument a compact set of the group (with non-empty interior), and is normalized so that the measure of the given set is 1.haar_measure_self: the Haar measure is normalized.is_left_invariant_haar_measure: the Haar measure is left invariant.regular_haar_measure: the Haar measure is a regular measure.is_haar_measure_haar_measure: the Haar measure satisfies theis_haar_measuretypeclass, i.e., it is invariant and gives finite mass to compact sets and positive mass to nonempty open sets.haar: some choice of a Haar measure, on a locally compact Hausdorff group, constructed ashaar_measure KwhereKis some arbitrary choice of a compact set with nonempty interior.haar_measure_unique: Every σ-finite left invariant measure on a locally compact Hausdorff group is a scalar multiple of the Haar measure.
References #
- Paul Halmos (1950), Measure Theory, §53
- Jonathan Gleason, Existence and Uniqueness of Haar Measure
- Note: step 9, page 8 contains a mistake: the last defined
μdoes not extend theμon compact sets, see Halmos (1950) p. 233, bottom of the page. This makes some other steps (like step 11) invalid.
- Note: step 9, page 8 contains a mistake: the last defined
- https://en.wikipedia.org/wiki/Haar_measure
We put the internal functions in the construction of the Haar measure in a namespace,
so that the chosen names don't clash with other declarations.
We first define a couple of the functions before proving the properties (that require that G
is a topological group).
The index or Haar covering number or ratio of K w.r.t. V, denoted (K : V):
it is the smallest number of (left) translates of V that is necessary to cover K.
It is defined to be 0 if no finite number of translates cover K.
Equations
- measure_theory.measure.haar.index K V = has_Inf.Inf (finset.card '' {t : finset G | K ⊆ ⋃ (g : G) (H : g ∈ t), (λ (h : G), g * h) ⁻¹' V})
noncomputable
def
measure_theory.measure.haar.add_index
{G : Type u_1}
[add_group G]
(K V : set G) :
additive version of measure_theory.measure.haar.index
Equations
- measure_theory.measure.haar.add_index K V = has_Inf.Inf (finset.card '' {t : finset G | K ⊆ ⋃ (g : G) (H : g ∈ t), (λ (h : G), g + h) ⁻¹' V})
noncomputable
def
measure_theory.measure.haar.add_prehaar
{G : Type u_1}
[add_group G]
[topological_space G]
(K₀ U : set G)
(K : topological_space.compacts G) :
additive version of measure_theory.measure.haar.prehaar
Equations
noncomputable
def
measure_theory.measure.haar.prehaar
{G : Type u_1}
[group G]
[topological_space G]
(K₀ U : set G)
(K : topological_space.compacts G) :
prehaar K₀ U K is a weighted version of the index, defined as (K : U)/(K₀ : U).
In the applications K₀ is compact with non-empty interior, U is open containing 1,
and K is any compact set.
The argument K is a (bundled) compact set, so that we can consider prehaar K₀ U as an
element of haar_product (below).
Equations
theorem
measure_theory.measure.haar.prehaar_empty
{G : Type u_1}
[group G]
[topological_space G]
(K₀ : topological_space.positive_compacts G)
{U : set G} :
theorem
measure_theory.measure.haar.add_prehaar_empty
{G : Type u_1}
[add_group G]
[topological_space G]
(K₀ : topological_space.positive_compacts G)
{U : set G} :
theorem
measure_theory.measure.haar.prehaar_nonneg
{G : Type u_1}
[group G]
[topological_space G]
(K₀ : topological_space.positive_compacts G)
{U : set G}
(K : topological_space.compacts G) :
0 ≤ measure_theory.measure.haar.prehaar ↑K₀ U K
theorem
measure_theory.measure.haar.add_prehaar_nonneg
{G : Type u_1}
[add_group G]
[topological_space G]
(K₀ : topological_space.positive_compacts G)
{U : set G}
(K : topological_space.compacts G) :
def
measure_theory.measure.haar.add_haar_product
{G : Type u_1}
[add_group G]
[topological_space G]
(K₀ : set G) :
additive version of measure_theory.measure.haar.haar_product
Equations
def
measure_theory.measure.haar.haar_product
{G : Type u_1}
[group G]
[topological_space G]
(K₀ : set G) :
haar_product K₀ is the product of intervals [0, (K : K₀)], for all compact sets K.
For all U, we can show that prehaar K₀ U ∈ haar_product K₀.
Equations
- measure_theory.measure.haar.haar_product K₀ = set.univ.pi (λ (K : topological_space.compacts G), set.Icc 0 ↑(measure_theory.measure.haar.index ↑K K₀))
@[simp]
theorem
measure_theory.measure.haar.mem_prehaar_empty
{G : Type u_1}
[group G]
[topological_space G]
{K₀ : set G}
{f : topological_space.compacts G → ℝ} :
f ∈ measure_theory.measure.haar.haar_product K₀ ↔ ∀ (K : topological_space.compacts G), f K ∈ set.Icc 0 ↑(measure_theory.measure.haar.index ↑K K₀)
@[simp]
theorem
measure_theory.measure.haar.mem_add_prehaar_empty
{G : Type u_1}
[add_group G]
[topological_space G]
{K₀ : set G}
{f : topological_space.compacts G → ℝ} :
f ∈ measure_theory.measure.haar.add_haar_product K₀ ↔ ∀ (K : topological_space.compacts G), f K ∈ set.Icc 0 ↑(measure_theory.measure.haar.add_index ↑K K₀)
def
measure_theory.measure.haar.cl_add_prehaar
{G : Type u_1}
[add_group G]
[topological_space G]
(K₀ : set G)
(V : topological_space.open_nhds_of 0) :
additive version of measure_theory.measure.haar.cl_prehaar
def
measure_theory.measure.haar.cl_prehaar
{G : Type u_1}
[group G]
[topological_space G]
(K₀ : set G)
(V : topological_space.open_nhds_of 1) :
The closure of the collection of elements of the form prehaar K₀ U,
for U open neighbourhoods of 1, contained in V. The closure is taken in the space
compacts G → ℝ, with the topology of pointwise convergence.
We show that the intersection of all these sets is nonempty, and the Haar measure
on compact sets is defined to be an element in the closure of this intersection.
Lemmas about index #
theorem
measure_theory.measure.haar.index_defined
{G : Type u_1}
[group G]
[topological_space G]
[topological_group G]
{K V : set G}
(hK : is_compact K)
(hV : (interior V).nonempty) :
If K is compact and V has nonempty interior, then the index (K : V) is well-defined,
there is a finite set t satisfying the desired properties.
theorem
measure_theory.measure.haar.add_index_defined
{G : Type u_1}
[add_group G]
[topological_space G]
[topological_add_group G]
{K V : set G}
(hK : is_compact K)
(hV : (interior V).nonempty) :
If K is compact and V has nonempty interior, then the index
(K : V) is well-defined, there is a finite set t satisfying the desired properties.
theorem
measure_theory.measure.haar.add_index_elim
{G : Type u_1}
[add_group G]
[topological_space G]
[topological_add_group G]
{K V : set G}
(hK : is_compact K)
(hV : (interior V).nonempty) :
theorem
measure_theory.measure.haar.index_elim
{G : Type u_1}
[group G]
[topological_space G]
[topological_group G]
{K V : set G}
(hK : is_compact K)
(hV : (interior V).nonempty) :
theorem
measure_theory.measure.haar.le_add_index_mul
{G : Type u_1}
[add_group G]
[topological_space G]
[topological_add_group G]
(K₀ : topological_space.positive_compacts G)
(K : topological_space.compacts G)
{V : set G}
(hV : (interior V).nonempty) :
theorem
measure_theory.measure.haar.le_index_mul
{G : Type u_1}
[group G]
[topological_space G]
[topological_group G]
(K₀ : topological_space.positive_compacts G)
(K : topological_space.compacts G)
{V : set G}
(hV : (interior V).nonempty) :
theorem
measure_theory.measure.haar.add_index_pos
{G : Type u_1}
[add_group G]
[topological_space G]
[topological_add_group G]
(K : topological_space.positive_compacts G)
{V : set G}
(hV : (interior V).nonempty) :
theorem
measure_theory.measure.haar.index_pos
{G : Type u_1}
[group G]
[topological_space G]
[topological_group G]
(K : topological_space.positive_compacts G)
{V : set G}
(hV : (interior V).nonempty) :
theorem
measure_theory.measure.haar.add_index_mono
{G : Type u_1}
[add_group G]
[topological_space G]
[topological_add_group G]
{K K' V : set G}
(hK' : is_compact K')
(h : K ⊆ K')
(hV : (interior V).nonempty) :
theorem
measure_theory.measure.haar.index_mono
{G : Type u_1}
[group G]
[topological_space G]
[topological_group G]
{K K' V : set G}
(hK' : is_compact K')
(h : K ⊆ K')
(hV : (interior V).nonempty) :
theorem
measure_theory.measure.haar.add_index_union_le
{G : Type u_1}
[add_group G]
[topological_space G]
[topological_add_group G]
(K₁ K₂ : topological_space.compacts G)
{V : set G}
(hV : (interior V).nonempty) :
theorem
measure_theory.measure.haar.index_union_le
{G : Type u_1}
[group G]
[topological_space G]
[topological_group G]
(K₁ K₂ : topological_space.compacts G)
{V : set G}
(hV : (interior V).nonempty) :
theorem
measure_theory.measure.haar.add_index_union_eq
{G : Type u_1}
[add_group G]
[topological_space G]
[topological_add_group G]
(K₁ K₂ : topological_space.compacts G)
{V : set G}
(hV : (interior V).nonempty)
(h : disjoint (K₁.carrier + -V) (K₂.carrier + -V)) :
theorem
measure_theory.measure.haar.index_union_eq
{G : Type u_1}
[group G]
[topological_space G]
[topological_group G]
(K₁ K₂ : topological_space.compacts G)
{V : set G}
(hV : (interior V).nonempty)
(h : disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)) :
theorem
measure_theory.measure.haar.mul_left_index_le
{G : Type u_1}
[group G]
[topological_space G]
[topological_group G]
{K : set G}
(hK : is_compact K)
{V : set G}
(hV : (interior V).nonempty)
(g : G) :
measure_theory.measure.haar.index ((λ (h : G), g * h) '' K) V ≤ measure_theory.measure.haar.index K V
theorem
measure_theory.measure.haar.add_left_add_index_le
{G : Type u_1}
[add_group G]
[topological_space G]
[topological_add_group G]
{K : set G}
(hK : is_compact K)
{V : set G}
(hV : (interior V).nonempty)
(g : G) :
measure_theory.measure.haar.add_index ((λ (h : G), g + h) '' K) V ≤ measure_theory.measure.haar.add_index K V
theorem
measure_theory.measure.haar.is_left_invariant_add_index
{G : Type u_1}
[add_group G]
[topological_space G]
[topological_add_group G]
{K : set G}
(hK : is_compact K)
(g : G)
{V : set G}
(hV : (interior V).nonempty) :
measure_theory.measure.haar.add_index ((λ (h : G), g + h) '' K) V = measure_theory.measure.haar.add_index K V
theorem
measure_theory.measure.haar.is_left_invariant_index
{G : Type u_1}
[group G]
[topological_space G]
[topological_group G]
{K : set G}
(hK : is_compact K)
(g : G)
{V : set G}
(hV : (interior V).nonempty) :
measure_theory.measure.haar.index ((λ (h : G), g * h) '' K) V = measure_theory.measure.haar.index K V
Lemmas about prehaar #
theorem
measure_theory.measure.haar.prehaar_le_index
{G : Type u_1}
[group G]
[topological_space G]
[topological_group G]
(K₀ : topological_space.positive_compacts G)
{U : set G}
(K : topological_space.compacts G)
(hU : (interior U).nonempty) :
theorem
measure_theory.measure.haar.add_prehaar_le_add_index
{G : Type u_1}
[add_group G]
[topological_space G]
[topological_add_group G]
(K₀ : topological_space.positive_compacts G)
{U : set G}
(K : topological_space.compacts G)
(hU : (interior U).nonempty) :
theorem
measure_theory.measure.haar.add_prehaar_pos
{G : Type u_1}
[add_group G]
[topological_space G]
[topological_add_group G]
(K₀ : topological_space.positive_compacts G)
{U : set G}
(hU : (interior U).nonempty)
{K : set G}
(h1K : is_compact K)
(h2K : (interior K).nonempty) :
0 < measure_theory.measure.haar.add_prehaar ↑K₀ U {carrier := K, is_compact' := h1K}
theorem
measure_theory.measure.haar.prehaar_pos
{G : Type u_1}
[group G]
[topological_space G]
[topological_group G]
(K₀ : topological_space.positive_compacts G)
{U : set G}
(hU : (interior U).nonempty)
{K : set G}
(h1K : is_compact K)
(h2K : (interior K).nonempty) :
0 < measure_theory.measure.haar.prehaar ↑K₀ U {carrier := K, is_compact' := h1K}
theorem
measure_theory.measure.haar.add_prehaar_mono
{G : Type u_1}
[add_group G]
[topological_space G]
[topological_add_group G]
{K₀ : topological_space.positive_compacts G}
{U : set G}
(hU : (interior U).nonempty)
{K₁ K₂ : topological_space.compacts G}
(h : ↑K₁ ⊆ K₂.carrier) :
theorem
measure_theory.measure.haar.prehaar_mono
{G : Type u_1}
[group G]
[topological_space G]
[topological_group G]
{K₀ : topological_space.positive_compacts G}
{U : set G}
(hU : (interior U).nonempty)
{K₁ K₂ : topological_space.compacts G}
(h : ↑K₁ ⊆ K₂.carrier) :
theorem
measure_theory.measure.haar.prehaar_self
{G : Type u_1}
[group G]
[topological_space G]
[topological_group G]
{K₀ : topological_space.positive_compacts G}
{U : set G}
(hU : (interior U).nonempty) :
measure_theory.measure.haar.prehaar ↑K₀ U K₀.to_compacts = 1
theorem
measure_theory.measure.haar.add_prehaar_self
{G : Type u_1}
[add_group G]
[topological_space G]
[topological_add_group G]
{K₀ : topological_space.positive_compacts G}
{U : set G}
(hU : (interior U).nonempty) :
theorem
measure_theory.measure.haar.add_prehaar_sup_le
{G : Type u_1}
[add_group G]
[topological_space G]
[topological_add_group G]
{K₀ : topological_space.positive_compacts G}
{U : set G}
(K₁ K₂ : topological_space.compacts G)
(hU : (interior U).nonempty) :
measure_theory.measure.haar.add_prehaar ↑K₀ U (K₁ ⊔ K₂) ≤ measure_theory.measure.haar.add_prehaar ↑K₀ U K₁ + measure_theory.measure.haar.add_prehaar ↑K₀ U K₂
theorem
measure_theory.measure.haar.prehaar_sup_le
{G : Type u_1}
[group G]
[topological_space G]
[topological_group G]
{K₀ : topological_space.positive_compacts G}
{U : set G}
(K₁ K₂ : topological_space.compacts G)
(hU : (interior U).nonempty) :
measure_theory.measure.haar.prehaar ↑K₀ U (K₁ ⊔ K₂) ≤ measure_theory.measure.haar.prehaar ↑K₀ U K₁ + measure_theory.measure.haar.prehaar ↑K₀ U K₂
theorem
measure_theory.measure.haar.prehaar_sup_eq
{G : Type u_1}
[group G]
[topological_space G]
[topological_group G]
{K₀ : topological_space.positive_compacts G}
{U : set G}
{K₁ K₂ : topological_space.compacts G}
(hU : (interior U).nonempty)
(h : disjoint (K₁.carrier * U⁻¹) (K₂.carrier * U⁻¹)) :
measure_theory.measure.haar.prehaar ↑K₀ U (K₁ ⊔ K₂) = measure_theory.measure.haar.prehaar ↑K₀ U K₁ + measure_theory.measure.haar.prehaar ↑K₀ U K₂
theorem
measure_theory.measure.haar.add_prehaar_sup_eq
{G : Type u_1}
[add_group G]
[topological_space G]
[topological_add_group G]
{K₀ : topological_space.positive_compacts G}
{U : set G}
{K₁ K₂ : topological_space.compacts G}
(hU : (interior U).nonempty)
(h : disjoint (K₁.carrier + -U) (K₂.carrier + -U)) :
measure_theory.measure.haar.add_prehaar ↑K₀ U (K₁ ⊔ K₂) = measure_theory.measure.haar.add_prehaar ↑K₀ U K₁ + measure_theory.measure.haar.add_prehaar ↑K₀ U K₂
theorem
measure_theory.measure.haar.is_left_invariant_prehaar
{G : Type u_1}
[group G]
[topological_space G]
[topological_group G]
{K₀ : topological_space.positive_compacts G}
{U : set G}
(hU : (interior U).nonempty)
(g : G)
(K : topological_space.compacts G) :
measure_theory.measure.haar.prehaar ↑K₀ U (topological_space.compacts.map (λ (b : G), g * b) _ K) = measure_theory.measure.haar.prehaar ↑K₀ U K
theorem
measure_theory.measure.haar.is_left_invariant_add_prehaar
{G : Type u_1}
[add_group G]
[topological_space G]
[topological_add_group G]
{K₀ : topological_space.positive_compacts G}
{U : set G}
(hU : (interior U).nonempty)
(g : G)
(K : topological_space.compacts G) :
measure_theory.measure.haar.add_prehaar ↑K₀ U (topological_space.compacts.map (λ (b : G), g + b) _ K) = measure_theory.measure.haar.add_prehaar ↑K₀ U K
Lemmas about haar_product #
theorem
measure_theory.measure.haar.prehaar_mem_haar_product
{G : Type u_1}
[group G]
[topological_space G]
[topological_group G]
(K₀ : topological_space.positive_compacts G)
{U : set G}
(hU : (interior U).nonempty) :
theorem
measure_theory.measure.haar.add_prehaar_mem_add_haar_product
{G : Type u_1}
[add_group G]
[topological_space G]
[topological_add_group G]
(K₀ : topological_space.positive_compacts G)
{U : set G}
(hU : (interior U).nonempty) :
theorem
measure_theory.measure.haar.nonempty_Inter_cl_add_prehaar
{G : Type u_1}
[add_group G]
[topological_space G]
[topological_add_group G]
(K₀ : topological_space.positive_compacts G) :
theorem
measure_theory.measure.haar.nonempty_Inter_cl_prehaar
{G : Type u_1}
[group G]
[topological_space G]
[topological_group G]
(K₀ : topological_space.positive_compacts G) :
Lemmas about chaar #
noncomputable
def
measure_theory.measure.haar.chaar
{G : Type u_1}
[group G]
[topological_space G]
[topological_group G]
(K₀ : topological_space.positive_compacts G)
(K : topological_space.compacts G) :
This is the "limit" of prehaar K₀ U K as U becomes a smaller and smaller open
neighborhood of (1 : G). More precisely, it is defined to be an arbitrary element
in the intersection of all the sets cl_prehaar K₀ V in haar_product K₀.
This is roughly equal to the Haar measure on compact sets,
but it can differ slightly. We do know that
haar_measure K₀ (interior K) ≤ chaar K₀ K ≤ haar_measure K₀ K.
Equations
noncomputable
def
measure_theory.measure.haar.add_chaar
{G : Type u_1}
[add_group G]
[topological_space G]
[topological_add_group G]
(K₀ : topological_space.positive_compacts G)
(K : topological_space.compacts G) :
additive version of measure_theory.measure.haar.chaar
Equations
theorem
measure_theory.measure.haar.chaar_mem_haar_product
{G : Type u_1}
[group G]
[topological_space G]
[topological_group G]
(K₀ : topological_space.positive_compacts G) :
theorem
measure_theory.measure.haar.chaar_mem_cl_prehaar
{G : Type u_1}
[group G]
[topological_space G]
[topological_group G]
(K₀ : topological_space.positive_compacts G)
(V : topological_space.open_nhds_of 1) :
theorem
measure_theory.measure.haar.add_chaar_mem_cl_add_prehaar
{G : Type u_1}
[add_group G]
[topological_space G]
[topological_add_group G]
(K₀ : topological_space.positive_compacts G)
(V : topological_space.open_nhds_of 0) :
theorem
measure_theory.measure.haar.add_chaar_nonneg
{G : Type u_1}
[add_group G]
[topological_space G]
[topological_add_group G]
(K₀ : topological_space.positive_compacts G)
(K : topological_space.compacts G) :
theorem
measure_theory.measure.haar.chaar_nonneg
{G : Type u_1}
[group G]
[topological_space G]
[topological_group G]
(K₀ : topological_space.positive_compacts G)
(K : topological_space.compacts G) :
theorem
measure_theory.measure.haar.chaar_empty
{G : Type u_1}
[group G]
[topological_space G]
[topological_group G]
(K₀ : topological_space.positive_compacts G) :
theorem
measure_theory.measure.haar.add_chaar_empty
{G : Type u_1}
[add_group G]
[topological_space G]
[topological_add_group G]
(K₀ : topological_space.positive_compacts G) :
theorem
measure_theory.measure.haar.add_chaar_self
{G : Type u_1}
[add_group G]
[topological_space G]
[topological_add_group G]
(K₀ : topological_space.positive_compacts G) :
theorem
measure_theory.measure.haar.chaar_self
{G : Type u_1}
[group G]
[topological_space G]
[topological_group G]
(K₀ : topological_space.positive_compacts G) :
theorem
measure_theory.measure.haar.chaar_mono
{G : Type u_1}
[group G]
[topological_space G]
[topological_group G]
{K₀ : topological_space.positive_compacts G}
{K₁ K₂ : topological_space.compacts G}
(h : ↑K₁ ⊆ ↑K₂) :
theorem
measure_theory.measure.haar.add_chaar_mono
{G : Type u_1}
[add_group G]
[topological_space G]
[topological_add_group G]
{K₀ : topological_space.positive_compacts G}
{K₁ K₂ : topological_space.compacts G}
(h : ↑K₁ ⊆ ↑K₂) :
theorem
measure_theory.measure.haar.add_chaar_sup_le
{G : Type u_1}
[add_group G]
[topological_space G]
[topological_add_group G]
{K₀ : topological_space.positive_compacts G}
(K₁ K₂ : topological_space.compacts G) :
theorem
measure_theory.measure.haar.chaar_sup_le
{G : Type u_1}
[group G]
[topological_space G]
[topological_group G]
{K₀ : topological_space.positive_compacts G}
(K₁ K₂ : topological_space.compacts G) :
measure_theory.measure.haar.chaar K₀ (K₁ ⊔ K₂) ≤ measure_theory.measure.haar.chaar K₀ K₁ + measure_theory.measure.haar.chaar K₀ K₂
theorem
measure_theory.measure.haar.chaar_sup_eq
{G : Type u_1}
[group G]
[topological_space G]
[topological_group G]
[t2_space G]
{K₀ : topological_space.positive_compacts G}
{K₁ K₂ : topological_space.compacts G}
(h : disjoint K₁.carrier K₂.carrier) :
measure_theory.measure.haar.chaar K₀ (K₁ ⊔ K₂) = measure_theory.measure.haar.chaar K₀ K₁ + measure_theory.measure.haar.chaar K₀ K₂
theorem
measure_theory.measure.haar.add_chaar_sup_eq
{G : Type u_1}
[add_group G]
[topological_space G]
[topological_add_group G]
[t2_space G]
{K₀ : topological_space.positive_compacts G}
{K₁ K₂ : topological_space.compacts G}
(h : disjoint K₁.carrier K₂.carrier) :
theorem
measure_theory.measure.haar.is_left_invariant_add_chaar
{G : Type u_1}
[add_group G]
[topological_space G]
[topological_add_group G]
{K₀ : topological_space.positive_compacts G}
(g : G)
(K : topological_space.compacts G) :
measure_theory.measure.haar.add_chaar K₀ (topological_space.compacts.map (λ (b : G), g + b) _ K) = measure_theory.measure.haar.add_chaar K₀ K
theorem
measure_theory.measure.haar.is_left_invariant_chaar
{G : Type u_1}
[group G]
[topological_space G]
[topological_group G]
{K₀ : topological_space.positive_compacts G}
(g : G)
(K : topological_space.compacts G) :
measure_theory.measure.haar.chaar K₀ (topological_space.compacts.map (λ (b : G), g * b) _ K) = measure_theory.measure.haar.chaar K₀ K
noncomputable
def
measure_theory.measure.haar.haar_content
{G : Type u_1}
[group G]
[topological_space G]
[topological_group G]
[t2_space G]
(K₀ : topological_space.positive_compacts G) :
The function chaar interpreted in ℝ≥0, as a content
Equations
- measure_theory.measure.haar.haar_content K₀ = {to_fun := λ (K : topological_space.compacts G), ⟨measure_theory.measure.haar.chaar K₀ K, _⟩, mono' := _, sup_disjoint' := _, sup_le' := _}
noncomputable
def
measure_theory.measure.haar.add_haar_content
{G : Type u_1}
[add_group G]
[topological_space G]
[topological_add_group G]
[t2_space G]
(K₀ : topological_space.positive_compacts G) :
additive version of measure_theory.measure.haar.haar_content
Equations
- measure_theory.measure.haar.add_haar_content K₀ = {to_fun := λ (K : topological_space.compacts G), ⟨measure_theory.measure.haar.add_chaar K₀ K, _⟩, mono' := _, sup_disjoint' := _, sup_le' := _}
We only prove the properties for haar_content that we use at least twice below.
theorem
measure_theory.measure.haar.haar_content_apply
{G : Type u_1}
[group G]
[topological_space G]
[topological_group G]
[t2_space G]
(K₀ : topological_space.positive_compacts G)
(K : topological_space.compacts G) :
⇑(measure_theory.measure.haar.haar_content K₀) K = ↑((λ (this : nnreal), this) ⟨measure_theory.measure.haar.chaar K₀ K, _⟩)
theorem
measure_theory.measure.haar.add_haar_content_apply
{G : Type u_1}
[add_group G]
[topological_space G]
[topological_add_group G]
[t2_space G]
(K₀ : topological_space.positive_compacts G)
(K : topological_space.compacts G) :
⇑(measure_theory.measure.haar.add_haar_content K₀) K = ↑((λ (this : nnreal), this) ⟨measure_theory.measure.haar.add_chaar K₀ K, _⟩)
theorem
measure_theory.measure.haar.add_haar_content_self
{G : Type u_1}
[add_group G]
[topological_space G]
[topological_add_group G]
[t2_space G]
{K₀ : topological_space.positive_compacts G} :
The variant of add_chaar_self for add_haar_content.
theorem
measure_theory.measure.haar.haar_content_self
{G : Type u_1}
[group G]
[topological_space G]
[topological_group G]
[t2_space G]
{K₀ : topological_space.positive_compacts G} :
The variant of chaar_self for haar_content
theorem
measure_theory.measure.haar.is_left_invariant_add_haar_content
{G : Type u_1}
[add_group G]
[topological_space G]
[topological_add_group G]
[t2_space G]
{K₀ : topological_space.positive_compacts G}
(g : G)
(K : topological_space.compacts G) :
⇑(measure_theory.measure.haar.add_haar_content K₀) (topological_space.compacts.map (λ (b : G), g + b) _ K) = ⇑(measure_theory.measure.haar.add_haar_content K₀) K
The variant of is_left_invariant_add_chaar for add_haar_content
theorem
measure_theory.measure.haar.is_left_invariant_haar_content
{G : Type u_1}
[group G]
[topological_space G]
[topological_group G]
[t2_space G]
{K₀ : topological_space.positive_compacts G}
(g : G)
(K : topological_space.compacts G) :
⇑(measure_theory.measure.haar.haar_content K₀) (topological_space.compacts.map (λ (b : G), g * b) _ K) = ⇑(measure_theory.measure.haar.haar_content K₀) K
The variant of is_left_invariant_chaar for haar_content
theorem
measure_theory.measure.haar.add_haar_content_outer_measure_self_pos
{G : Type u_1}
[add_group G]
[topological_space G]
[topological_add_group G]
[t2_space G]
{K₀ : topological_space.positive_compacts G} :
theorem
measure_theory.measure.haar.haar_content_outer_measure_self_pos
{G : Type u_1}
[group G]
[topological_space G]
[topological_group G]
[t2_space G]
{K₀ : topological_space.positive_compacts G} :
0 < ⇑((measure_theory.measure.haar.haar_content K₀).outer_measure) ↑K₀
The Haar measure #
noncomputable
def
measure_theory.measure.haar_measure
{G : Type u_1}
[group G]
[topological_space G]
[t2_space G]
[topological_group G]
[measurable_space G]
[borel_space G]
(K₀ : topological_space.positive_compacts G) :
The Haar measure on the locally compact group G, scaled so that haar_measure K₀ K₀ = 1.
Equations
Instances for measure_theory.measure.haar_measure
noncomputable
def
measure_theory.measure.add_haar_measure
{G : Type u_1}
[add_group G]
[topological_space G]
[t2_space G]
[topological_add_group G]
[measurable_space G]
[borel_space G]
(K₀ : topological_space.positive_compacts G) :
The Haar measure on the locally compact additive group G,
scaled so that add_haar_measure K₀ K₀ = 1.
Equations
Instances for measure_theory.measure.add_haar_measure
theorem
measure_theory.measure.add_haar_measure_apply
{G : Type u_1}
[add_group G]
[topological_space G]
[t2_space G]
[topological_add_group G]
[measurable_space G]
[borel_space G]
{K₀ : topological_space.positive_compacts G}
{s : set G}
(hs : measurable_set s) :
theorem
measure_theory.measure.haar_measure_apply
{G : Type u_1}
[group G]
[topological_space G]
[t2_space G]
[topological_group G]
[measurable_space G]
[borel_space G]
{K₀ : topological_space.positive_compacts G}
{s : set G}
(hs : measurable_set s) :
@[protected, instance]
def
measure_theory.measure.is_add_left_invariant_add_haar_measure
{G : Type u_1}
[add_group G]
[topological_space G]
[t2_space G]
[topological_add_group G]
[measurable_space G]
[borel_space G]
(K₀ : topological_space.positive_compacts G) :
@[protected, instance]
def
measure_theory.measure.is_mul_left_invariant_haar_measure
{G : Type u_1}
[group G]
[topological_space G]
[t2_space G]
[topological_group G]
[measurable_space G]
[borel_space G]
(K₀ : topological_space.positive_compacts G) :
theorem
measure_theory.measure.haar_measure_self
{G : Type u_1}
[group G]
[topological_space G]
[t2_space G]
[topological_group G]
[measurable_space G]
[borel_space G]
{K₀ : topological_space.positive_compacts G} :
⇑(measure_theory.measure.haar_measure K₀) ↑K₀ = 1
theorem
measure_theory.measure.add_haar_measure_self
{G : Type u_1}
[add_group G]
[topological_space G]
[t2_space G]
[topological_add_group G]
[measurable_space G]
[borel_space G]
{K₀ : topological_space.positive_compacts G} :
⇑(measure_theory.measure.add_haar_measure K₀) ↑K₀ = 1
@[protected, instance]
def
measure_theory.measure.regular_haar_measure
{G : Type u_1}
[group G]
[topological_space G]
[t2_space G]
[topological_group G]
[measurable_space G]
[borel_space G]
{K₀ : topological_space.positive_compacts G} :
The Haar measure is regular.
@[protected, instance]
def
measure_theory.measure.regular_add_haar_measure
{G : Type u_1}
[add_group G]
[topological_space G]
[t2_space G]
[topological_add_group G]
[measurable_space G]
[borel_space G]
{K₀ : topological_space.positive_compacts G} :
The additive Haar measure is regular.
@[protected, instance]
def
measure_theory.measure.sigma_finite_haar_measure
{G : Type u_1}
[group G]
[topological_space G]
[t2_space G]
[topological_group G]
[measurable_space G]
[borel_space G]
[topological_space.second_countable_topology G]
{K₀ : topological_space.positive_compacts G} :
The Haar measure is sigma-finite in a second countable group.
@[protected, instance]
def
measure_theory.measure.sigma_finite_add_haar_measure
{G : Type u_1}
[add_group G]
[topological_space G]
[t2_space G]
[topological_add_group G]
[measurable_space G]
[borel_space G]
[topological_space.second_countable_topology G]
{K₀ : topological_space.positive_compacts G} :
The additive Haar measure is sigma-finite in a second countable group.
@[protected, instance]
def
measure_theory.measure.is_add_haar_measure_add_haar_measure
{G : Type u_1}
[add_group G]
[topological_space G]
[t2_space G]
[topological_add_group G]
[measurable_space G]
[borel_space G]
(K₀ : topological_space.positive_compacts G) :
The additive Haar measure is an additive Haar measure, i.e., it is invariant and gives finite mass to compact sets and positive mass to nonempty open sets.
@[protected, instance]
def
measure_theory.measure.is_haar_measure_haar_measure
{G : Type u_1}
[group G]
[topological_space G]
[t2_space G]
[topological_group G]
[measurable_space G]
[borel_space G]
(K₀ : topological_space.positive_compacts G) :
The Haar measure is a Haar measure, i.e., it is invariant and gives finite mass to compact sets and positive mass to nonempty open sets.
@[reducible]
noncomputable
def
measure_theory.measure.add_haar
{G : Type u_1}
[add_group G]
[topological_space G]
[t2_space G]
[topological_add_group G]
[measurable_space G]
[borel_space G]
[locally_compact_space G] :
add_haar is some choice of a Haar measure, on a locally compact
additive group.
@[reducible]
noncomputable
def
measure_theory.measure.haar
{G : Type u_1}
[group G]
[topological_space G]
[t2_space G]
[topological_group G]
[measurable_space G]
[borel_space G]
[locally_compact_space G] :
haar is some choice of a Haar measure, on a locally compact group.
theorem
measure_theory.measure.add_haar_measure_unique
{G : Type u_1}
[add_group G]
[topological_space G]
[t2_space G]
[topological_add_group G]
[measurable_space G]
[borel_space G]
[topological_space.second_countable_topology G]
(μ : measure_theory.measure G)
[measure_theory.sigma_finite μ]
[μ.is_add_left_invariant]
(K₀ : topological_space.positive_compacts G) :
μ = ⇑μ ↑K₀ • measure_theory.measure.add_haar_measure K₀
The additive Haar measure is unique up to scaling. More precisely: every σ-finite
left invariant measure is a scalar multiple of the additive Haar measure. This is slightly weaker
than assuming that μ is an additive Haar measure (in particular we don't require μ ≠ 0).
theorem
measure_theory.measure.haar_measure_unique
{G : Type u_1}
[group G]
[topological_space G]
[t2_space G]
[topological_group G]
[measurable_space G]
[borel_space G]
[topological_space.second_countable_topology G]
(μ : measure_theory.measure G)
[measure_theory.sigma_finite μ]
[μ.is_mul_left_invariant]
(K₀ : topological_space.positive_compacts G) :
μ = ⇑μ ↑K₀ • measure_theory.measure.haar_measure K₀
The Haar measure is unique up to scaling. More precisely: every σ-finite left invariant measure
is a scalar multiple of the Haar measure.
This is slightly weaker than assuming that μ is a Haar measure (in particular we don't require
μ ≠ 0).
theorem
measure_theory.measure.regular_of_is_add_left_invariant
{G : Type u_1}
[add_group G]
[topological_space G]
[t2_space G]
[topological_add_group G]
[measurable_space G]
[borel_space G]
[topological_space.second_countable_topology G]
{μ : measure_theory.measure G}
[measure_theory.sigma_finite μ]
[μ.is_add_left_invariant]
{K : set G}
(hK : is_compact K)
(h2K : (interior K).nonempty)
(hμK : ⇑μ K ≠ ⊤) :
μ.regular
To show that an invariant σ-finite measure is regular it is sufficient to show that it is finite on some compact set with non-empty interior.
theorem
measure_theory.measure.regular_of_is_mul_left_invariant
{G : Type u_1}
[group G]
[topological_space G]
[t2_space G]
[topological_group G]
[measurable_space G]
[borel_space G]
[topological_space.second_countable_topology G]
{μ : measure_theory.measure G}
[measure_theory.sigma_finite μ]
[μ.is_mul_left_invariant]
{K : set G}
(hK : is_compact K)
(h2K : (interior K).nonempty)
(hμK : ⇑μ K ≠ ⊤) :
μ.regular
To show that an invariant σ-finite measure is regular it is sufficient to show that it is finite on some compact set with non-empty interior.
theorem
measure_theory.measure.is_add_haar_measure_eq_smul_is_add_haar_measure
{G : Type u_1}
[add_group G]
[topological_space G]
[t2_space G]
[topological_add_group G]
[measurable_space G]
[borel_space G]
[topological_space.second_countable_topology G]
[locally_compact_space G]
(μ ν : measure_theory.measure G)
[μ.is_add_haar_measure]
[ν.is_add_haar_measure] :
theorem
measure_theory.measure.is_haar_measure_eq_smul_is_haar_measure
{G : Type u_1}
[group G]
[topological_space G]
[t2_space G]
[topological_group G]
[measurable_space G]
[borel_space G]
[topological_space.second_countable_topology G]
[locally_compact_space G]
(μ ν : measure_theory.measure G)
[μ.is_haar_measure]
[ν.is_haar_measure] :
@[protected, instance]
def
measure_theory.measure.regular_of_is_haar_measure
{G : Type u_1}
[group G]
[topological_space G]
[t2_space G]
[topological_group G]
[measurable_space G]
[borel_space G]
[topological_space.second_countable_topology G]
[locally_compact_space G]
(μ : measure_theory.measure G)
[μ.is_haar_measure] :
μ.regular
@[protected, instance]
def
measure_theory.measure.regular_of_is_add_haar_measure
{G : Type u_1}
[add_group G]
[topological_space G]
[t2_space G]
[topological_add_group G]
[measurable_space G]
[borel_space G]
[topological_space.second_countable_topology G]
[locally_compact_space G]
(μ : measure_theory.measure G)
[μ.is_add_haar_measure] :
μ.regular
theorem
measure_theory.measure.div_mem_nhds_one_of_haar_pos
{G : Type u_1}
[group G]
[topological_space G]
[t2_space G]
[topological_group G]
[measurable_space G]
[borel_space G]
[topological_space.second_countable_topology G]
(μ : measure_theory.measure G)
[μ.is_haar_measure]
[locally_compact_space G]
(E : set G)
(hE : measurable_set E)
(hEpos : 0 < ⇑μ E) :
Steinhaus Theorem In any locally compact group G with a haar measure μ, for any
measurable set E of positive measure, the set E / E is a neighbourhood of 1.
theorem
measure_theory.measure.sub_mem_nhds_zero_of_add_haar_pos
{G : Type u_1}
[add_group G]
[topological_space G]
[t2_space G]
[topological_add_group G]
[measurable_space G]
[borel_space G]
[topological_space.second_countable_topology G]
(μ : measure_theory.measure G)
[μ.is_add_haar_measure]
[locally_compact_space G]
(E : set G)
(hE : measurable_set E)
(hEpos : 0 < ⇑μ E) :
Steinhaus Theorem In any locally compact group G with a haar measure μ,
for any measurable set E of positive measure, the set E - E is a neighbourhood of 0.
@[protected, instance]
def
measure_theory.measure.is_haar_measure.is_inv_invariant
{G : Type u_1}
[comm_group G]
[topological_space G]
[topological_group G]
[t2_space G]
[measurable_space G]
[borel_space G]
[topological_space.second_countable_topology G]
(μ : measure_theory.measure G)
[μ.is_haar_measure]
[locally_compact_space G] :
Any Haar measure is invariant under inversion in an abelian group.
@[protected, instance]
def
measure_theory.measure.is_add_haar_measure.is_neg_invariant
{G : Type u_1}
[add_comm_group G]
[topological_space G]
[topological_add_group G]
[t2_space G]
[measurable_space G]
[borel_space G]
[topological_space.second_countable_topology G]
(μ : measure_theory.measure G)
[μ.is_add_haar_measure]
[locally_compact_space G] :
Any additive Haar measure is invariant under negation in an abelian group.
theorem
measure_theory.measure.measure_preserving_zpow
{G : Type u_1}
[comm_group G]
[topological_space G]
[topological_group G]
[t2_space G]
[measurable_space G]
[borel_space G]
[topological_space.second_countable_topology G]
(μ : measure_theory.measure G)
[μ.is_haar_measure]
[compact_space G]
[rootable_by G ℤ]
{n : ℤ}
(hn : n ≠ 0) :
measure_theory.measure_preserving (λ (g : G), g ^ n) μ μ
theorem
measure_theory.measure.measure_preserving_zsmul
{G : Type u_1}
[add_comm_group G]
[topological_space G]
[topological_add_group G]
[t2_space G]
[measurable_space G]
[borel_space G]
[topological_space.second_countable_topology G]
(μ : measure_theory.measure G)
[μ.is_add_haar_measure]
[compact_space G]
[divisible_by G ℤ]
{n : ℤ}
(hn : n ≠ 0) :
measure_theory.measure_preserving (λ (g : G), n • g) μ μ
theorem
measure_theory.measure.measure_preserving.zsmul
{G : Type u_1}
[add_comm_group G]
[topological_space G]
[topological_add_group G]
[t2_space G]
[measurable_space G]
[borel_space G]
[topological_space.second_countable_topology G]
(μ : measure_theory.measure G)
[μ.is_add_haar_measure]
[compact_space G]
[divisible_by G ℤ]
{n : ℤ}
(hn : n ≠ 0)
{X : Type u_2}
[measurable_space X]
{μ' : measure_theory.measure X}
{f : X → G}
(hf : measure_theory.measure_preserving f μ' μ) :
measure_theory.measure_preserving (λ (x : X), n • f x) μ' μ
theorem
measure_theory.measure.measure_preserving.zpow
{G : Type u_1}
[comm_group G]
[topological_space G]
[topological_group G]
[t2_space G]
[measurable_space G]
[borel_space G]
[topological_space.second_countable_topology G]
(μ : measure_theory.measure G)
[μ.is_haar_measure]
[compact_space G]
[rootable_by G ℤ]
{n : ℤ}
(hn : n ≠ 0)
{X : Type u_2}
[measurable_space X]
{μ' : measure_theory.measure X}
{f : X → G}
(hf : measure_theory.measure_preserving f μ' μ) :
measure_theory.measure_preserving (λ (x : X), f x ^ n) μ' μ