Measures on Groups

We develop some properties of measures on (topological) groups

• We define properties on measures: measures that are left or right invariant w.r.t. multiplication.
• We define the measure μ.inv : A ↦ μ(A⁻¹) and show that it is right invariant iff μ is left invariant.
• We define a class IsHaarMeasure μ, requiring that the measure μ is left-invariant, finite on compact sets, and positive on open sets.

We also give analogues of all these notions in the additive world.

theorem MeasureTheory.map_mul_left_eq_self {G : Type u_1} [MeasurableSpace G] [Mul G] (μ : MeasureTheory.Measure G) [μ.IsMulLeftInvariant] (g : G) : MeasureTheory.Measure.map (fun (x : G) => g * x) μ = μ

theorem MeasureTheory.map_add_left_eq_self {G : Type u_1} [MeasurableSpace G] [Add G] (μ : MeasureTheory.Measure G) [μ.IsAddLeftInvariant] (g : G) : MeasureTheory.Measure.map (fun (x : G) => g + x) μ = μ

theorem MeasureTheory.map_mul_right_eq_self {G : Type u_1} [MeasurableSpace G] [Mul G] (μ : MeasureTheory.Measure G) [μ.IsMulRightInvariant] (g : G) : MeasureTheory.Measure.map (fun (x : G) => x * g) μ = μ

theorem MeasureTheory.map_add_right_eq_self {G : Type u_1} [MeasurableSpace G] [Add G] (μ : MeasureTheory.Measure G) [μ.IsAddRightInvariant] (g : G) : MeasureTheory.Measure.map (fun (x : G) => x + g) μ = μ

instance MeasureTheory.isMulLeftInvariant_smul {G : Type u_1} [MeasurableSpace G] [Mul G] {μ : MeasureTheory.Measure G} [μ.IsMulLeftInvariant] (c : ENNReal) : (c • μ).IsMulLeftInvariant

Equations

• ⋯ = ⋯

instance MeasureTheory.isAddLeftInvariant_smul {G : Type u_1} [MeasurableSpace G] [Add G] {μ : MeasureTheory.Measure G} [μ.IsAddLeftInvariant] (c : ENNReal) : (c • μ).IsAddLeftInvariant

Equations

• ⋯ = ⋯

instance MeasureTheory.isMulRightInvariant_smul {G : Type u_1} [MeasurableSpace G] [Mul G] {μ : MeasureTheory.Measure G} [μ.IsMulRightInvariant] (c : ENNReal) : (c • μ).IsMulRightInvariant

Equations

• ⋯ = ⋯

instance MeasureTheory.isAddRightInvariant_smul {G : Type u_1} [MeasurableSpace G] [Add G] {μ : MeasureTheory.Measure G} [μ.IsAddRightInvariant] (c : ENNReal) : (c • μ).IsAddRightInvariant

Equations

• ⋯ = ⋯

instance MeasureTheory.isMulLeftInvariant_smul_nnreal {G : Type u_1} [MeasurableSpace G] [Mul G] {μ : MeasureTheory.Measure G} [μ.IsMulLeftInvariant] (c : NNReal) : (c • μ).IsMulLeftInvariant

Equations

• ⋯ = ⋯

instance MeasureTheory.isAddLeftInvariant_smul_nnreal {G : Type u_1} [MeasurableSpace G] [Add G] {μ : MeasureTheory.Measure G} [μ.IsAddLeftInvariant] (c : NNReal) : (c • μ).IsAddLeftInvariant

Equations

• ⋯ = ⋯

instance MeasureTheory.isMulRightInvariant_smul_nnreal {G : Type u_1} [MeasurableSpace G] [Mul G] {μ : MeasureTheory.Measure G} [μ.IsMulRightInvariant] (c : NNReal) : (c • μ).IsMulRightInvariant

Equations

• ⋯ = ⋯

instance MeasureTheory.isAddRightInvariant_smul_nnreal {G : Type u_1} [MeasurableSpace G] [Add G] {μ : MeasureTheory.Measure G} [μ.IsAddRightInvariant] (c : NNReal) : (c • μ).IsAddRightInvariant

Equations

• ⋯ = ⋯

theorem MeasureTheory.measurePreserving_mul_left {G : Type u_1} [MeasurableSpace G] [Mul G] [MeasurableMul G] (μ : MeasureTheory.Measure G) [μ.IsMulLeftInvariant] (g : G) : MeasureTheory.MeasurePreserving (fun (x : G) => g * x) μ μ

theorem MeasureTheory.measurePreserving_add_left {G : Type u_1} [MeasurableSpace G] [Add G] [MeasurableAdd G] (μ : MeasureTheory.Measure G) [μ.IsAddLeftInvariant] (g : G) : MeasureTheory.MeasurePreserving (fun (x : G) => g + x) μ μ

theorem MeasureTheory.MeasurePreserving.mul_left {G : Type u_1} [MeasurableSpace G] [Mul G] [MeasurableMul G] (μ : MeasureTheory.Measure G) [μ.IsMulLeftInvariant] (g : G) {X : Type u_3} [MeasurableSpace X] {μ' : MeasureTheory.Measure X} {f : X → G} (hf : MeasureTheory.MeasurePreserving f μ' μ) : MeasureTheory.MeasurePreserving (fun (x : X) => g * f x) μ' μ

theorem MeasureTheory.MeasurePreserving.add_left {G : Type u_1} [MeasurableSpace G] [Add G] [MeasurableAdd G] (μ : MeasureTheory.Measure G) [μ.IsAddLeftInvariant] (g : G) {X : Type u_3} [MeasurableSpace X] {μ' : MeasureTheory.Measure X} {f : X → G} (hf : MeasureTheory.MeasurePreserving f μ' μ) : MeasureTheory.MeasurePreserving (fun (x : X) => g + f x) μ' μ

theorem MeasureTheory.measurePreserving_mul_right {G : Type u_1} [MeasurableSpace G] [Mul G] [MeasurableMul G] (μ : MeasureTheory.Measure G) [μ.IsMulRightInvariant] (g : G) : MeasureTheory.MeasurePreserving (fun (x : G) => x * g) μ μ

theorem MeasureTheory.measurePreserving_add_right {G : Type u_1} [MeasurableSpace G] [Add G] [MeasurableAdd G] (μ : MeasureTheory.Measure G) [μ.IsAddRightInvariant] (g : G) : MeasureTheory.MeasurePreserving (fun (x : G) => x + g) μ μ

theorem MeasureTheory.MeasurePreserving.mul_right {G : Type u_1} [MeasurableSpace G] [Mul G] [MeasurableMul G] (μ : MeasureTheory.Measure G) [μ.IsMulRightInvariant] (g : G) {X : Type u_3} [MeasurableSpace X] {μ' : MeasureTheory.Measure X} {f : X → G} (hf : MeasureTheory.MeasurePreserving f μ' μ) : MeasureTheory.MeasurePreserving (fun (x : X) => f x * g) μ' μ

theorem MeasureTheory.MeasurePreserving.add_right {G : Type u_1} [MeasurableSpace G] [Add G] [MeasurableAdd G] (μ : MeasureTheory.Measure G) [μ.IsAddRightInvariant] (g : G) {X : Type u_3} [MeasurableSpace X] {μ' : MeasureTheory.Measure X} {f : X → G} (hf : MeasureTheory.MeasurePreserving f μ' μ) : MeasureTheory.MeasurePreserving (fun (x : X) => f x + g) μ' μ

instance MeasureTheory.Subgroup.smulInvariantMeasure {G : Type u_3} {α : Type u_4} [Group G] [MulAction G α] [MeasurableSpace α] {μ : MeasureTheory.Measure α} [MeasureTheory.SMulInvariantMeasure G α μ] (H : Subgroup G) : MeasureTheory.SMulInvariantMeasure (↥H) α μ

Equations

• ⋯ = ⋯

instance MeasureTheory.Subgroup.vaddInvariantMeasure {G : Type u_3} {α : Type u_4} [AddGroup G] [AddAction G α] [MeasurableSpace α] {μ : MeasureTheory.Measure α} [MeasureTheory.VAddInvariantMeasure G α μ] (H : AddSubgroup G) : MeasureTheory.VAddInvariantMeasure (↥H) α μ

Equations

• ⋯ = ⋯

theorem MeasureTheory.forall_measure_preimage_mul_iff {G : Type u_1} [MeasurableSpace G] [Mul G] [MeasurableMul G] (μ : MeasureTheory.Measure G) : (∀ (g : G) (A : Set G), MeasurableSet A → μ ((fun (h : G) => g * h) ⁻¹' A) = μ A) ↔ μ.IsMulLeftInvariant

An alternative way to prove that μ is left invariant under multiplication.

theorem MeasureTheory.forall_measure_preimage_add_iff {G : Type u_1} [MeasurableSpace G] [Add G] [MeasurableAdd G] (μ : MeasureTheory.Measure G) : (∀ (g : G) (A : Set G), MeasurableSet A → μ ((fun (h : G) => g + h) ⁻¹' A) = μ A) ↔ μ.IsAddLeftInvariant

An alternative way to prove that μ is left invariant under addition.

theorem MeasureTheory.forall_measure_preimage_mul_right_iff {G : Type u_1} [MeasurableSpace G] [Mul G] [MeasurableMul G] (μ : MeasureTheory.Measure G) : (∀ (g : G) (A : Set G), MeasurableSet A → μ ((fun (h : G) => h * g) ⁻¹' A) = μ A) ↔ μ.IsMulRightInvariant

An alternative way to prove that μ is right invariant under multiplication.

theorem MeasureTheory.forall_measure_preimage_add_right_iff {G : Type u_1} [MeasurableSpace G] [Add G] [MeasurableAdd G] (μ : MeasureTheory.Measure G) : (∀ (g : G) (A : Set G), MeasurableSet A → μ ((fun (h : G) => h + g) ⁻¹' A) = μ A) ↔ μ.IsAddRightInvariant

An alternative way to prove that μ is right invariant under addition.

instance MeasureTheory.Measure.prod.instIsMulLeftInvariant {G : Type u_1} [MeasurableSpace G] [Mul G] {μ : MeasureTheory.Measure G} [MeasurableMul G] [μ.IsMulLeftInvariant] [MeasureTheory.SFinite μ] {H : Type u_3} [Mul H] {mH : MeasurableSpace H} {ν : MeasureTheory.Measure H} [MeasurableMul H] [ν.IsMulLeftInvariant] [MeasureTheory.SFinite ν] : (μ.prod ν).IsMulLeftInvariant

Equations

• ⋯ = ⋯

instance MeasureTheory.Measure.prod.instIsAddLeftInvariant {G : Type u_1} [MeasurableSpace G] [Add G] {μ : MeasureTheory.Measure G} [MeasurableAdd G] [μ.IsAddLeftInvariant] [MeasureTheory.SFinite μ] {H : Type u_3} [Add H] {mH : MeasurableSpace H} {ν : MeasureTheory.Measure H} [MeasurableAdd H] [ν.IsAddLeftInvariant] [MeasureTheory.SFinite ν] : (μ.prod ν).IsAddLeftInvariant

Equations

• ⋯ = ⋯

instance MeasureTheory.Measure.prod.instIsMulRightInvariant {G : Type u_1} [MeasurableSpace G] [Mul G] {μ : MeasureTheory.Measure G} [MeasurableMul G] [μ.IsMulRightInvariant] [MeasureTheory.SFinite μ] {H : Type u_3} [Mul H] {mH : MeasurableSpace H} {ν : MeasureTheory.Measure H} [MeasurableMul H] [ν.IsMulRightInvariant] [MeasureTheory.SFinite ν] : (μ.prod ν).IsMulRightInvariant

Equations

• ⋯ = ⋯

instance MeasureTheory.Measure.prod.instIsAddRightInvariant {G : Type u_1} [MeasurableSpace G] [Add G] {μ : MeasureTheory.Measure G} [MeasurableAdd G] [μ.IsAddRightInvariant] [MeasureTheory.SFinite μ] {H : Type u_3} [Add H] {mH : MeasurableSpace H} {ν : MeasureTheory.Measure H} [MeasurableAdd H] [ν.IsAddRightInvariant] [MeasureTheory.SFinite ν] : (μ.prod ν).IsAddRightInvariant

Equations

• ⋯ = ⋯

theorem MeasureTheory.isMulLeftInvariant_map {G : Type u_1} [MeasurableSpace G] [Mul G] {μ : MeasureTheory.Measure G} [MeasurableMul G] {H : Type u_3} [MeasurableSpace H] [Mul H] [MeasurableMul H] [μ.IsMulLeftInvariant] (f : G →ₙ* H) (hf : Measurable ⇑f) (h_surj : Function.Surjective ⇑f) : (MeasureTheory.Measure.map (⇑f) μ).IsMulLeftInvariant

theorem MeasureTheory.isAddLeftInvariant_map {G : Type u_1} [MeasurableSpace G] [Add G] {μ : MeasureTheory.Measure G} [MeasurableAdd G] {H : Type u_3} [MeasurableSpace H] [Add H] [MeasurableAdd H] [μ.IsAddLeftInvariant] (f : AddHom G H) (hf : Measurable ⇑f) (h_surj : Function.Surjective ⇑f) : (MeasureTheory.Measure.map (⇑f) μ).IsAddLeftInvariant

theorem MeasureTheory.isMulLeftInvariant_map_smul {G : Type u_1} [MeasurableSpace G] [Semigroup G] [MeasurableMul G] {μ : MeasureTheory.Measure G} {α : Type u_3} [SMul α G] [SMulCommClass α G G] [MeasurableSpace α] [MeasurableSMul α G] [μ.IsMulLeftInvariant] (a : α) : (MeasureTheory.Measure.map (fun (x : G) => a • x) μ).IsMulLeftInvariant

The image of a left invariant measure under a left action is left invariant, assuming that the action preserves multiplication.

theorem MeasureTheory.isAddLeftInvariant_map_vadd {G : Type u_1} [MeasurableSpace G] [AddSemigroup G] [MeasurableAdd G] {μ : MeasureTheory.Measure G} {α : Type u_3} [VAdd α G] [VAddCommClass α G G] [MeasurableSpace α] [MeasurableVAdd α G] [μ.IsAddLeftInvariant] (a : α) : (MeasureTheory.Measure.map (fun (x : G) => a +ᵥ x) μ).IsAddLeftInvariant

The image of a left invariant measure under a left additive action is left invariant, assuming that the action preserves addition.

theorem MeasureTheory.isMulRightInvariant_map_smul {G : Type u_1} [MeasurableSpace G] [Semigroup G] [MeasurableMul G] {μ : MeasureTheory.Measure G} {α : Type u_3} [SMul α G] [SMulCommClass α Gᵐᵒᵖ G] [MeasurableSpace α] [MeasurableSMul α G] [μ.IsMulRightInvariant] (a : α) : (MeasureTheory.Measure.map (fun (x : G) => a • x) μ).IsMulRightInvariant

The image of a right invariant measure under a left action is right invariant, assuming that the action preserves multiplication.

theorem MeasureTheory.isAddRightInvariant_map_vadd {G : Type u_1} [MeasurableSpace G] [AddSemigroup G] [MeasurableAdd G] {μ : MeasureTheory.Measure G} {α : Type u_3} [VAdd α G] [VAddCommClass α Gᵃᵒᵖ G] [MeasurableSpace α] [MeasurableVAdd α G] [μ.IsAddRightInvariant] (a : α) : (MeasureTheory.Measure.map (fun (x : G) => a +ᵥ x) μ).IsAddRightInvariant

The image of a right invariant measure under a left additive action is right invariant, assuming that the action preserves addition.

instance MeasureTheory.isMulLeftInvariant_map_mul_right {G : Type u_1} [MeasurableSpace G] [Semigroup G] [MeasurableMul G] {μ : MeasureTheory.Measure G} [μ.IsMulLeftInvariant] (g : G) : (MeasureTheory.Measure.map (fun (x : G) => x * g) μ).IsMulLeftInvariant

The image of a left invariant measure under right multiplication is left invariant.

Equations

• ⋯ = ⋯

instance MeasureTheory.isMulLeftInvariant_map_add_right {G : Type u_1} [MeasurableSpace G] [AddSemigroup G] [MeasurableAdd G] {μ : MeasureTheory.Measure G} [μ.IsAddLeftInvariant] (g : G) : (MeasureTheory.Measure.map (fun (x : G) => x + g) μ).IsAddLeftInvariant

The image of a left invariant measure under right addition is left invariant.

Equations

• ⋯ = ⋯

instance MeasureTheory.isMulRightInvariant_map_mul_left {G : Type u_1} [MeasurableSpace G] [Semigroup G] [MeasurableMul G] {μ : MeasureTheory.Measure G} [μ.IsMulRightInvariant] (g : G) : (MeasureTheory.Measure.map (fun (x : G) => g * x) μ).IsMulRightInvariant

The image of a right invariant measure under left multiplication is right invariant.

Equations

• ⋯ = ⋯

instance MeasureTheory.isMulRightInvariant_map_add_left {G : Type u_1} [MeasurableSpace G] [AddSemigroup G] [MeasurableAdd G] {μ : MeasureTheory.Measure G} [μ.IsAddRightInvariant] (g : G) : (MeasureTheory.Measure.map (fun (x : G) => g + x) μ).IsAddRightInvariant

The image of a right invariant measure under left addition is right invariant.

Equations

• ⋯ = ⋯

theorem MeasureTheory.map_div_right_eq_self {G : Type u_1} [MeasurableSpace G] [DivInvMonoid G] (μ : MeasureTheory.Measure G) [μ.IsMulRightInvariant] (g : G) : MeasureTheory.Measure.map (fun (x : G) => x / g) μ = μ

theorem MeasureTheory.map_sub_right_eq_self {G : Type u_1} [MeasurableSpace G] [SubNegMonoid G] (μ : MeasureTheory.Measure G) [μ.IsAddRightInvariant] (g : G) : MeasureTheory.Measure.map (fun (x : G) => x - g) μ = μ

theorem MeasureTheory.measurePreserving_div_right {G : Type u_1} [MeasurableSpace G] [Group G] [MeasurableMul G] (μ : MeasureTheory.Measure G) [μ.IsMulRightInvariant] (g : G) : MeasureTheory.MeasurePreserving (fun (x : G) => x / g) μ μ

theorem MeasureTheory.measurePreserving_sub_right {G : Type u_1} [MeasurableSpace G] [AddGroup G] [MeasurableAdd G] (μ : MeasureTheory.Measure G) [μ.IsAddRightInvariant] (g : G) : MeasureTheory.MeasurePreserving (fun (x : G) => x - g) μ μ

@[simp]

theorem MeasureTheory.measure_preimage_mul {G : Type u_1} [MeasurableSpace G] [Group G] [MeasurableMul G] (μ : MeasureTheory.Measure G) [μ.IsMulLeftInvariant] (g : G) (A : Set G) : μ ((fun (h : G) => g * h) ⁻¹' A) = μ A

We shorten this from measure_preimage_mul_left, since left invariant is the preferred option for measures in this formalization.

@[simp]

theorem MeasureTheory.measure_preimage_add {G : Type u_1} [MeasurableSpace G] [AddGroup G] [MeasurableAdd G] (μ : MeasureTheory.Measure G) [μ.IsAddLeftInvariant] (g : G) (A : Set G) : μ ((fun (h : G) => g + h) ⁻¹' A) = μ A

We shorten this from measure_preimage_add_left, since left invariant is the preferred option for measures in this formalization.

@[simp]

theorem MeasureTheory.measure_preimage_mul_right {G : Type u_1} [MeasurableSpace G] [Group G] [MeasurableMul G] (μ : MeasureTheory.Measure G) [μ.IsMulRightInvariant] (g : G) (A : Set G) : μ ((fun (h : G) => h * g) ⁻¹' A) = μ A

@[simp]

theorem MeasureTheory.measure_preimage_add_right {G : Type u_1} [MeasurableSpace G] [AddGroup G] [MeasurableAdd G] (μ : MeasureTheory.Measure G) [μ.IsAddRightInvariant] (g : G) (A : Set G) : μ ((fun (h : G) => h + g) ⁻¹' A) = μ A

theorem MeasureTheory.map_mul_left_ae {G : Type u_1} [MeasurableSpace G] [Group G] [MeasurableMul G] (μ : MeasureTheory.Measure G) [μ.IsMulLeftInvariant] (x : G) : Filter.map (fun (h : G) => x * h) (MeasureTheory.ae μ) = MeasureTheory.ae μ

theorem MeasureTheory.map_add_left_ae {G : Type u_1} [MeasurableSpace G] [AddGroup G] [MeasurableAdd G] (μ : MeasureTheory.Measure G) [μ.IsAddLeftInvariant] (x : G) : Filter.map (fun (h : G) => x + h) (MeasureTheory.ae μ) = MeasureTheory.ae μ

theorem MeasureTheory.map_mul_right_ae {G : Type u_1} [MeasurableSpace G] [Group G] [MeasurableMul G] (μ : MeasureTheory.Measure G) [μ.IsMulRightInvariant] (x : G) : Filter.map (fun (h : G) => h * x) (MeasureTheory.ae μ) = MeasureTheory.ae μ

theorem MeasureTheory.map_add_right_ae {G : Type u_1} [MeasurableSpace G] [AddGroup G] [MeasurableAdd G] (μ : MeasureTheory.Measure G) [μ.IsAddRightInvariant] (x : G) : Filter.map (fun (h : G) => h + x) (MeasureTheory.ae μ) = MeasureTheory.ae μ

theorem MeasureTheory.map_div_right_ae {G : Type u_1} [MeasurableSpace G] [Group G] [MeasurableMul G] (μ : MeasureTheory.Measure G) [μ.IsMulRightInvariant] (x : G) : Filter.map (fun (t : G) => t / x) (MeasureTheory.ae μ) = MeasureTheory.ae μ

theorem MeasureTheory.map_sub_right_ae {G : Type u_1} [MeasurableSpace G] [AddGroup G] [MeasurableAdd G] (μ : MeasureTheory.Measure G) [μ.IsAddRightInvariant] (x : G) : Filter.map (fun (t : G) => t - x) (MeasureTheory.ae μ) = MeasureTheory.ae μ

theorem MeasureTheory.eventually_mul_left_iff {G : Type u_1} [MeasurableSpace G] [Group G] [MeasurableMul G] (μ : MeasureTheory.Measure G) [μ.IsMulLeftInvariant] (t : G) {p : G → Prop} : (∀ᵐ (x : G) ∂μ, p (t * x)) ↔ ∀ᵐ (x : G) ∂μ, p x

theorem MeasureTheory.eventually_add_left_iff {G : Type u_1} [MeasurableSpace G] [AddGroup G] [MeasurableAdd G] (μ : MeasureTheory.Measure G) [μ.IsAddLeftInvariant] (t : G) {p : G → Prop} : (∀ᵐ (x : G) ∂μ, p (t + x)) ↔ ∀ᵐ (x : G) ∂μ, p x

theorem MeasureTheory.eventually_mul_right_iff {G : Type u_1} [MeasurableSpace G] [Group G] [MeasurableMul G] (μ : MeasureTheory.Measure G) [μ.IsMulRightInvariant] (t : G) {p : G → Prop} : (∀ᵐ (x : G) ∂μ, p (x * t)) ↔ ∀ᵐ (x : G) ∂μ, p x

theorem MeasureTheory.eventually_add_right_iff {G : Type u_1} [MeasurableSpace G] [AddGroup G] [MeasurableAdd G] (μ : MeasureTheory.Measure G) [μ.IsAddRightInvariant] (t : G) {p : G → Prop} : (∀ᵐ (x : G) ∂μ, p (x + t)) ↔ ∀ᵐ (x : G) ∂μ, p x

theorem MeasureTheory.eventually_div_right_iff {G : Type u_1} [MeasurableSpace G] [Group G] [MeasurableMul G] (μ : MeasureTheory.Measure G) [μ.IsMulRightInvariant] (t : G) {p : G → Prop} : (∀ᵐ (x : G) ∂μ, p (x / t)) ↔ ∀ᵐ (x : G) ∂μ, p x

theorem MeasureTheory.eventually_sub_right_iff {G : Type u_1} [MeasurableSpace G] [AddGroup G] [MeasurableAdd G] (μ : MeasureTheory.Measure G) [μ.IsAddRightInvariant] (t : G) {p : G → Prop} : (∀ᵐ (x : G) ∂μ, p (x - t)) ↔ ∀ᵐ (x : G) ∂μ, p x

noncomputable def MeasureTheory.Measure.inv {G : Type u_1} [MeasurableSpace G] [Inv G] (μ : MeasureTheory.Measure G) : MeasureTheory.Measure G

The measure A ↦ μ (A⁻¹), where A⁻¹ is the pointwise inverse of A.

Equations

• μ.inv = MeasureTheory.Measure.map Inv.inv μ

noncomputable def MeasureTheory.Measure.neg {G : Type u_1} [MeasurableSpace G] [Neg G] (μ : MeasureTheory.Measure G) : MeasureTheory.Measure G

The measure A ↦ μ (- A), where - A is the pointwise negation of A.

Equations

• μ.neg = MeasureTheory.Measure.map Neg.neg μ

class MeasureTheory.Measure.IsNegInvariant {G : Type u_1} [MeasurableSpace G] [Neg G] (μ : MeasureTheory.Measure G) : Prop

A measure is invariant under negation if - μ = μ. Equivalently, this means that for all measurable A we have μ (- A) = μ A, where - A is the pointwise negation of A.

• neg_eq_self : μ.neg = μ

theorem MeasureTheory.Measure.IsNegInvariant.neg_eq_self {G : Type u_1} : ∀ {inst : MeasurableSpace G} {inst_1 : Neg G} {μ : MeasureTheory.Measure G} [self : μ.IsNegInvariant], μ.neg = μ

class MeasureTheory.Measure.IsInvInvariant {G : Type u_1} [MeasurableSpace G] [Inv G] (μ : MeasureTheory.Measure G) : Prop

A measure is invariant under inversion if μ⁻¹ = μ. Equivalently, this means that for all measurable A we have μ (A⁻¹) = μ A, where A⁻¹ is the pointwise inverse of A.

• inv_eq_self : μ.inv = μ

theorem MeasureTheory.Measure.IsInvInvariant.inv_eq_self {G : Type u_1} : ∀ {inst : MeasurableSpace G} {inst_1 : Inv G} {μ : MeasureTheory.Measure G} [self : μ.IsInvInvariant], μ.inv = μ

theorem MeasureTheory.Measure.inv_def {G : Type u_1} [MeasurableSpace G] [Inv G] (μ : MeasureTheory.Measure G) : μ.inv = MeasureTheory.Measure.map Inv.inv μ

theorem MeasureTheory.Measure.neg_def {G : Type u_1} [MeasurableSpace G] [Neg G] (μ : MeasureTheory.Measure G) : μ.neg = MeasureTheory.Measure.map Neg.neg μ

@[simp]

theorem MeasureTheory.Measure.inv_eq_self {G : Type u_1} [MeasurableSpace G] [Inv G] (μ : MeasureTheory.Measure G) [μ.IsInvInvariant] : μ.inv = μ

@[simp]

theorem MeasureTheory.Measure.neg_eq_self {G : Type u_1} [MeasurableSpace G] [Neg G] (μ : MeasureTheory.Measure G) [μ.IsNegInvariant] : μ.neg = μ

@[simp]

theorem MeasureTheory.Measure.map_inv_eq_self {G : Type u_1} [MeasurableSpace G] [Inv G] (μ : MeasureTheory.Measure G) [μ.IsInvInvariant] : MeasureTheory.Measure.map Inv.inv μ = μ

@[simp]

theorem MeasureTheory.Measure.map_neg_eq_self {G : Type u_1} [MeasurableSpace G] [Neg G] (μ : MeasureTheory.Measure G) [μ.IsNegInvariant] : MeasureTheory.Measure.map Neg.neg μ = μ

theorem MeasureTheory.Measure.measurePreserving_inv {G : Type u_1} [MeasurableSpace G] [Inv G] [MeasurableInv G] (μ : MeasureTheory.Measure G) [μ.IsInvInvariant] : MeasureTheory.MeasurePreserving Inv.inv μ μ

theorem MeasureTheory.Measure.measurePreserving_neg {G : Type u_1} [MeasurableSpace G] [Neg G] [MeasurableNeg G] (μ : MeasureTheory.Measure G) [μ.IsNegInvariant] : MeasureTheory.MeasurePreserving Neg.neg μ μ

instance MeasureTheory.Measure.inv.instSFinite {G : Type u_1} [MeasurableSpace G] [Inv G] (μ : MeasureTheory.Measure G) [MeasureTheory.SFinite μ] : MeasureTheory.SFinite μ.inv

Equations

• ⋯ = ⋯

instance MeasureTheory.Measure.neg.instSFinite {G : Type u_1} [MeasurableSpace G] [Neg G] (μ : MeasureTheory.Measure G) [MeasureTheory.SFinite μ] : MeasureTheory.SFinite μ.neg

Equations

• ⋯ = ⋯

@[simp]

theorem MeasureTheory.Measure.inv_apply {G : Type u_1} [MeasurableSpace G] [InvolutiveInv G] [MeasurableInv G] (μ : MeasureTheory.Measure G) (s : Set G) : μ.inv s = μ s⁻¹

@[simp]

theorem MeasureTheory.Measure.neg_apply {G : Type u_1} [MeasurableSpace G] [InvolutiveNeg G] [MeasurableNeg G] (μ : MeasureTheory.Measure G) (s : Set G) : μ.neg s = μ (-s)

@[simp]

theorem MeasureTheory.Measure.inv_inv {G : Type u_1} [MeasurableSpace G] [InvolutiveInv G] [MeasurableInv G] (μ : MeasureTheory.Measure G) : μ.inv.inv = μ

@[simp]

theorem MeasureTheory.Measure.neg_neg {G : Type u_1} [MeasurableSpace G] [InvolutiveNeg G] [MeasurableNeg G] (μ : MeasureTheory.Measure G) : μ.neg.neg = μ

@[simp]

theorem MeasureTheory.Measure.measure_inv {G : Type u_1} [MeasurableSpace G] [InvolutiveInv G] [MeasurableInv G] (μ : MeasureTheory.Measure G) [μ.IsInvInvariant] (A : Set G) : μ A⁻¹ = μ A

@[simp]

theorem MeasureTheory.Measure.measure_neg {G : Type u_1} [MeasurableSpace G] [InvolutiveNeg G] [MeasurableNeg G] (μ : MeasureTheory.Measure G) [μ.IsNegInvariant] (A : Set G) : μ (-A) = μ A

theorem MeasureTheory.Measure.measure_preimage_inv {G : Type u_1} [MeasurableSpace G] [InvolutiveInv G] [MeasurableInv G] (μ : MeasureTheory.Measure G) [μ.IsInvInvariant] (A : Set G) : μ (Inv.inv ⁻¹' A) = μ A

theorem MeasureTheory.Measure.measure_preimage_neg {G : Type u_1} [MeasurableSpace G] [InvolutiveNeg G] [MeasurableNeg G] (μ : MeasureTheory.Measure G) [μ.IsNegInvariant] (A : Set G) : μ (Neg.neg ⁻¹' A) = μ A

instance MeasureTheory.Measure.inv.instSigmaFinite {G : Type u_1} [MeasurableSpace G] [InvolutiveInv G] [MeasurableInv G] (μ : MeasureTheory.Measure G) [MeasureTheory.SigmaFinite μ] : MeasureTheory.SigmaFinite μ.inv

Equations

• ⋯ = ⋯

instance MeasureTheory.Measure.neg.instSigmaFinite {G : Type u_1} [MeasurableSpace G] [InvolutiveNeg G] [MeasurableNeg G] (μ : MeasureTheory.Measure G) [MeasureTheory.SigmaFinite μ] : MeasureTheory.SigmaFinite μ.neg

Equations

• ⋯ = ⋯

instance MeasureTheory.Measure.inv.instIsMulRightInvariant {G : Type u_1} [MeasurableSpace G] [DivisionMonoid G] [MeasurableMul G] [MeasurableInv G] {μ : MeasureTheory.Measure G} [μ.IsMulLeftInvariant] : μ.inv.IsMulRightInvariant

Equations

• ⋯ = ⋯

instance MeasureTheory.Measure.neg.instIsAddRightInvariant {G : Type u_1} [MeasurableSpace G] [SubtractionMonoid G] [MeasurableAdd G] [MeasurableNeg G] {μ : MeasureTheory.Measure G} [μ.IsAddLeftInvariant] : μ.neg.IsAddRightInvariant

Equations

• ⋯ = ⋯

instance MeasureTheory.Measure.inv.instIsMulLeftInvariant {G : Type u_1} [MeasurableSpace G] [DivisionMonoid G] [MeasurableMul G] [MeasurableInv G] {μ : MeasureTheory.Measure G} [μ.IsMulRightInvariant] : μ.inv.IsMulLeftInvariant

Equations

• ⋯ = ⋯

instance MeasureTheory.Measure.neg.instIsAddLeftInvariant {G : Type u_1} [MeasurableSpace G] [SubtractionMonoid G] [MeasurableAdd G] [MeasurableNeg G] {μ : MeasureTheory.Measure G} [μ.IsAddRightInvariant] : μ.neg.IsAddLeftInvariant

Equations

• ⋯ = ⋯

theorem MeasureTheory.Measure.measurePreserving_div_left {G : Type u_1} [MeasurableSpace G] [DivisionMonoid G] [MeasurableMul G] [MeasurableInv G] (μ : MeasureTheory.Measure G) [μ.IsInvInvariant] [μ.IsMulLeftInvariant] (g : G) : MeasureTheory.MeasurePreserving (fun (t : G) => g / t) μ μ

theorem MeasureTheory.Measure.measurePreserving_sub_left {G : Type u_1} [MeasurableSpace G] [SubtractionMonoid G] [MeasurableAdd G] [MeasurableNeg G] (μ : MeasureTheory.Measure G) [μ.IsNegInvariant] [μ.IsAddLeftInvariant] (g : G) : MeasureTheory.MeasurePreserving (fun (t : G) => g - t) μ μ

theorem MeasureTheory.Measure.map_div_left_eq_self {G : Type u_1} [MeasurableSpace G] [DivisionMonoid G] [MeasurableMul G] [MeasurableInv G] (μ : MeasureTheory.Measure G) [μ.IsInvInvariant] [μ.IsMulLeftInvariant] (g : G) : MeasureTheory.Measure.map (fun (t : G) => g / t) μ = μ

theorem MeasureTheory.Measure.map_sub_left_eq_self {G : Type u_1} [MeasurableSpace G] [SubtractionMonoid G] [MeasurableAdd G] [MeasurableNeg G] (μ : MeasureTheory.Measure G) [μ.IsNegInvariant] [μ.IsAddLeftInvariant] (g : G) : MeasureTheory.Measure.map (fun (t : G) => g - t) μ = μ

theorem MeasureTheory.Measure.measurePreserving_mul_right_inv {G : Type u_1} [MeasurableSpace G] [DivisionMonoid G] [MeasurableMul G] [MeasurableInv G] (μ : MeasureTheory.Measure G) [μ.IsInvInvariant] [μ.IsMulLeftInvariant] (g : G) : MeasureTheory.MeasurePreserving (fun (t : G) => (g * t)⁻¹) μ μ

theorem MeasureTheory.Measure.measurePreserving_add_right_neg {G : Type u_1} [MeasurableSpace G] [SubtractionMonoid G] [MeasurableAdd G] [MeasurableNeg G] (μ : MeasureTheory.Measure G) [μ.IsNegInvariant] [μ.IsAddLeftInvariant] (g : G) : MeasureTheory.MeasurePreserving (fun (t : G) => -(g + t)) μ μ

theorem MeasureTheory.Measure.map_mul_right_inv_eq_self {G : Type u_1} [MeasurableSpace G] [DivisionMonoid G] [MeasurableMul G] [MeasurableInv G] (μ : MeasureTheory.Measure G) [μ.IsInvInvariant] [μ.IsMulLeftInvariant] (g : G) : MeasureTheory.Measure.map (fun (t : G) => (g * t)⁻¹) μ = μ

theorem MeasureTheory.Measure.map_add_right_neg_eq_self {G : Type u_1} [MeasurableSpace G] [SubtractionMonoid G] [MeasurableAdd G] [MeasurableNeg G] (μ : MeasureTheory.Measure G) [μ.IsNegInvariant] [μ.IsAddLeftInvariant] (g : G) : MeasureTheory.Measure.map (fun (t : G) => -(g + t)) μ = μ

theorem MeasureTheory.Measure.map_div_left_ae {G : Type u_1} [MeasurableSpace G] [Group G] [MeasurableMul G] [MeasurableInv G] (μ : MeasureTheory.Measure G) [μ.IsMulLeftInvariant] [μ.IsInvInvariant] (x : G) : Filter.map (fun (t : G) => x / t) (MeasureTheory.ae μ) = MeasureTheory.ae μ

theorem MeasureTheory.Measure.map_sub_left_ae {G : Type u_1} [MeasurableSpace G] [AddGroup G] [MeasurableAdd G] [MeasurableNeg G] (μ : MeasureTheory.Measure G) [μ.IsAddLeftInvariant] [μ.IsNegInvariant] (x : G) : Filter.map (fun (t : G) => x - t) (MeasureTheory.ae μ) = MeasureTheory.ae μ

instance MeasureTheory.Measure.IsFiniteMeasureOnCompacts.inv {G : Type u_1} [MeasurableSpace G] [TopologicalSpace G] [BorelSpace G] {μ : MeasureTheory.Measure G} [Group G] [ContinuousInv G] [MeasureTheory.IsFiniteMeasureOnCompacts μ] : MeasureTheory.IsFiniteMeasureOnCompacts μ.inv

Equations

• ⋯ = ⋯

instance MeasureTheory.Measure.IsFiniteMeasureOnCompacts.neg {G : Type u_1} [MeasurableSpace G] [TopologicalSpace G] [BorelSpace G] {μ : MeasureTheory.Measure G} [AddGroup G] [ContinuousNeg G] [MeasureTheory.IsFiniteMeasureOnCompacts μ] : MeasureTheory.IsFiniteMeasureOnCompacts μ.neg

Equations

• ⋯ = ⋯

instance MeasureTheory.Measure.IsOpenPosMeasure.inv {G : Type u_1} [MeasurableSpace G] [TopologicalSpace G] [BorelSpace G] {μ : MeasureTheory.Measure G} [Group G] [ContinuousInv G] [μ.IsOpenPosMeasure] : μ.inv.IsOpenPosMeasure

Equations

• ⋯ = ⋯

instance MeasureTheory.Measure.IsOpenPosMeasure.neg {G : Type u_1} [MeasurableSpace G] [TopologicalSpace G] [BorelSpace G] {μ : MeasureTheory.Measure G} [AddGroup G] [ContinuousNeg G] [μ.IsOpenPosMeasure] : μ.neg.IsOpenPosMeasure

Equations

• ⋯ = ⋯

instance MeasureTheory.Measure.Regular.inv {G : Type u_1} [MeasurableSpace G] [TopologicalSpace G] [BorelSpace G] {μ : MeasureTheory.Measure G} [Group G] [ContinuousInv G] [μ.Regular] : μ.inv.Regular

Equations

• ⋯ = ⋯

instance MeasureTheory.Measure.Regular.neg {G : Type u_1} [MeasurableSpace G] [TopologicalSpace G] [BorelSpace G] {μ : MeasureTheory.Measure G} [AddGroup G] [ContinuousNeg G] [μ.Regular] : μ.neg.Regular

Equations

• ⋯ = ⋯

instance MeasureTheory.Measure.InnerRegular.inv {G : Type u_1} [MeasurableSpace G] [TopologicalSpace G] [BorelSpace G] {μ : MeasureTheory.Measure G} [Group G] [ContinuousInv G] [μ.InnerRegular] : μ.inv.InnerRegular

Equations

• ⋯ = ⋯

instance MeasureTheory.Measure.InnerRegular.neg {G : Type u_1} [MeasurableSpace G] [TopologicalSpace G] [BorelSpace G] {μ : MeasureTheory.Measure G} [AddGroup G] [ContinuousNeg G] [μ.InnerRegular] : μ.neg.InnerRegular

Equations

• ⋯ = ⋯

instance MeasureTheory.innerRegular_map_smul {G : Type u_1} [MeasurableSpace G] [TopologicalSpace G] [BorelSpace G] {μ : MeasureTheory.Measure G} {α : Type u_3} [Monoid α] [MulAction α G] [ContinuousConstSMul α G] [μ.InnerRegular] (a : α) : (MeasureTheory.Measure.map (fun (x : G) => a • x) μ).InnerRegular

The image of an inner regular measure under map of a left action is again inner regular.

Equations

• ⋯ = ⋯

instance MeasureTheory.innerRegular_map_vadd {G : Type u_1} [MeasurableSpace G] [TopologicalSpace G] [BorelSpace G] {μ : MeasureTheory.Measure G} {α : Type u_3} [AddMonoid α] [AddAction α G] [ContinuousConstVAdd α G] [μ.InnerRegular] (a : α) : (MeasureTheory.Measure.map (fun (x : G) => a +ᵥ x) μ).InnerRegular

The image of a inner regular measure under map of a left additive action is again inner regular

Equations

• ⋯ = ⋯

instance MeasureTheory.innerRegular_map_mul_left {G : Type u_1} [MeasurableSpace G] [TopologicalSpace G] [BorelSpace G] {μ : MeasureTheory.Measure G} [Group G] [TopologicalGroup G] [μ.InnerRegular] (g : G) : (MeasureTheory.Measure.map (fun (x : G) => g * x) μ).InnerRegular

The image of an inner regular measure under left multiplication is again inner regular.

Equations

• ⋯ = ⋯

instance MeasureTheory.innerRegular_map_add_left {G : Type u_1} [MeasurableSpace G] [TopologicalSpace G] [BorelSpace G] {μ : MeasureTheory.Measure G} [AddGroup G] [TopologicalAddGroup G] [μ.InnerRegular] (g : G) : (MeasureTheory.Measure.map (fun (x : G) => g + x) μ).InnerRegular

The image of an inner regular measure under left addition is again inner regular.

Equations

• ⋯ = ⋯

instance MeasureTheory.innerRegular_map_mul_right {G : Type u_1} [MeasurableSpace G] [TopologicalSpace G] [BorelSpace G] {μ : MeasureTheory.Measure G} [Group G] [TopologicalGroup G] [μ.InnerRegular] (g : G) : (MeasureTheory.Measure.map (fun (x : G) => x * g) μ).InnerRegular

The image of an inner regular measure under right multiplication is again inner regular.

Equations

• ⋯ = ⋯

instance MeasureTheory.innerRegular_map_add_right {G : Type u_1} [MeasurableSpace G] [TopologicalSpace G] [BorelSpace G] {μ : MeasureTheory.Measure G} [AddGroup G] [TopologicalAddGroup G] [μ.InnerRegular] (g : G) : (MeasureTheory.Measure.map (fun (x : G) => x + g) μ).InnerRegular

The image of an inner regular measure under right addition is again inner regular.

Equations

• ⋯ = ⋯

theorem MeasureTheory.regular_inv_iff {G : Type u_1} [MeasurableSpace G] [TopologicalSpace G] [BorelSpace G] {μ : MeasureTheory.Measure G} [Group G] [TopologicalGroup G] : μ.inv.Regular ↔ μ.Regular

theorem MeasureTheory.regular_neg_iff {G : Type u_1} [MeasurableSpace G] [TopologicalSpace G] [BorelSpace G] {μ : MeasureTheory.Measure G} [AddGroup G] [TopologicalAddGroup G] : μ.neg.Regular ↔ μ.Regular

theorem MeasureTheory.innerRegular_inv_iff {G : Type u_1} [MeasurableSpace G] [TopologicalSpace G] [BorelSpace G] {μ : MeasureTheory.Measure G} [Group G] [TopologicalGroup G] : μ.inv.InnerRegular ↔ μ.InnerRegular

theorem MeasureTheory.innerRegular_neg_iff {G : Type u_1} [MeasurableSpace G] [TopologicalSpace G] [BorelSpace G] {μ : MeasureTheory.Measure G} [AddGroup G] [TopologicalAddGroup G] : μ.neg.InnerRegular ↔ μ.InnerRegular

theorem MeasureTheory.eventually_nhds_one_measure_smul_diff_lt {G : Type u_1} [MeasurableSpace G] [TopologicalSpace G] [BorelSpace G] {μ : MeasureTheory.Measure G} [Group G] [TopologicalGroup G] [LocallyCompactSpace G] [MeasureTheory.IsFiniteMeasureOnCompacts μ] [μ.InnerRegularCompactLTTop] {k : Set G} (hk : IsCompact k) (h'k : IsClosed k) {ε : ENNReal} (hε : ε ≠ 0) : ∀ᶠ (g : G) in nhds 1, μ (g • k \ k) < ε

Continuity of the measure of translates of a compact set: Given a compact set k in a topological group, for g close enough to the origin, μ (g • k \ k) is arbitrarily small.

theorem MeasureTheory.eventually_nhds_zero_measure_vadd_diff_lt {G : Type u_1} [MeasurableSpace G] [TopologicalSpace G] [BorelSpace G] {μ : MeasureTheory.Measure G} [AddGroup G] [TopologicalAddGroup G] [LocallyCompactSpace G] [MeasureTheory.IsFiniteMeasureOnCompacts μ] [μ.InnerRegularCompactLTTop] {k : Set G} (hk : IsCompact k) (h'k : IsClosed k) {ε : ENNReal} (hε : ε ≠ 0) : ∀ᶠ (g : G) in nhds 0, μ ((g +ᵥ k) \ k) < ε

theorem MeasureTheory.tendsto_measure_smul_diff_isCompact_isClosed {G : Type u_1} [MeasurableSpace G] [TopologicalSpace G] [BorelSpace G] {μ : MeasureTheory.Measure G} [Group G] [TopologicalGroup G] [LocallyCompactSpace G] [MeasureTheory.IsFiniteMeasureOnCompacts μ] [μ.InnerRegularCompactLTTop] {k : Set G} (hk : IsCompact k) (h'k : IsClosed k) : Filter.Tendsto (fun (g : G) => μ (g • k \ k)) (nhds 1) (nhds 0)

Continuity of the measure of translates of a compact set: Given a closed compact set k in a topological group, the measure of g • k \ k tends to zero as g tends to 1.

theorem MeasureTheory.tendsto_measure_vadd_diff_isCompact_isClosed {G : Type u_1} [MeasurableSpace G] [TopologicalSpace G] [BorelSpace G] {μ : MeasureTheory.Measure G} [AddGroup G] [TopologicalAddGroup G] [LocallyCompactSpace G] [MeasureTheory.IsFiniteMeasureOnCompacts μ] [μ.InnerRegularCompactLTTop] {k : Set G} (hk : IsCompact k) (h'k : IsClosed k) : Filter.Tendsto (fun (g : G) => μ ((g +ᵥ k) \ k)) (nhds 0) (nhds 0)

theorem MeasureTheory.isOpenPosMeasure_of_mulLeftInvariant_of_compact {G : Type u_1} [MeasurableSpace G] [TopologicalSpace G] [BorelSpace G] {μ : MeasureTheory.Measure G} [Group G] [TopologicalGroup G] [μ.IsMulLeftInvariant] (K : Set G) (hK : IsCompact K) (h : μ K ≠ 0) : μ.IsOpenPosMeasure

If a left-invariant measure gives positive mass to a compact set, then it gives positive mass to any open set.

theorem MeasureTheory.isOpenPosMeasure_of_addLeftInvariant_of_compact {G : Type u_1} [MeasurableSpace G] [TopologicalSpace G] [BorelSpace G] {μ : MeasureTheory.Measure G} [AddGroup G] [TopologicalAddGroup G] [μ.IsAddLeftInvariant] (K : Set G) (hK : IsCompact K) (h : μ K ≠ 0) : μ.IsOpenPosMeasure

If a left-invariant measure gives positive mass to a compact set, then it gives positive mass to any open set.

@[instance 80]

instance MeasureTheory.isOpenPosMeasure_of_mulLeftInvariant_of_regular {G : Type u_1} [MeasurableSpace G] [TopologicalSpace G] [BorelSpace G] {μ : MeasureTheory.Measure G} [Group G] [TopologicalGroup G] [μ.IsMulLeftInvariant] [μ.Regular] [NeZero μ] : μ.IsOpenPosMeasure

A nonzero left-invariant regular measure gives positive mass to any open set.

Equations

• ⋯ = ⋯

@[instance 80]

instance MeasureTheory.isOpenPosMeasure_of_addLeftInvariant_of_regular {G : Type u_1} [MeasurableSpace G] [TopologicalSpace G] [BorelSpace G] {μ : MeasureTheory.Measure G} [AddGroup G] [TopologicalAddGroup G] [μ.IsAddLeftInvariant] [μ.Regular] [NeZero μ] : μ.IsOpenPosMeasure

A nonzero left-invariant regular measure gives positive mass to any open set.

Equations

• ⋯ = ⋯

@[instance 80]

instance MeasureTheory.isOpenPosMeasure_of_mulLeftInvariant_of_innerRegular {G : Type u_1} [MeasurableSpace G] [TopologicalSpace G] [BorelSpace G] {μ : MeasureTheory.Measure G} [Group G] [TopologicalGroup G] [μ.IsMulLeftInvariant] [μ.InnerRegular] [NeZero μ] : μ.IsOpenPosMeasure

A nonzero left-invariant inner regular measure gives positive mass to any open set.

Equations

• ⋯ = ⋯

@[instance 80]

instance MeasureTheory.isOpenPosMeasure_of_addLeftInvariant_of_innerRegular {G : Type u_1} [MeasurableSpace G] [TopologicalSpace G] [BorelSpace G] {μ : MeasureTheory.Measure G} [AddGroup G] [TopologicalAddGroup G] [μ.IsAddLeftInvariant] [μ.InnerRegular] [NeZero μ] : μ.IsOpenPosMeasure

A nonzero left-invariant inner regular measure gives positive mass to any open set.

Equations

• ⋯ = ⋯

theorem MeasureTheory.null_iff_of_isMulLeftInvariant {G : Type u_1} [MeasurableSpace G] [TopologicalSpace G] [BorelSpace G] {μ : MeasureTheory.Measure G} [Group G] [TopologicalGroup G] [μ.IsMulLeftInvariant] [μ.Regular] {s : Set G} (hs : IsOpen s) : μ s = 0 ↔ s = ∅ ∨ μ = 0

theorem MeasureTheory.null_iff_of_isAddLeftInvariant {G : Type u_1} [MeasurableSpace G] [TopologicalSpace G] [BorelSpace G] {μ : MeasureTheory.Measure G} [AddGroup G] [TopologicalAddGroup G] [μ.IsAddLeftInvariant] [μ.Regular] {s : Set G} (hs : IsOpen s) : μ s = 0 ↔ s = ∅ ∨ μ = 0

theorem MeasureTheory.measure_ne_zero_iff_nonempty_of_isMulLeftInvariant {G : Type u_1} [MeasurableSpace G] [TopologicalSpace G] [BorelSpace G] {μ : MeasureTheory.Measure G} [Group G] [TopologicalGroup G] [μ.IsMulLeftInvariant] [μ.Regular] (hμ : μ ≠ 0) {s : Set G} (hs : IsOpen s) : μ s ≠ 0 ↔ s.Nonempty

theorem MeasureTheory.measure_ne_zero_iff_nonempty_of_isAddLeftInvariant {G : Type u_1} [MeasurableSpace G] [TopologicalSpace G] [BorelSpace G] {μ : MeasureTheory.Measure G} [AddGroup G] [TopologicalAddGroup G] [μ.IsAddLeftInvariant] [μ.Regular] (hμ : μ ≠ 0) {s : Set G} (hs : IsOpen s) : μ s ≠ 0 ↔ s.Nonempty

theorem MeasureTheory.measure_pos_iff_nonempty_of_isMulLeftInvariant {G : Type u_1} [MeasurableSpace G] [TopologicalSpace G] [BorelSpace G] {μ : MeasureTheory.Measure G} [Group G] [TopologicalGroup G] [μ.IsMulLeftInvariant] [μ.Regular] (h3μ : μ ≠ 0) {s : Set G} (hs : IsOpen s) : 0 < μ s ↔ s.Nonempty

theorem MeasureTheory.measure_pos_iff_nonempty_of_isAddLeftInvariant {G : Type u_1} [MeasurableSpace G] [TopologicalSpace G] [BorelSpace G] {μ : MeasureTheory.Measure G} [AddGroup G] [TopologicalAddGroup G] [μ.IsAddLeftInvariant] [μ.Regular] (h3μ : μ ≠ 0) {s : Set G} (hs : IsOpen s) : 0 < μ s ↔ s.Nonempty

theorem MeasureTheory.measure_lt_top_of_isCompact_of_isMulLeftInvariant {G : Type u_1} [MeasurableSpace G] [TopologicalSpace G] [BorelSpace G] {μ : MeasureTheory.Measure G} [Group G] [TopologicalGroup G] [μ.IsMulLeftInvariant] (U : Set G) (hU : IsOpen U) (h'U : U.Nonempty) (h : μ U ≠ ⊤) {K : Set G} (hK : IsCompact K) : μ K < ⊤

If a left-invariant measure gives finite mass to a nonempty open set, then it gives finite mass to any compact set.

theorem MeasureTheory.measure_lt_top_of_isCompact_of_isAddLeftInvariant {G : Type u_1} [MeasurableSpace G] [TopologicalSpace G] [BorelSpace G] {μ : MeasureTheory.Measure G} [AddGroup G] [TopologicalAddGroup G] [μ.IsAddLeftInvariant] (U : Set G) (hU : IsOpen U) (h'U : U.Nonempty) (h : μ U ≠ ⊤) {K : Set G} (hK : IsCompact K) : μ K < ⊤

If a left-invariant measure gives finite mass to a nonempty open set, then it gives finite mass to any compact set.

theorem MeasureTheory.measure_lt_top_of_isCompact_of_isMulLeftInvariant' {G : Type u_1} [MeasurableSpace G] [TopologicalSpace G] [BorelSpace G] {μ : MeasureTheory.Measure G} [Group G] [TopologicalGroup G] [μ.IsMulLeftInvariant] {U : Set G} (hU : (interior U).Nonempty) (h : μ U ≠ ⊤) {K : Set G} (hK : IsCompact K) : μ K < ⊤

If a left-invariant measure gives finite mass to a set with nonempty interior, then it gives finite mass to any compact set.

theorem MeasureTheory.measure_lt_top_of_isCompact_of_isAddLeftInvariant' {G : Type u_1} [MeasurableSpace G] [TopologicalSpace G] [BorelSpace G] {μ : MeasureTheory.Measure G} [AddGroup G] [TopologicalAddGroup G] [μ.IsAddLeftInvariant] {U : Set G} (hU : (interior U).Nonempty) (h : μ U ≠ ⊤) {K : Set G} (hK : IsCompact K) : μ K < ⊤

If a left-invariant measure gives finite mass to a set with nonempty interior, then it gives finite mass to any compact set.

@[simp]

theorem MeasureTheory.measure_univ_of_isMulLeftInvariant {G : Type u_1} [MeasurableSpace G] [TopologicalSpace G] [BorelSpace G] [Group G] [TopologicalGroup G] [WeaklyLocallyCompactSpace G] [NoncompactSpace G] (μ : MeasureTheory.Measure G) [μ.IsOpenPosMeasure] [μ.IsMulLeftInvariant] : μ Set.univ = ⊤

In a noncompact locally compact group, a left-invariant measure which is positive on open sets has infinite mass.

@[simp]

theorem MeasureTheory.measure_univ_of_isAddLeftInvariant {G : Type u_1} [MeasurableSpace G] [TopologicalSpace G] [BorelSpace G] [AddGroup G] [TopologicalAddGroup G] [WeaklyLocallyCompactSpace G] [NoncompactSpace G] (μ : MeasureTheory.Measure G) [μ.IsOpenPosMeasure] [μ.IsAddLeftInvariant] : μ Set.univ = ⊤

In a noncompact locally compact additive group, a left-invariant measure which is positive on open sets has infinite mass.

theorem MeasurableSet.mul_closure_one_eq {G : Type u_1} [MeasurableSpace G] [TopologicalSpace G] [BorelSpace G] [Group G] [TopologicalGroup G] {s : Set G} (hs : MeasurableSet s) : s * closure {1} = s

theorem MeasurableSet.add_closure_zero_eq {G : Type u_1} [MeasurableSpace G] [TopologicalSpace G] [BorelSpace G] [AddGroup G] [TopologicalAddGroup G] {s : Set G} (hs : MeasurableSet s) : s + closure {0} = s

@[deprecated IsCompact.closure_subset_measurableSet]

theorem IsCompact.closure_subset_of_measurableSet_of_group {G : Type u_1} [MeasurableSpace G] [TopologicalSpace G] [BorelSpace G] [Group G] [TopologicalGroup G] {k : Set G} {s : Set G} (hk : IsCompact k) (hs : MeasurableSet s) (h : k ⊆ s) : closure k ⊆ s

If a compact set is included in a measurable set, then so is its closure.

@[deprecated IsCompact.closure_subset_measurableSet]

theorem IsCompact.closure_subset_of_measurableSet_of_addGroup {G : Type u_1} [MeasurableSpace G] [TopologicalSpace G] [BorelSpace G] [AddGroup G] [TopologicalAddGroup G] {k : Set G} {s : Set G} (hk : IsCompact k) (hs : MeasurableSet s) (h : k ⊆ s) : closure k ⊆ s

@[simp]

theorem MeasureTheory.measure_mul_closure_one {G : Type u_1} [MeasurableSpace G] [TopologicalSpace G] [BorelSpace G] [Group G] [TopologicalGroup G] (s : Set G) (μ : MeasureTheory.Measure G) : μ (s * closure {1}) = μ s

@[simp]

theorem MeasureTheory.measure_add_closure_zero {G : Type u_1} [MeasurableSpace G] [TopologicalSpace G] [BorelSpace G] [AddGroup G] [TopologicalAddGroup G] (s : Set G) (μ : MeasureTheory.Measure G) : μ (s + closure {0}) = μ s

@[deprecated IsCompact.measure_closure]

theorem IsCompact.measure_closure_eq_of_group {G : Type u_1} [MeasurableSpace G] [TopologicalSpace G] [BorelSpace G] [Group G] [TopologicalGroup G] {k : Set G} (hk : IsCompact k) (μ : MeasureTheory.Measure G) : μ (closure k) = μ k

@[deprecated IsCompact.measure_closure]

theorem IsCompact.measure_closure_eq_of_addGroup {G : Type u_1} [MeasurableSpace G] [TopologicalSpace G] [BorelSpace G] [AddGroup G] [TopologicalAddGroup G] {k : Set G} (hk : IsCompact k) (μ : MeasureTheory.Measure G) : μ (closure k) = μ k

theorem MeasureTheory.innerRegularWRT_isCompact_isClosed_measure_ne_top_of_group {G : Type u_1} [MeasurableSpace G] [TopologicalSpace G] [BorelSpace G] {μ : MeasureTheory.Measure G} [Group G] [TopologicalGroup G] [h : μ.InnerRegularCompactLTTop] : μ.InnerRegularWRT (fun (s : Set G) => IsCompact s ∧ IsClosed s) fun (s : Set G) => MeasurableSet s ∧ μ s ≠ ⊤

theorem MeasureTheory.innerRegularWRT_isCompact_isClosed_measure_ne_top_of_addGroup {G : Type u_1} [MeasurableSpace G] [TopologicalSpace G] [BorelSpace G] {μ : MeasureTheory.Measure G} [AddGroup G] [TopologicalAddGroup G] [h : μ.InnerRegularCompactLTTop] : μ.InnerRegularWRT (fun (s : Set G) => IsCompact s ∧ IsClosed s) fun (s : Set G) => MeasurableSet s ∧ μ s ≠ ⊤

@[instance 100]

instance MeasureTheory.IsMulLeftInvariant.isMulRightInvariant {G : Type u_1} [MeasurableSpace G] [CommSemigroup G] {μ : MeasureTheory.Measure G} [μ.IsMulLeftInvariant] : μ.IsMulRightInvariant

In an abelian group every left invariant measure is also right-invariant. We don't declare the converse as an instance, since that would loop type-class inference, and we use IsMulLeftInvariant as the default hypothesis in abelian groups.

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@[instance 100]

instance MeasureTheory.IsAddLeftInvariant.isAddRightInvariant {G : Type u_1} [MeasurableSpace G] [AddCommSemigroup G] {μ : MeasureTheory.Measure G} [μ.IsAddLeftInvariant] : μ.IsAddRightInvariant

In an abelian additive group every left invariant measure is also right-invariant. We don't declare the converse as an instance, since that would loop type-class inference, and we use IsAddLeftInvariant as the default hypothesis in abelian groups.

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class MeasureTheory.Measure.IsAddHaarMeasure {G : Type u_3} [AddGroup G] [TopologicalSpace G] [MeasurableSpace G] (μ : MeasureTheory.Measure G) extends MeasureTheory.IsFiniteMeasureOnCompacts , MeasureTheory.Measure.IsAddLeftInvariant , MeasureTheory.Measure.IsOpenPosMeasure : Prop

A measure on an additive group is an additive Haar measure if it is left-invariant, and gives finite mass to compact sets and positive mass to open sets.

Textbooks generally require an additional regularity assumption to ensure nice behavior on arbitrary locally compact groups. Use [IsAddHaarMeasure μ] [Regular μ] or [IsAddHaarMeasure μ] [InnerRegular μ] in these situations. Note that a Haar measure in our sense is automatically regular and inner regular on second countable locally compact groups, as checked just below this definition.

class MeasureTheory.Measure.IsHaarMeasure {G : Type u_3} [Group G] [TopologicalSpace G] [MeasurableSpace G] (μ : MeasureTheory.Measure G) extends MeasureTheory.IsFiniteMeasureOnCompacts , MeasureTheory.Measure.IsMulLeftInvariant , MeasureTheory.Measure.IsOpenPosMeasure : Prop

A measure on a group is a Haar measure if it is left-invariant, and gives finite mass to compact sets and positive mass to open sets.

Textbooks generally require an additional regularity assumption to ensure nice behavior on arbitrary locally compact groups. Use [IsHaarMeasure μ] [Regular μ] or [IsHaarMeasure μ] [InnerRegular μ] in these situations. Note that a Haar measure in our sense is automatically regular and inner regular on second countable locally compact groups, as checked just below this definition.

@[simp]

theorem MeasureTheory.Measure.haar_singleton {G : Type u_1} [MeasurableSpace G] [Group G] [TopologicalSpace G] (μ : MeasureTheory.Measure G) [μ.IsHaarMeasure] [TopologicalGroup G] [BorelSpace G] (g : G) : μ {g} = μ {1}

@[simp]

theorem MeasureTheory.Measure.addHaar_singleton {G : Type u_1} [MeasurableSpace G] [AddGroup G] [TopologicalSpace G] (μ : MeasureTheory.Measure G) [μ.IsAddHaarMeasure] [TopologicalAddGroup G] [BorelSpace G] (g : G) : μ {g} = μ {0}

theorem MeasureTheory.Measure.IsHaarMeasure.smul {G : Type u_1} [MeasurableSpace G] [Group G] [TopologicalSpace G] (μ : MeasureTheory.Measure G) [μ.IsHaarMeasure] {c : ENNReal} (cpos : c ≠ 0) (ctop : c ≠ ⊤) : (c • μ).IsHaarMeasure

theorem MeasureTheory.Measure.IsAddHaarMeasure.smul {G : Type u_1} [MeasurableSpace G] [AddGroup G] [TopologicalSpace G] (μ : MeasureTheory.Measure G) [μ.IsAddHaarMeasure] {c : ENNReal} (cpos : c ≠ 0) (ctop : c ≠ ⊤) : (c • μ).IsAddHaarMeasure

theorem MeasureTheory.Measure.isHaarMeasure_of_isCompact_nonempty_interior {G : Type u_1} [MeasurableSpace G] [Group G] [TopologicalSpace G] [TopologicalGroup G] [BorelSpace G] (μ : MeasureTheory.Measure G) [μ.IsMulLeftInvariant] (K : Set G) (hK : IsCompact K) (h'K : (interior K).Nonempty) (h : μ K ≠ 0) (h' : μ K ≠ ⊤) : μ.IsHaarMeasure

If a left-invariant measure gives positive mass to some compact set with nonempty interior, then it is a Haar measure.

theorem MeasureTheory.Measure.isAddHaarMeasure_of_isCompact_nonempty_interior {G : Type u_1} [MeasurableSpace G] [AddGroup G] [TopologicalSpace G] [TopologicalAddGroup G] [BorelSpace G] (μ : MeasureTheory.Measure G) [μ.IsAddLeftInvariant] (K : Set G) (hK : IsCompact K) (h'K : (interior K).Nonempty) (h : μ K ≠ 0) (h' : μ K ≠ ⊤) : μ.IsAddHaarMeasure

If a left-invariant measure gives positive mass to some compact set with nonempty interior, then it is an additive Haar measure.

theorem MeasureTheory.Measure.isHaarMeasure_map {G : Type u_1} [MeasurableSpace G] [Group G] [TopologicalSpace G] (μ : MeasureTheory.Measure G) [μ.IsHaarMeasure] [BorelSpace G] [TopologicalGroup G] {H : Type u_3} [Group H] [TopologicalSpace H] [MeasurableSpace H] [BorelSpace H] [TopologicalGroup H] (f : G →* H) (hf : Continuous ⇑f) (h_surj : Function.Surjective ⇑f) (h_prop : Filter.Tendsto (⇑f) (Filter.cocompact G) (Filter.cocompact H)) : (MeasureTheory.Measure.map (⇑f) μ).IsHaarMeasure

The image of a Haar measure under a continuous surjective proper group homomorphism is again a Haar measure. See also MulEquiv.isHaarMeasure_map.

theorem MeasureTheory.Measure.isAddHaarMeasure_map {G : Type u_1} [MeasurableSpace G] [AddGroup G] [TopologicalSpace G] (μ : MeasureTheory.Measure G) [μ.IsAddHaarMeasure] [BorelSpace G] [TopologicalAddGroup G] {H : Type u_3} [AddGroup H] [TopologicalSpace H] [MeasurableSpace H] [BorelSpace H] [TopologicalAddGroup H] (f : G →+ H) (hf : Continuous ⇑f) (h_surj : Function.Surjective ⇑f) (h_prop : Filter.Tendsto (⇑f) (Filter.cocompact G) (Filter.cocompact H)) : (MeasureTheory.Measure.map (⇑f) μ).IsAddHaarMeasure

The image of an additive Haar measure under a continuous surjective proper additive group homomorphism is again an additive Haar measure. See also AddEquiv.isAddHaarMeasure_map.

theorem MeasureTheory.Measure.isHaarMeasure_map_of_isFiniteMeasure {G : Type u_1} [MeasurableSpace G] [Group G] [TopologicalSpace G] (μ : MeasureTheory.Measure G) [μ.IsHaarMeasure] [BorelSpace G] [TopologicalGroup G] {H : Type u_3} [Group H] [TopologicalSpace H] [MeasurableSpace H] [BorelSpace H] [TopologicalGroup H] [MeasureTheory.IsFiniteMeasure μ] (f : G →* H) (hf : Continuous ⇑f) (h_surj : Function.Surjective ⇑f) : (MeasureTheory.Measure.map (⇑f) μ).IsHaarMeasure

The image of a finite Haar measure under a continuous surjective group homomorphism is again a Haar measure. See also isHaarMeasure_map.

theorem MeasureTheory.Measure.isAddHaarMeasure_map_of_isFiniteMeasure {G : Type u_1} [MeasurableSpace G] [AddGroup G] [TopologicalSpace G] (μ : MeasureTheory.Measure G) [μ.IsAddHaarMeasure] [BorelSpace G] [TopologicalAddGroup G] {H : Type u_3} [AddGroup H] [TopologicalSpace H] [MeasurableSpace H] [BorelSpace H] [TopologicalAddGroup H] [MeasureTheory.IsFiniteMeasure μ] (f : G →+ H) (hf : Continuous ⇑f) (h_surj : Function.Surjective ⇑f) : (MeasureTheory.Measure.map (⇑f) μ).IsAddHaarMeasure

The image of a finite additive Haar measure under a continuous surjective additive group homomorphism is again an additive Haar measure. See also isAddHaarMeasure_map.

instance MeasureTheory.Measure.isHaarMeasure_map_smul {G : Type u_1} [MeasurableSpace G] [Group G] [TopologicalSpace G] (μ : MeasureTheory.Measure G) [μ.IsHaarMeasure] {α : Type u_3} [BorelSpace G] [TopologicalGroup G] [Group α] [MulAction α G] [SMulCommClass α G G] [MeasurableSpace α] [MeasurableSMul α G] [ContinuousConstSMul α G] (a : α) : (MeasureTheory.Measure.map (fun (x : G) => a • x) μ).IsHaarMeasure

The image of a Haar measure under map of a left action is again a Haar measure.

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instance MeasureTheory.Measure.isAddHaarMeasure_map_vadd {G : Type u_1} [MeasurableSpace G] [AddGroup G] [TopologicalSpace G] (μ : MeasureTheory.Measure G) [μ.IsAddHaarMeasure] {α : Type u_3} [BorelSpace G] [TopologicalAddGroup G] [AddGroup α] [AddAction α G] [VAddCommClass α G G] [MeasurableSpace α] [MeasurableVAdd α G] [ContinuousConstVAdd α G] (a : α) : (MeasureTheory.Measure.map (fun (x : G) => a +ᵥ x) μ).IsAddHaarMeasure

The image of a Haar measure under map of a left additive action is again a Haar measure

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instance MeasureTheory.Measure.isHaarMeasure_map_mul_right {G : Type u_1} [MeasurableSpace G] [Group G] [TopologicalSpace G] (μ : MeasureTheory.Measure G) [μ.IsHaarMeasure] [BorelSpace G] [TopologicalGroup G] (g : G) : (MeasureTheory.Measure.map (fun (x : G) => x * g) μ).IsHaarMeasure

The image of a Haar measure under right multiplication is again a Haar measure.

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instance MeasureTheory.Measure.isHaarMeasure_map_add_right {G : Type u_1} [MeasurableSpace G] [AddGroup G] [TopologicalSpace G] (μ : MeasureTheory.Measure G) [μ.IsAddHaarMeasure] [BorelSpace G] [TopologicalAddGroup G] (g : G) : (MeasureTheory.Measure.map (fun (x : G) => x + g) μ).IsAddHaarMeasure

The image of a Haar measure under right addition is again a Haar measure.

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theorem MulEquiv.isHaarMeasure_map {G : Type u_1} [MeasurableSpace G] [Group G] [TopologicalSpace G] (μ : MeasureTheory.Measure G) [μ.IsHaarMeasure] [BorelSpace G] [TopologicalGroup G] {H : Type u_3} [Group H] [TopologicalSpace H] [MeasurableSpace H] [BorelSpace H] [TopologicalGroup H] (e : G ≃* H) (he : Continuous ⇑e) (hesymm : Continuous ⇑e.symm) : (MeasureTheory.Measure.map (⇑e) μ).IsHaarMeasure

A convenience wrapper for MeasureTheory.Measure.isHaarMeasure_map.

theorem AddEquiv.isAddHaarMeasure_map {G : Type u_1} [MeasurableSpace G] [AddGroup G] [TopologicalSpace G] (μ : MeasureTheory.Measure G) [μ.IsAddHaarMeasure] [BorelSpace G] [TopologicalAddGroup G] {H : Type u_3} [AddGroup H] [TopologicalSpace H] [MeasurableSpace H] [BorelSpace H] [TopologicalAddGroup H] (e : G ≃+ H) (he : Continuous ⇑e) (hesymm : Continuous ⇑e.symm) : (MeasureTheory.Measure.map (⇑e) μ).IsAddHaarMeasure

A convenience wrapper for MeasureTheory.Measure.isAddHaarMeasure_map.

instance ContinuousLinearEquiv.isAddHaarMeasure_map {E : Type u_3} {F : Type u_4} {R : Type u_5} {S : Type u_6} [Semiring R] [Semiring S] [AddCommGroup E] [Module R E] [AddCommGroup F] [Module S F] [TopologicalSpace E] [TopologicalAddGroup E] [TopologicalSpace F] [TopologicalAddGroup F] {σ : R →+* S} {σ' : S →+* R} [RingHomInvPair σ σ'] [RingHomInvPair σ' σ] [MeasurableSpace E] [BorelSpace E] [MeasurableSpace F] [BorelSpace F] (L : E ≃SL[σ] F) (μ : MeasureTheory.Measure E) [μ.IsAddHaarMeasure] : (MeasureTheory.Measure.map (⇑L) μ).IsAddHaarMeasure

A convenience wrapper for MeasureTheory.Measure.isAddHaarMeasure_map`.

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@[instance 100]

instance MeasureTheory.Measure.IsHaarMeasure.sigmaFinite {G : Type u_1} [MeasurableSpace G] [Group G] [TopologicalSpace G] (μ : MeasureTheory.Measure G) [μ.IsHaarMeasure] [SigmaCompactSpace G] : MeasureTheory.SigmaFinite μ

A Haar measure on a σ-compact space is σ-finite.

See Note [lower instance priority]

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@[instance 100]

instance MeasureTheory.Measure.IsAddHaarMeasure.sigmaFinite {G : Type u_1} [MeasurableSpace G] [AddGroup G] [TopologicalSpace G] (μ : MeasureTheory.Measure G) [μ.IsAddHaarMeasure] [SigmaCompactSpace G] : MeasureTheory.SigmaFinite μ

A Haar measure on a σ-compact space is σ-finite.

See Note [lower instance priority]

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instance MeasureTheory.Measure.prod.instIsHaarMeasure {G : Type u_3} [Group G] [TopologicalSpace G] : ∀ {x : MeasurableSpace G} {H : Type u_4} [inst : Group H] [inst_1 : TopologicalSpace H] {x_1 : MeasurableSpace H} (μ : MeasureTheory.Measure G) (ν : MeasureTheory.Measure H) [inst_2 : μ.IsHaarMeasure] [inst_3 : ν.IsHaarMeasure] [inst_4 : MeasureTheory.SFinite μ] [inst_5 : MeasureTheory.SFinite ν] [inst_6 : MeasurableMul G] [inst_7 : MeasurableMul H], (μ.prod ν).IsHaarMeasure

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instance MeasureTheory.Measure.prod.instIsAddHaarMeasure {G : Type u_3} [AddGroup G] [TopologicalSpace G] : ∀ {x : MeasurableSpace G} {H : Type u_4} [inst : AddGroup H] [inst_1 : TopologicalSpace H] {x_1 : MeasurableSpace H} (μ : MeasureTheory.Measure G) (ν : MeasureTheory.Measure H) [inst_2 : μ.IsAddHaarMeasure] [inst_3 : ν.IsAddHaarMeasure] [inst_4 : MeasureTheory.SFinite μ] [inst_5 : MeasureTheory.SFinite ν] [inst_6 : MeasurableAdd G] [inst_7 : MeasurableAdd H], (μ.prod ν).IsAddHaarMeasure

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@[instance 100]

instance MeasureTheory.Measure.IsHaarMeasure.noAtoms {G : Type u_1} [MeasurableSpace G] [Group G] [TopologicalSpace G] [TopologicalGroup G] [BorelSpace G] [T1Space G] [WeaklyLocallyCompactSpace G] [(nhdsWithin 1 {1}ᶜ).NeBot] (μ : MeasureTheory.Measure G) [μ.IsHaarMeasure] : MeasureTheory.NoAtoms μ

If the neutral element of a group is not isolated, then a Haar measure on this group has no atoms.

The additive version of this instance applies in particular to show that an additive Haar measure on a nontrivial finite-dimensional real vector space has no atom.

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@[instance 100]

instance MeasureTheory.Measure.IsAddHaarMeasure.noAtoms {G : Type u_1} [MeasurableSpace G] [AddGroup G] [TopologicalSpace G] [TopologicalAddGroup G] [BorelSpace G] [T1Space G] [WeaklyLocallyCompactSpace G] [(nhdsWithin 0 {0}ᶜ).NeBot] (μ : MeasureTheory.Measure G) [μ.IsAddHaarMeasure] : MeasureTheory.NoAtoms μ

If the zero element of an additive group is not isolated, then an additive Haar measure on this group has no atoms.

This applies in particular to show that an additive Haar measure on a nontrivial finite-dimensional real vector space has no atom.

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