Action of DomMulAct and DomAddAct on α →ₘ[μ] β

If M acts on α by measure-preserving transformations, then Mᵈᵐᵃ acts on α →ₘ[μ] β by sending each function f to f ∘ (DomMulAct.mk.symm c • ·). We define this action and basic instances about it.

Implementation notes

In fact, it suffices to require that (c • ·) is only quasi-measure-preserving but we do not have a typeclass for quasi-measure-preserving actions yet.

Keywords

@[instance_reducible]

instance DomMulAct.instSMulAEEqFun {M : Type u_1} {α : Type u_2} {β : Type u_3} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace β] [SMul M α] [MeasurableConstSMul M α] [MeasureTheory.SMulInvariantMeasure M α μ] : SMul Mᵈᵐᵃ (α →ₘ[μ] β)

Equations

• DomMulAct.instSMulAEEqFun = { smul := fun (c : Mᵈᵐᵃ) (f : α →ₘ[μ] β) => f.compMeasurePreserving (fun (x : α) => DomMulAct.mk.symm c • x) ⋯ }

@[instance_reducible]

instance DomAddAct.instVAddAEEqFun {M : Type u_1} {α : Type u_2} {β : Type u_3} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace β] [VAdd M α] [MeasurableConstVAdd M α] [MeasureTheory.VAddInvariantMeasure M α μ] : VAdd Mᵈᵃᵃ (α →ₘ[μ] β)

Equations

• DomAddAct.instVAddAEEqFun = { vadd := fun (c : Mᵈᵃᵃ) (f : α →ₘ[μ] β) => f.compMeasurePreserving (fun (x : α) => DomAddAct.mk.symm c +ᵥ x) ⋯ }

theorem DomMulAct.smul_aeeqFun_aeeq {M : Type u_1} {α : Type u_2} {β : Type u_3} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace β] [SMul M α] [MeasurableConstSMul M α] [MeasureTheory.SMulInvariantMeasure M α μ] (c : Mᵈᵐᵃ) (f : α →ₘ[μ] β) : ↑(c • f) =ᵐ[μ] fun (x : α) => ↑f (mk.symm c • x)

theorem DomAddAct.vadd_aeeqFun_aeeq {M : Type u_1} {α : Type u_2} {β : Type u_3} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace β] [VAdd M α] [MeasurableConstVAdd M α] [MeasureTheory.VAddInvariantMeasure M α μ] (c : Mᵈᵃᵃ) (f : α →ₘ[μ] β) : ↑(c +ᵥ f) =ᵐ[μ] fun (x : α) => ↑f (mk.symm c +ᵥ x)

@[simp]

theorem DomMulAct.mk_smul_mk_aeeqFun {M : Type u_3} {α : Type u_1} {β : Type u_2} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace β] [SMul M α] [MeasurableConstSMul M α] [MeasureTheory.SMulInvariantMeasure M α μ] (c : M) (f : α → β) (hf : MeasureTheory.AEStronglyMeasurable f μ) : mk c • MeasureTheory.AEEqFun.mk f hf = MeasureTheory.AEEqFun.mk (fun (x : α) => f (c • x)) ⋯

@[simp]

theorem DomAddAct.mk_vadd_mk_aeeqFun {M : Type u_3} {α : Type u_1} {β : Type u_2} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace β] [VAdd M α] [MeasurableConstVAdd M α] [MeasureTheory.VAddInvariantMeasure M α μ] (c : M) (f : α → β) (hf : MeasureTheory.AEStronglyMeasurable f μ) : mk c +ᵥ MeasureTheory.AEEqFun.mk f hf = MeasureTheory.AEEqFun.mk (fun (x : α) => f (c +ᵥ x)) ⋯

@[simp]

theorem DomMulAct.smul_aeeqFun_const {M : Type u_1} {α : Type u_2} {β : Type u_3} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace β] [SMul M α] [MeasurableConstSMul M α] [MeasureTheory.SMulInvariantMeasure M α μ] (c : Mᵈᵐᵃ) (b : β) : c • MeasureTheory.AEEqFun.const α b = MeasureTheory.AEEqFun.const α b

@[simp]

theorem DomAddAct.vadd_aeeqFun_const {M : Type u_1} {α : Type u_2} {β : Type u_3} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace β] [VAdd M α] [MeasurableConstVAdd M α] [MeasureTheory.VAddInvariantMeasure M α μ] (c : Mᵈᵃᵃ) (b : β) : c +ᵥ MeasureTheory.AEEqFun.const α b = MeasureTheory.AEEqFun.const α b

instance DomMulAct.instSMulCommClassAEEqFun {M : Type u_3} {N : Type u_1} {α : Type u_4} {β : Type u_2} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace β] [SMul M α] [MeasurableConstSMul M α] [MeasureTheory.SMulInvariantMeasure M α μ] [SMul N β] [ContinuousConstSMul N β] : SMulCommClass Mᵈᵐᵃ N (α →ₘ[μ] β)

instance DomMulAct.instSMulCommClassAEEqFun_1 {M : Type u_3} {N : Type u_1} {α : Type u_4} {β : Type u_2} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace β] [SMul M α] [MeasurableConstSMul M α] [MeasureTheory.SMulInvariantMeasure M α μ] [SMul N β] [ContinuousConstSMul N β] : SMulCommClass N Mᵈᵐᵃ (α →ₘ[μ] β)

instance DomMulAct.instSMulCommClassAEEqFun_2 {M : Type u_3} {N : Type u_1} {α : Type u_2} {β : Type u_4} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace β] [SMul M α] [MeasurableConstSMul M α] [MeasureTheory.SMulInvariantMeasure M α μ] [SMul N α] [MeasurableConstSMul N α] [MeasureTheory.SMulInvariantMeasure N α μ] [SMulCommClass M N α] : SMulCommClass Mᵈᵐᵃ Nᵈᵐᵃ (α →ₘ[μ] β)

instance DomAddAct.instVAddCommClassAEEqFun_2 {M : Type u_3} {N : Type u_1} {α : Type u_2} {β : Type u_4} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace β] [VAdd M α] [MeasurableConstVAdd M α] [MeasureTheory.VAddInvariantMeasure M α μ] [VAdd N α] [MeasurableConstVAdd N α] [MeasureTheory.VAddInvariantMeasure N α μ] [VAddCommClass M N α] : VAddCommClass Mᵈᵃᵃ Nᵈᵃᵃ (α →ₘ[μ] β)

@[instance_reducible]

instance DomMulAct.instSMulZeroClassAEEqFun {M : Type u_1} {α : Type u_2} {β : Type u_3} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace β] [SMul M α] [MeasurableConstSMul M α] [MeasureTheory.SMulInvariantMeasure M α μ] [Zero β] : SMulZeroClass Mᵈᵐᵃ (α →ₘ[μ] β)

Equations

• DomMulAct.instSMulZeroClassAEEqFun = { toSMul := DomMulAct.instSMulAEEqFun, smul_zero := ⋯ }

@[instance_reducible]

instance DomMulAct.instDistribSMulAEEqFun {M : Type u_1} {α : Type u_2} {β : Type u_3} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace β] [SMul M α] [MeasurableConstSMul M α] [MeasureTheory.SMulInvariantMeasure M α μ] [AddMonoid β] [ContinuousAdd β] : DistribSMul Mᵈᵐᵃ (α →ₘ[μ] β)

Equations

• DomMulAct.instDistribSMulAEEqFun = { toSMulZeroClass := DomMulAct.instSMulZeroClassAEEqFun, smul_add := ⋯ }

@[instance_reducible]

instance DomMulAct.instMulActionAEEqFun {M : Type u_1} {α : Type u_2} {β : Type u_3} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace β] [Monoid M] [MulAction M α] [MeasurableConstSMul M α] [MeasureTheory.SMulInvariantMeasure M α μ] : MulAction Mᵈᵐᵃ (α →ₘ[μ] β)

Equations

• DomMulAct.instMulActionAEEqFun = { toSMul := DomMulAct.instSMulAEEqFun, mul_smul := ⋯, one_smul := ⋯ }

@[instance_reducible]

instance DomAddAct.instAddActionAEEqFun {M : Type u_1} {α : Type u_2} {β : Type u_3} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace β] [AddMonoid M] [AddAction M α] [MeasurableConstVAdd M α] [MeasureTheory.VAddInvariantMeasure M α μ] : AddAction Mᵈᵃᵃ (α →ₘ[μ] β)

Equations

• DomAddAct.instAddActionAEEqFun = { toVAdd := DomAddAct.instVAddAEEqFun, add_vadd := ⋯, zero_vadd := ⋯ }

@[instance_reducible]

instance DomMulAct.instMulDistribMulActionAEEqFun {M : Type u_1} {α : Type u_2} {β : Type u_3} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace β] [Monoid M] [MulAction M α] [MeasurableConstSMul M α] [MeasureTheory.SMulInvariantMeasure M α μ] [Monoid β] [ContinuousMul β] : MulDistribMulAction Mᵈᵐᵃ (α →ₘ[μ] β)

Equations

• DomMulAct.instMulDistribMulActionAEEqFun = { toMulAction := DomMulAct.instMulActionAEEqFun, smul_one := ⋯, smul_mul := ⋯ }

@[instance_reducible]

instance DomMulAct.instDistribMulActionAEEqFun {M : Type u_1} {α : Type u_2} {β : Type u_3} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace β] [Monoid M] [MulAction M α] [MeasurableConstSMul M α] [MeasureTheory.SMulInvariantMeasure M α μ] [AddMonoid β] [ContinuousAdd β] : DistribMulAction Mᵈᵐᵃ (α →ₘ[μ] β)

Equations

• DomMulAct.instDistribMulActionAEEqFun = { toMulAction := DomMulAct.instMulActionAEEqFun, smul_zero := ⋯, smul_add := ⋯ }