Measurable embeddings and equivalences

A measurable equivalence between measurable spaces is an equivalence which respects the σ-algebras, that is, for which both directions of the equivalence are measurable functions.

Main definitions

• MeasurableEmbedding: a map f : α → β is called a measurable embedding if it is injective, measurable, and sends measurable sets to measurable sets.
• MeasurableEquiv: an equivalence α ≃ β is a measurable equivalence if its forward and inverse functions are measurable.

We prove a multitude of elementary lemmas about these, and one more substantial theorem:

• MeasurableEmbedding.schroederBernstein: the measurable Schröder-Bernstein Theorem: given measurable embeddings α → β and β → α, we can find a measurable equivalence α ≃ᵐ β.

Notation

• We write α ≃ᵐ β for measurable equivalences between the measurable spaces α and β. This should not be confused with ≃ₘ which is used for diffeomorphisms between manifolds.

Tags

measurable equivalence, measurable embedding

structure MeasurableEmbedding {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] (f : α → β) : Prop

A map f : α → β is called a measurable embedding if it is injective, measurable, and sends measurable sets to measurable sets. The latter assumption can be replaced with “f has measurable inverse g : Set.range f → α”, see MeasurableEmbedding.measurable_rangeSplitting, MeasurableEmbedding.of_measurable_inverse_range, and MeasurableEmbedding.of_measurable_inverse.

One more interpretation: f is a measurable embedding if it defines a measurable equivalence to its range and the range is a measurable set. One implication is formalized as MeasurableEmbedding.equivRange; the other one follows from MeasurableEquiv.measurableEmbedding, MeasurableEmbedding.subtype_coe, and MeasurableEmbedding.comp.

• injective : Function.Injective f

  A measurable embedding is injective.

• measurable : Measurable f

  A measurable embedding is a measurable function.

• measurableSet_image' : ∀ ⦃s : Set α⦄, MeasurableSet s → MeasurableSet (f '' s)

  The image of a measurable set under a measurable embedding is a measurable set.

theorem MeasurableEmbedding.injective {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] {f : α → β} (self : MeasurableEmbedding f) : Function.Injective f

A measurable embedding is injective.

theorem MeasurableEmbedding.measurable {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] {f : α → β} (self : MeasurableEmbedding f) : Measurable f

A measurable embedding is a measurable function.

theorem MeasurableEmbedding.measurableSet_image' {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] {f : α → β} (self : MeasurableEmbedding f) ⦃s : Set α⦄ : MeasurableSet s → MeasurableSet (f '' s)

The image of a measurable set under a measurable embedding is a measurable set.

theorem MeasurableEmbedding.measurableSet_image {α : Type u_1} {β : Type u_2} {s : Set α} {mα : MeasurableSpace α} [MeasurableSpace β] {f : α → β} (hf : MeasurableEmbedding f) : MeasurableSet (f '' s) ↔ MeasurableSet s

theorem MeasurableEmbedding.id {α : Type u_1} {mα : MeasurableSpace α} : MeasurableEmbedding id

theorem MeasurableEmbedding.comp {α : Type u_1} {β : Type u_2} {γ : Type u_3} {mα : MeasurableSpace α} [MeasurableSpace β] [MeasurableSpace γ] {f : α → β} {g : β → γ} (hg : MeasurableEmbedding g) (hf : MeasurableEmbedding f) : MeasurableEmbedding (g ∘ f)

theorem MeasurableEmbedding.subtype_coe {α : Type u_1} {s : Set α} {mα : MeasurableSpace α} (hs : MeasurableSet s) : MeasurableEmbedding Subtype.val

theorem MeasurableEmbedding.measurableSet_range {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} [MeasurableSpace β] {f : α → β} (hf : MeasurableEmbedding f) : MeasurableSet (Set.range f)

theorem MeasurableEmbedding.measurableSet_preimage {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} [MeasurableSpace β] {f : α → β} (hf : MeasurableEmbedding f) {s : Set β} : MeasurableSet (f ⁻¹' s) ↔ MeasurableSet (s ∩ Set.range f)

theorem MeasurableEmbedding.measurable_rangeSplitting {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} [MeasurableSpace β] {f : α → β} (hf : MeasurableEmbedding f) : Measurable (Set.rangeSplitting f)

theorem MeasurableEmbedding.measurable_extend {α : Type u_1} {β : Type u_2} {γ : Type u_3} {mα : MeasurableSpace α} [MeasurableSpace β] [MeasurableSpace γ] {f : α → β} (hf : MeasurableEmbedding f) {g : α → γ} {g' : β → γ} (hg : Measurable g) (hg' : Measurable g') : Measurable (Function.extend f g g')

theorem MeasurableEmbedding.exists_measurable_extend {α : Type u_1} {β : Type u_2} {γ : Type u_3} {mα : MeasurableSpace α} [MeasurableSpace β] [MeasurableSpace γ] {f : α → β} (hf : MeasurableEmbedding f) {g : α → γ} (hg : Measurable g) (hne : β → Nonempty γ) : ∃ (g' : β → γ), Measurable g' ∧ g' ∘ f = g

theorem MeasurableEmbedding.measurable_comp_iff {α : Type u_1} {β : Type u_2} {γ : Type u_3} {mα : MeasurableSpace α} [MeasurableSpace β] [MeasurableSpace γ] {f : α → β} {g : β → γ} (hg : MeasurableEmbedding g) : Measurable (g ∘ f) ↔ Measurable f

theorem MeasurableSet.of_union_range_cover {α : Type u_1} {α₁ : Type u_6} {α₂ : Type u_7} {mα : MeasurableSpace α} {mα₁ : MeasurableSpace α₁} {mα₂ : MeasurableSpace α₂} {i₁ : α₁ → α} {i₂ : α₂ → α} {s : Set α} (hi₁ : MeasurableEmbedding i₁) (hi₂ : MeasurableEmbedding i₂) (h : Set.univ ⊆ Set.range i₁ ∪ Set.range i₂) (hs₁ : MeasurableSet (i₁ ⁻¹' s)) (hs₂ : MeasurableSet (i₂ ⁻¹' s)) : MeasurableSet s

theorem MeasurableSet.of_union₃_range_cover {α : Type u_1} {α₁ : Type u_6} {α₂ : Type u_7} {α₃ : Type u_8} {mα : MeasurableSpace α} {mα₁ : MeasurableSpace α₁} {mα₂ : MeasurableSpace α₂} {mα₃ : MeasurableSpace α₃} {i₁ : α₁ → α} {i₂ : α₂ → α} {i₃ : α₃ → α} {s : Set α} (hi₁ : MeasurableEmbedding i₁) (hi₂ : MeasurableEmbedding i₂) (hi₃ : MeasurableEmbedding i₃) (h : Set.univ ⊆ Set.range i₁ ∪ Set.range i₂ ∪ Set.range i₃) (hs₁ : MeasurableSet (i₁ ⁻¹' s)) (hs₂ : MeasurableSet (i₂ ⁻¹' s)) (hs₃ : MeasurableSet (i₃ ⁻¹' s)) : MeasurableSet s

theorem Measurable.of_union_range_cover {α : Type u_1} {β : Type u_2} {α₁ : Type u_6} {α₂ : Type u_7} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {mα₁ : MeasurableSpace α₁} {mα₂ : MeasurableSpace α₂} {i₁ : α₁ → α} {i₂ : α₂ → α} {f : α → β} (hi₁ : MeasurableEmbedding i₁) (hi₂ : MeasurableEmbedding i₂) (h : Set.univ ⊆ Set.range i₁ ∪ Set.range i₂) (hf₁ : Measurable (f ∘ i₁)) (hf₂ : Measurable (f ∘ i₂)) : Measurable f

theorem Measurable.of_union₃_range_cover {α : Type u_1} {β : Type u_2} {α₁ : Type u_6} {α₂ : Type u_7} {α₃ : Type u_8} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {mα₁ : MeasurableSpace α₁} {mα₂ : MeasurableSpace α₂} {mα₃ : MeasurableSpace α₃} {i₁ : α₁ → α} {i₂ : α₂ → α} {i₃ : α₃ → α} {f : α → β} (hi₁ : MeasurableEmbedding i₁) (hi₂ : MeasurableEmbedding i₂) (hi₃ : MeasurableEmbedding i₃) (h : Set.univ ⊆ Set.range i₁ ∪ Set.range i₂ ∪ Set.range i₃) (hf₁ : Measurable (f ∘ i₁)) (hf₂ : Measurable (f ∘ i₂)) (hf₃ : Measurable (f ∘ i₃)) : Measurable f

theorem MeasurableSet.exists_measurable_proj {α : Type u_1} {s : Set α} : ∀ {x : MeasurableSpace α}, MeasurableSet s → s.Nonempty → ∃ (f : α → ↑s), Measurable f ∧ ∀ (x : ↑s), f ↑x = x

structure MeasurableEquiv (α : Type u_6) (β : Type u_7) [MeasurableSpace α] [MeasurableSpace β] extends Equiv : Type (max u_6 u_7)

Equivalences between measurable spaces. Main application is the simplification of measurability statements along measurable equivalences.

• toFun : α → β
• invFun : β → α
• left_inv : Function.LeftInverse self.invFun self.toFun
• right_inv : Function.RightInverse self.invFun self.toFun
• measurable_toFun : Measurable ⇑self.toEquiv

  The forward function of a measurable equivalence is measurable.

• measurable_invFun : Measurable ⇑self.symm

  The inverse function of a measurable equivalence is measurable.

theorem MeasurableEquiv.measurable_toFun {α : Type u_6} {β : Type u_7} [MeasurableSpace α] [MeasurableSpace β] (self : α ≃ᵐ β) : Measurable ⇑self.toEquiv

The forward function of a measurable equivalence is measurable.

theorem MeasurableEquiv.measurable_invFun {α : Type u_6} {β : Type u_7} [MeasurableSpace α] [MeasurableSpace β] (self : α ≃ᵐ β) : Measurable ⇑self.symm

The inverse function of a measurable equivalence is measurable.

def «term_≃ᵐ_» : Lean.TrailingParserDescr

Equivalences between measurable spaces. Main application is the simplification of measurability statements along measurable equivalences.

Equations

• «term_≃ᵐ_» = Lean.ParserDescr.trailingNode `«term_≃ᵐ_» 25 25 (Lean.ParserDescr.binary `andthen (Lean.ParserDescr.symbol " ≃ᵐ ") (Lean.ParserDescr.cat `term 26))

theorem MeasurableEquiv.toEquiv_injective {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] : Function.Injective MeasurableEquiv.toEquiv

instance MeasurableEquiv.instEquivLike {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] : EquivLike (α ≃ᵐ β) α β

Equations

• MeasurableEquiv.instEquivLike = { coe := fun (e : α ≃ᵐ β) => ⇑e.toEquiv, inv := fun (e : α ≃ᵐ β) => ⇑e.symm, left_inv := ⋯, right_inv := ⋯, coe_injective' := ⋯ }

@[simp]

theorem MeasurableEquiv.coe_toEquiv {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] (e : α ≃ᵐ β) : ⇑e.toEquiv = ⇑e

theorem MeasurableEquiv.measurable {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] (e : α ≃ᵐ β) : Measurable ⇑e

@[simp]

theorem MeasurableEquiv.coe_mk {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] (e : α ≃ β) (h1 : Measurable ⇑e) (h2 : Measurable ⇑e.symm) : ⇑{ toEquiv := e, measurable_toFun := h1, measurable_invFun := h2 } = ⇑e

def MeasurableEquiv.refl (α : Type u_6) [MeasurableSpace α] : α ≃ᵐ α

Any measurable space is equivalent to itself.

Equations

• MeasurableEquiv.refl α = { toEquiv := Equiv.refl α, measurable_toFun := ⋯, measurable_invFun := ⋯ }

instance MeasurableEquiv.instInhabited {α : Type u_1} [MeasurableSpace α] : Inhabited (α ≃ᵐ α)

Equations

• MeasurableEquiv.instInhabited = { default := MeasurableEquiv.refl α }

def MeasurableEquiv.trans {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ] (ab : α ≃ᵐ β) (bc : β ≃ᵐ γ) : α ≃ᵐ γ

The composition of equivalences between measurable spaces.

Equations

• ab.trans bc = { toEquiv := ab.trans bc.toEquiv, measurable_toFun := ⋯, measurable_invFun := ⋯ }

theorem MeasurableEquiv.coe_trans {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ] (ab : α ≃ᵐ β) (bc : β ≃ᵐ γ) : ⇑(ab.trans bc) = ⇑bc ∘ ⇑ab

def MeasurableEquiv.symm {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] (ab : α ≃ᵐ β) : β ≃ᵐ α

The inverse of an equivalence between measurable spaces.

Equations

• ab.symm = { toEquiv := ab.symm, measurable_toFun := ⋯, measurable_invFun := ⋯ }

@[simp]

theorem MeasurableEquiv.coe_toEquiv_symm {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] (e : α ≃ᵐ β) : ⇑e.symm = ⇑e.symm

def MeasurableEquiv.Simps.apply {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] (h : α ≃ᵐ β) : α → β

See Note [custom simps projection]. We need to specify this projection explicitly in this case, because it is a composition of multiple projections.

Equations

• MeasurableEquiv.Simps.apply h = ⇑h

def MeasurableEquiv.Simps.symm_apply {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] (h : α ≃ᵐ β) : β → α

See Note [custom simps projection]

Equations

• MeasurableEquiv.Simps.symm_apply h = ⇑h.symm

theorem MeasurableEquiv.ext {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] {e₁ : α ≃ᵐ β} {e₂ : α ≃ᵐ β} (h : ⇑e₁ = ⇑e₂) : e₁ = e₂

@[simp]

theorem MeasurableEquiv.symm_mk {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] (e : α ≃ β) (h1 : Measurable ⇑e) (h2 : Measurable ⇑e.symm) : { toEquiv := e, measurable_toFun := h1, measurable_invFun := h2 }.symm = { toEquiv := e.symm, measurable_toFun := h2, measurable_invFun := h1 }

@[simp]

theorem MeasurableEquiv.trans_apply {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ] (ab : α ≃ᵐ β) (bc : β ≃ᵐ γ) : ∀ (a : α), (ab.trans bc) a = bc (ab a)

@[simp]

theorem MeasurableEquiv.refl_toEquiv (α : Type u_6) [MeasurableSpace α] : (MeasurableEquiv.refl α).toEquiv = Equiv.refl α

@[simp]

theorem MeasurableEquiv.refl_apply (α : Type u_6) [MeasurableSpace α] (a : α) : (MeasurableEquiv.refl α) a = a

@[simp]

theorem MeasurableEquiv.trans_toEquiv {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ] (ab : α ≃ᵐ β) (bc : β ≃ᵐ γ) : (ab.trans bc).toEquiv = ab.trans bc.toEquiv

@[simp]

theorem MeasurableEquiv.symm_symm {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] (e : α ≃ᵐ β) : e.symm.symm = e

theorem MeasurableEquiv.symm_bijective {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] : Function.Bijective MeasurableEquiv.symm

@[simp]

theorem MeasurableEquiv.symm_refl (α : Type u_6) [MeasurableSpace α] : (MeasurableEquiv.refl α).symm = MeasurableEquiv.refl α

@[simp]

theorem MeasurableEquiv.symm_comp_self {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] (e : α ≃ᵐ β) : ⇑e.symm ∘ ⇑e = id

@[simp]

theorem MeasurableEquiv.self_comp_symm {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] (e : α ≃ᵐ β) : ⇑e ∘ ⇑e.symm = id

@[simp]

theorem MeasurableEquiv.apply_symm_apply {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] (e : α ≃ᵐ β) (y : β) : e (e.symm y) = y

@[simp]

theorem MeasurableEquiv.symm_apply_apply {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] (e : α ≃ᵐ β) (x : α) : e.symm (e x) = x

@[simp]

theorem MeasurableEquiv.symm_trans_self {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] (e : α ≃ᵐ β) : e.symm.trans e = MeasurableEquiv.refl β

@[simp]

theorem MeasurableEquiv.self_trans_symm {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] (e : α ≃ᵐ β) : e.trans e.symm = MeasurableEquiv.refl α

theorem MeasurableEquiv.surjective {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] (e : α ≃ᵐ β) : Function.Surjective ⇑e

theorem MeasurableEquiv.bijective {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] (e : α ≃ᵐ β) : Function.Bijective ⇑e

theorem MeasurableEquiv.injective {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] (e : α ≃ᵐ β) : Function.Injective ⇑e

@[simp]

theorem MeasurableEquiv.symm_preimage_preimage {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] (e : α ≃ᵐ β) (s : Set β) : ⇑e.symm ⁻¹' (⇑e ⁻¹' s) = s

theorem MeasurableEquiv.image_eq_preimage {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] (e : α ≃ᵐ β) (s : Set α) : ⇑e '' s = ⇑e.symm ⁻¹' s

theorem MeasurableEquiv.preimage_symm {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] (e : α ≃ᵐ β) (s : Set α) : ⇑e.symm ⁻¹' s = ⇑e '' s

theorem MeasurableEquiv.image_symm {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] (e : α ≃ᵐ β) (s : Set β) : ⇑e.symm '' s = ⇑e ⁻¹' s

theorem MeasurableEquiv.eq_image_iff_symm_image_eq {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] (e : α ≃ᵐ β) (s : Set β) (t : Set α) : s = ⇑e '' t ↔ ⇑e.symm '' s = t

@[simp]

theorem MeasurableEquiv.image_preimage {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] (e : α ≃ᵐ β) (s : Set β) : ⇑e '' (⇑e ⁻¹' s) = s

@[simp]

theorem MeasurableEquiv.preimage_image {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] (e : α ≃ᵐ β) (s : Set α) : ⇑e ⁻¹' (⇑e '' s) = s

@[simp]

theorem MeasurableEquiv.measurableSet_preimage {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] (e : α ≃ᵐ β) {s : Set β} : MeasurableSet (⇑e ⁻¹' s) ↔ MeasurableSet s

@[simp]

theorem MeasurableEquiv.measurableSet_image {α : Type u_1} {β : Type u_2} {s : Set α} [MeasurableSpace α] [MeasurableSpace β] (e : α ≃ᵐ β) : MeasurableSet (⇑e '' s) ↔ MeasurableSet s

@[simp]

theorem MeasurableEquiv.map_eq {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] (e : α ≃ᵐ β) : MeasurableSpace.map (⇑e) inst✝¹ = inst✝

theorem MeasurableEquiv.measurableEmbedding {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] (e : α ≃ᵐ β) : MeasurableEmbedding ⇑e

A measurable equivalence is a measurable embedding.

def MeasurableEquiv.cast {α : Type u_6} {β : Type u_6} [i₁ : MeasurableSpace α] [i₂ : MeasurableSpace β] (h : α = β) (hi : HEq i₁ i₂) : α ≃ᵐ β

Equal measurable spaces are equivalent.

Equations

• MeasurableEquiv.cast h hi = { toEquiv := Equiv.cast h, measurable_toFun := ⋯, measurable_invFun := ⋯ }

def MeasurableEquiv.ulift {α : Type u} [MeasurableSpace α] : ULift.{v, u} α ≃ᵐ α

Measurable equivalence between ULift α and α.

Equations

• MeasurableEquiv.ulift = { toEquiv := Equiv.ulift, measurable_toFun := ⋯, measurable_invFun := ⋯ }

theorem MeasurableEquiv.measurable_comp_iff {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ] {f : β → γ} (e : α ≃ᵐ β) : Measurable (f ∘ ⇑e) ↔ Measurable f

def MeasurableEquiv.ofUniqueOfUnique (α : Type u_6) (β : Type u_7) [MeasurableSpace α] [MeasurableSpace β] [Unique α] [Unique β] : α ≃ᵐ β

Any two types with unique elements are measurably equivalent.

Equations

• MeasurableEquiv.ofUniqueOfUnique α β = { toEquiv := Equiv.equivOfUnique α β, measurable_toFun := ⋯, measurable_invFun := ⋯ }

def MeasurableEquiv.prodCongr {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ] [MeasurableSpace δ] (ab : α ≃ᵐ β) (cd : γ ≃ᵐ δ) : α × γ ≃ᵐ β × δ

Products of equivalent measurable spaces are equivalent.

Equations

• ab.prodCongr cd = { toEquiv := ab.prodCongr cd.toEquiv, measurable_toFun := ⋯, measurable_invFun := ⋯ }

def MeasurableEquiv.prodComm {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] : α × β ≃ᵐ β × α

Products of measurable spaces are symmetric.

Equations

• MeasurableEquiv.prodComm = { toEquiv := Equiv.prodComm α β, measurable_toFun := ⋯, measurable_invFun := ⋯ }

def MeasurableEquiv.prodAssoc {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ] : (α × β) × γ ≃ᵐ α × β × γ

Products of measurable spaces are associative.

Equations

• MeasurableEquiv.prodAssoc = { toEquiv := Equiv.prodAssoc α β γ, measurable_toFun := ⋯, measurable_invFun := ⋯ }

def MeasurableEquiv.punitProd {α : Type u_1} [MeasurableSpace α] : PUnit.{u_6 + 1} × α ≃ᵐ α

PUnit is a left identity for product of measurable spaces up to a measurable equivalence.

Equations

• MeasurableEquiv.punitProd = { toEquiv := Equiv.punitProd α, measurable_toFun := ⋯, measurable_invFun := ⋯ }

def MeasurableEquiv.prodPUnit {α : Type u_1} [MeasurableSpace α] : α × PUnit.{u_6 + 1} ≃ᵐ α

PUnit is a right identity for product of measurable spaces up to a measurable equivalence.

Equations

• MeasurableEquiv.prodPUnit = { toEquiv := Equiv.prodPUnit α, measurable_toFun := ⋯, measurable_invFun := ⋯ }

def MeasurableEquiv.sumCongr {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ] [MeasurableSpace δ] (ab : α ≃ᵐ β) (cd : γ ≃ᵐ δ) : α ⊕ γ ≃ᵐ β ⊕ δ

Sums of measurable spaces are symmetric.

Equations

• ab.sumCongr cd = { toEquiv := ab.sumCongr cd.toEquiv, measurable_toFun := ⋯, measurable_invFun := ⋯ }

def MeasurableEquiv.Set.prod {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] (s : Set α) (t : Set β) : ↑(s ×ˢ t) ≃ᵐ ↑s × ↑t

s ×ˢ t ≃ (s × t) as measurable spaces.

Equations

• MeasurableEquiv.Set.prod s t = { toEquiv := Equiv.Set.prod s t, measurable_toFun := ⋯, measurable_invFun := ⋯ }

def MeasurableEquiv.Set.univ (α : Type u_6) [MeasurableSpace α] : ↑Set.univ ≃ᵐ α

univ α ≃ α as measurable spaces.

Equations

• MeasurableEquiv.Set.univ α = { toEquiv := Equiv.Set.univ α, measurable_toFun := ⋯, measurable_invFun := ⋯ }

def MeasurableEquiv.Set.singleton {α : Type u_1} [MeasurableSpace α] (a : α) : ↑{a} ≃ᵐ Unit

{a} ≃ Unit as measurable spaces.

Equations

• MeasurableEquiv.Set.singleton a = { toEquiv := Equiv.Set.singleton a, measurable_toFun := ⋯, measurable_invFun := ⋯ }

def MeasurableEquiv.Set.rangeInl {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] : ↑(Set.range Sum.inl) ≃ᵐ α

α is equivalent to its image in α ⊕ β as measurable spaces.

Equations

• MeasurableEquiv.Set.rangeInl = { toEquiv := Equiv.Set.rangeInl α β, measurable_toFun := ⋯, measurable_invFun := ⋯ }

def MeasurableEquiv.Set.rangeInr {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] : ↑(Set.range Sum.inr) ≃ᵐ β

β is equivalent to its image in α ⊕ β as measurable spaces.

Equations

• MeasurableEquiv.Set.rangeInr = { toEquiv := Equiv.Set.rangeInr α β, measurable_toFun := ⋯, measurable_invFun := ⋯ }

def MeasurableEquiv.sumProdDistrib (α : Type u_6) (β : Type u_7) (γ : Type u_8) [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ] : (α ⊕ β) × γ ≃ᵐ α × γ ⊕ β × γ

Products distribute over sums (on the right) as measurable spaces.

Equations

• MeasurableEquiv.sumProdDistrib α β γ = { toEquiv := Equiv.sumProdDistrib α β γ, measurable_toFun := ⋯, measurable_invFun := ⋯ }

def MeasurableEquiv.prodSumDistrib (α : Type u_6) (β : Type u_7) (γ : Type u_8) [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ] : α × (β ⊕ γ) ≃ᵐ α × β ⊕ α × γ

Products distribute over sums (on the left) as measurable spaces.

Equations

• MeasurableEquiv.prodSumDistrib α β γ = MeasurableEquiv.prodComm.trans ((MeasurableEquiv.sumProdDistrib β γ α).trans (MeasurableEquiv.prodComm.sumCongr MeasurableEquiv.prodComm))

def MeasurableEquiv.sumProdSum (α : Type u_6) (β : Type u_7) (γ : Type u_8) (δ : Type u_9) [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ] [MeasurableSpace δ] : (α ⊕ β) × (γ ⊕ δ) ≃ᵐ (α × γ ⊕ α × δ) ⊕ β × γ ⊕ β × δ

Products distribute over sums as measurable spaces.

Equations

• MeasurableEquiv.sumProdSum α β γ δ = (MeasurableEquiv.sumProdDistrib α β (γ ⊕ δ)).trans ((MeasurableEquiv.prodSumDistrib α γ δ).sumCongr (MeasurableEquiv.prodSumDistrib β γ δ))

def MeasurableEquiv.piCongrRight {δ' : Type u_5} {π : δ' → Type u_6} {π' : δ' → Type u_7} [(x : δ') → MeasurableSpace (π x)] [(x : δ') → MeasurableSpace (π' x)] (e : (a : δ') → π a ≃ᵐ π' a) : ((a : δ') → π a) ≃ᵐ ((a : δ') → π' a)

A family of measurable equivalences Π a, β₁ a ≃ᵐ β₂ a generates a measurable equivalence between Π a, β₁ a and Π a, β₂ a.

Equations

• MeasurableEquiv.piCongrRight e = { toEquiv := Equiv.piCongrRight fun (a : δ') => (e a).toEquiv, measurable_toFun := ⋯, measurable_invFun := ⋯ }

def MeasurableEquiv.piCongrLeft {δ : Type u_4} {δ' : Type u_5} (π : δ' → Type u_6) [(x : δ') → MeasurableSpace (π x)] (f : δ ≃ δ') : ((b : δ) → π (f b)) ≃ᵐ ((a : δ') → π a)

Moving a dependent type along an equivalence of coordinates, as a measurable equivalence.

Equations

• MeasurableEquiv.piCongrLeft π f = { toEquiv := Equiv.piCongrLeft π f, measurable_toFun := ⋯, measurable_invFun := ⋯ }

theorem MeasurableEquiv.coe_piCongrLeft {δ : Type u_4} {δ' : Type u_5} {π : δ' → Type u_6} [(x : δ') → MeasurableSpace (π x)] (f : δ ≃ δ') : ⇑(MeasurableEquiv.piCongrLeft π f) = ⇑(Equiv.piCongrLeft π f)

theorem MeasurableEquiv.piCongrLeft_apply_apply {ι : Type u_8} {ι' : Type u_9} (e : ι ≃ ι') {β : ι' → Type u_10} [(i' : ι') → MeasurableSpace (β i')] (x : (i : ι) → β (e i)) (i : ι) : (MeasurableEquiv.piCongrLeft (fun (i' : ι') => β i') e) x (e i) = x i

def MeasurableEquiv.piMeasurableEquivTProd {δ' : Type u_5} {π : δ' → Type u_6} [(x : δ') → MeasurableSpace (π x)] [DecidableEq δ'] {l : List δ'} (hnd : l.Nodup) (h : ∀ (i : δ'), i ∈ l) : ((i : δ') → π i) ≃ᵐ List.TProd π l

Pi-types are measurably equivalent to iterated products.

Equations

• MeasurableEquiv.piMeasurableEquivTProd hnd h = { toEquiv := List.TProd.piEquivTProd hnd h, measurable_toFun := ⋯, measurable_invFun := ⋯ }

@[simp]

theorem MeasurableEquiv.piMeasurableEquivTProd_apply {δ' : Type u_5} {π : δ' → Type u_6} [(x : δ') → MeasurableSpace (π x)] [DecidableEq δ'] {l : List δ'} (hnd : l.Nodup) (h : ∀ (i : δ'), i ∈ l) : ⇑(MeasurableEquiv.piMeasurableEquivTProd hnd h) = List.TProd.mk l

@[simp]

theorem MeasurableEquiv.piMeasurableEquivTProd_symm_apply {δ' : Type u_5} {π : δ' → Type u_6} [(x : δ') → MeasurableSpace (π x)] [DecidableEq δ'] {l : List δ'} (hnd : l.Nodup) (h : ∀ (i : δ'), i ∈ l) : ⇑(MeasurableEquiv.piMeasurableEquivTProd hnd h).symm = List.TProd.elim' h

def MeasurableEquiv.piUnique {δ' : Type u_5} (π : δ' → Type u_6) [(x : δ') → MeasurableSpace (π x)] [Unique δ'] : ((i : δ') → π i) ≃ᵐ π default

The measurable equivalence (∀ i, π i) ≃ᵐ π ⋆ when the domain of π only contains ⋆

Equations

• MeasurableEquiv.piUnique π = { toEquiv := Equiv.piUnique π, measurable_toFun := ⋯, measurable_invFun := ⋯ }

@[simp]

theorem MeasurableEquiv.piUnique_symm_apply {δ' : Type u_5} (π : δ' → Type u_6) [(x : δ') → MeasurableSpace (π x)] [Unique δ'] : ⇑(MeasurableEquiv.piUnique π).symm = uniqueElim

@[simp]

theorem MeasurableEquiv.piUnique_apply {δ' : Type u_5} (π : δ' → Type u_6) [(x : δ') → MeasurableSpace (π x)] [Unique δ'] : ⇑(MeasurableEquiv.piUnique π) = fun (f : (i : δ') → π i) => f default

def MeasurableEquiv.funUnique (α : Type u_8) (β : Type u_9) [Unique α] [MeasurableSpace β] : (α → β) ≃ᵐ β

If α has a unique term, then the type of function α → β is measurably equivalent to β.

Equations

• MeasurableEquiv.funUnique α β = MeasurableEquiv.piUnique fun (a : α) => β

@[simp]

theorem MeasurableEquiv.funUnique_apply (α : Type u_8) (β : Type u_9) [Unique α] [MeasurableSpace β] : ⇑(MeasurableEquiv.funUnique α β) = fun (f : α → β) => f default

@[simp]

theorem MeasurableEquiv.funUnique_symm_apply (α : Type u_8) (β : Type u_9) [Unique α] [MeasurableSpace β] : ⇑(MeasurableEquiv.funUnique α β).symm = uniqueElim

def MeasurableEquiv.piFinTwo (α : Fin 2 → Type u_8) [(i : Fin 2) → MeasurableSpace (α i)] : ((i : Fin 2) → α i) ≃ᵐ α 0 × α 1

The space Π i : Fin 2, α i is measurably equivalent to α 0 × α 1.

Equations

• MeasurableEquiv.piFinTwo α = { toEquiv := piFinTwoEquiv α, measurable_toFun := ⋯, measurable_invFun := ⋯ }

@[simp]

theorem MeasurableEquiv.piFinTwo_apply (α : Fin 2 → Type u_8) [(i : Fin 2) → MeasurableSpace (α i)] : ⇑(MeasurableEquiv.piFinTwo α) = fun (f : (i : Fin 2) → α i) => (f 0, f 1)

@[simp]

theorem MeasurableEquiv.piFinTwo_symm_apply (α : Fin 2 → Type u_8) [(i : Fin 2) → MeasurableSpace (α i)] : ⇑(MeasurableEquiv.piFinTwo α).symm = fun (p : α 0 × α 1) => Fin.cons p.1 (Fin.cons p.2 finZeroElim)

def MeasurableEquiv.finTwoArrow {α : Type u_1} [MeasurableSpace α] : (Fin 2 → α) ≃ᵐ α × α

The space Fin 2 → α is measurably equivalent to α × α.

Equations

• MeasurableEquiv.finTwoArrow = MeasurableEquiv.piFinTwo fun (x : Fin 2) => α

@[simp]

theorem MeasurableEquiv.finTwoArrow_symm_apply {α : Type u_1} [MeasurableSpace α] : ⇑MeasurableEquiv.finTwoArrow.symm = fun (p : α × α) => Fin.cons p.1 (Fin.cons p.2 finZeroElim)

@[simp]

theorem MeasurableEquiv.finTwoArrow_apply {α : Type u_1} [MeasurableSpace α] : ⇑MeasurableEquiv.finTwoArrow = fun (f : Fin 2 → α) => (f 0, f 1)

def MeasurableEquiv.piFinSuccAbove {n : ℕ} (α : Fin (n + 1) → Type u_8) [(i : Fin (n + 1)) → MeasurableSpace (α i)] (i : Fin (n + 1)) : ((j : Fin (n + 1)) → α j) ≃ᵐ α i × ((j : Fin n) → α (i.succAbove j))

Measurable equivalence between Π j : Fin (n + 1), α j and α i × Π j : Fin n, α (Fin.succAbove i j).

Measurable version of Fin.insertNthEquiv.

Equations

• MeasurableEquiv.piFinSuccAbove α i = { toEquiv := (Fin.insertNthEquiv α i).symm, measurable_toFun := ⋯, measurable_invFun := ⋯ }

@[simp]

theorem MeasurableEquiv.piFinSuccAbove_symm_apply {n : ℕ} (α : Fin (n + 1) → Type u_8) [(i : Fin (n + 1)) → MeasurableSpace (α i)] (i : Fin (n + 1)) : ⇑(MeasurableEquiv.piFinSuccAbove α i).symm = ⇑(Fin.insertNthEquiv α i)

@[simp]

theorem MeasurableEquiv.piFinSuccAbove_apply {n : ℕ} (α : Fin (n + 1) → Type u_8) [(i : Fin (n + 1)) → MeasurableSpace (α i)] (i : Fin (n + 1)) : ⇑(MeasurableEquiv.piFinSuccAbove α i) = ⇑(Fin.insertNthEquiv α i).symm

def MeasurableEquiv.piEquivPiSubtypeProd {δ' : Type u_5} (π : δ' → Type u_6) [(x : δ') → MeasurableSpace (π x)] (p : δ' → Prop) [DecidablePred p] : ((i : δ') → π i) ≃ᵐ ((i : Subtype p) → π ↑i) × ((i : { i : δ' // ¬p i }) → π ↑i)

Measurable equivalence between (dependent) functions on a type and pairs of functions on {i // p i} and {i // ¬p i}. See also Equiv.piEquivPiSubtypeProd.

Equations

• MeasurableEquiv.piEquivPiSubtypeProd π p = { toEquiv := Equiv.piEquivPiSubtypeProd p π, measurable_toFun := ⋯, measurable_invFun := ⋯ }

@[simp]

theorem MeasurableEquiv.piEquivPiSubtypeProd_apply {δ' : Type u_5} (π : δ' → Type u_6) [(x : δ') → MeasurableSpace (π x)] (p : δ' → Prop) [DecidablePred p] : ⇑(MeasurableEquiv.piEquivPiSubtypeProd π p) = fun (f : (i : δ') → π i) => (fun (x : { x : δ' // p x }) => f ↑x, fun (x : { x : δ' // ¬p x }) => f ↑x)

@[simp]

theorem MeasurableEquiv.piEquivPiSubtypeProd_symm_apply {δ' : Type u_5} (π : δ' → Type u_6) [(x : δ') → MeasurableSpace (π x)] (p : δ' → Prop) [DecidablePred p] : ⇑(MeasurableEquiv.piEquivPiSubtypeProd π p).symm = fun (f : ((i : { x : δ' // p x }) → π ↑i) × ((i : { x : δ' // ¬p x }) → π ↑i)) (x : δ') => if h : p x then f.1 ⟨x, h⟩ else f.2 ⟨x, h⟩

def MeasurableEquiv.sumPiEquivProdPi {δ : Type u_4} {δ' : Type u_5} (α : δ ⊕ δ' → Type u_8) [(i : δ ⊕ δ') → MeasurableSpace (α i)] : ((i : δ ⊕ δ') → α i) ≃ᵐ ((i : δ) → α (Sum.inl i)) × ((i' : δ') → α (Sum.inr i'))

The measurable equivalence between the pi type over a sum type and a product of pi-types. This is similar to MeasurableEquiv.piEquivPiSubtypeProd.

Equations

• MeasurableEquiv.sumPiEquivProdPi α = { toEquiv := Equiv.sumPiEquivProdPi α, measurable_toFun := ⋯, measurable_invFun := ⋯ }

theorem MeasurableEquiv.coe_sumPiEquivProdPi {δ : Type u_4} {δ' : Type u_5} (α : δ ⊕ δ' → Type u_8) [(i : δ ⊕ δ') → MeasurableSpace (α i)] : ⇑(MeasurableEquiv.sumPiEquivProdPi α) = ⇑(Equiv.sumPiEquivProdPi α)

theorem MeasurableEquiv.coe_sumPiEquivProdPi_symm {δ : Type u_4} {δ' : Type u_5} (α : δ ⊕ δ' → Type u_8) [(i : δ ⊕ δ') → MeasurableSpace (α i)] : ⇑(MeasurableEquiv.sumPiEquivProdPi α).symm = ⇑(Equiv.sumPiEquivProdPi α).symm

def MeasurableEquiv.piOptionEquivProd {δ : Type u_8} (α : Option δ → Type u_9) [(i : Option δ) → MeasurableSpace (α i)] : ((i : Option δ) → α i) ≃ᵐ ((i : δ) → α (some i)) × α none

The measurable equivalence for (dependent) functions on an Option type (∀ i : Option δ, α i) ≃ᵐ (∀ (i : δ), α i) × α none.

Equations

• One or more equations did not get rendered due to their size.

def MeasurableEquiv.piFinsetUnion {δ' : Type u_5} (π : δ' → Type u_6) [(x : δ') → MeasurableSpace (π x)] [DecidableEq δ'] {s : Finset δ'} {t : Finset δ'} (h : Disjoint s t) : ((i : { x : δ' // x ∈ s }) → π ↑i) × ((i : { x : δ' // x ∈ t }) → π ↑i) ≃ᵐ ((i : { x : δ' // x ∈ s ∪ t }) → π ↑i)

The measurable equivalence (∀ i : s, π i) × (∀ i : t, π i) ≃ᵐ (∀ i : s ∪ t, π i) for disjoint finsets s and t. Equiv.piFinsetUnion as a measurable equivalence.

Equations

• One or more equations did not get rendered due to their size.

def MeasurableEquiv.sumCompl {α : Type u_1} [MeasurableSpace α] {s : Set α} [DecidablePred fun (x : α) => x ∈ s] (hs : MeasurableSet s) : ↑s ⊕ ↑sᶜ ≃ᵐ α

If s is a measurable set in a measurable space, that space is equivalent to the sum of s and sᶜ.

Equations

• MeasurableEquiv.sumCompl hs = { toEquiv := Equiv.sumCompl fun (x : α) => x ∈ s, measurable_toFun := ⋯, measurable_invFun := ⋯ }

def MeasurableEquiv.ofInvolutive {α : Type u_1} [MeasurableSpace α] (f : α → α) (hf : Function.Involutive f) (hf' : Measurable f) : α ≃ᵐ α

Convert a measurable involutive function f to a measurable permutation with toFun = invFun = f. See also Function.Involutive.toPerm.

Equations

• MeasurableEquiv.ofInvolutive f hf hf' = { toEquiv := Function.Involutive.toPerm f hf, measurable_toFun := hf', measurable_invFun := hf' }

@[simp]

theorem MeasurableEquiv.ofInvolutive_toEquiv {α : Type u_1} [MeasurableSpace α] (f : α → α) (hf : Function.Involutive f) (hf' : Measurable f) : (MeasurableEquiv.ofInvolutive f hf hf').toEquiv = Function.Involutive.toPerm f hf

@[simp]

theorem MeasurableEquiv.ofInvolutive_apply {α : Type u_1} [MeasurableSpace α] (f : α → α) (hf : Function.Involutive f) (hf' : Measurable f) (a : α) : (MeasurableEquiv.ofInvolutive f hf hf') a = f a

@[simp]

theorem MeasurableEquiv.ofInvolutive_symm {α : Type u_1} [MeasurableSpace α] (f : α → α) (hf : Function.Involutive f) (hf' : Measurable f) : (MeasurableEquiv.ofInvolutive f hf hf').symm = MeasurableEquiv.ofInvolutive f hf hf'

@[simp]

theorem MeasurableEmbedding.comap_eq {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] {f : α → β} (hf : MeasurableEmbedding f) : MeasurableSpace.comap f inst✝ = inst✝¹

theorem MeasurableEmbedding.iff_comap_eq {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] {f : α → β} : MeasurableEmbedding f ↔ Function.Injective f ∧ MeasurableSpace.comap f inst✝ = inst✝¹ ∧ MeasurableSet (Set.range f)

noncomputable def MeasurableEmbedding.equivImage {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] {f : α → β} (s : Set α) (hf : MeasurableEmbedding f) : ↑s ≃ᵐ ↑(f '' s)

A set is equivalent to its image under a function f as measurable spaces, if f is a measurable embedding

Equations

• MeasurableEmbedding.equivImage s hf = { toEquiv := Equiv.Set.image f s ⋯, measurable_toFun := ⋯, measurable_invFun := ⋯ }

noncomputable def MeasurableEmbedding.equivRange {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] {f : α → β} (hf : MeasurableEmbedding f) : α ≃ᵐ ↑(Set.range f)

The domain of f is equivalent to its range as measurable spaces, if f is a measurable embedding

Equations

• hf.equivRange = (MeasurableEquiv.Set.univ α).symm.trans ((MeasurableEmbedding.equivImage Set.univ hf).trans (MeasurableEquiv.cast ⋯ ⋯))

theorem MeasurableEmbedding.of_measurable_inverse_on_range {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] {f : α → β} {g : ↑(Set.range f) → α} (hf₁ : Measurable f) (hf₂ : MeasurableSet (Set.range f)) (hg : Measurable g) (H : Function.LeftInverse g (Set.rangeFactorization f)) : MeasurableEmbedding f

theorem MeasurableEmbedding.of_measurable_inverse {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] {f : α → β} {g : β → α} (hf₁ : Measurable f) (hf₂ : MeasurableSet (Set.range f)) (hg : Measurable g) (H : Function.LeftInverse g f) : MeasurableEmbedding f

noncomputable def MeasurableEmbedding.schroederBernstein {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] {f : α → β} {g : β → α} (hf : MeasurableEmbedding f) (hg : MeasurableEmbedding g) : α ≃ᵐ β

The measurable Schröder-Bernstein Theorem: given measurable embeddings α → β and β → α, we can find a measurable equivalence α ≃ᵐ β.

Equations

• One or more equations did not get rendered due to their size.

theorem MeasurableSpace.comap_compl {α : Type u_1} {β : Type u_2} {m' : MeasurableSpace β} [BooleanAlgebra β] (h : Measurable compl) (f : α → β) : MeasurableSpace.comap (fun (a : α) => (f a)ᶜ) inferInstance = MeasurableSpace.comap f inferInstance

@[simp]

theorem MeasurableSpace.comap_not {α : Type u_1} (p : α → Prop) : MeasurableSpace.comap (fun (a : α) => ¬p a) inferInstance = MeasurableSpace.comap p inferInstance