Borel (measurable) space #

Main definitions #

borel α : the least σ-algebra that contains all open sets;
class BorelSpace : a space with TopologicalSpace and MeasurableSpace structures such that ‹MeasurableSpace α› = borel α;
class OpensMeasurableSpace : a space with TopologicalSpace and MeasurableSpace structures such that all open sets are measurable; equivalently, borel α ≤ ‹MeasurableSpace α›.
BorelSpace instances on Empty, Unit, Bool, Nat, Int, Rat;
MeasurableSpace and BorelSpace instances on ℝ, ℝ≥0, ℝ≥0∞.

Main statements #

IsOpen.measurableSet, IsClosed.measurableSet: open and closed sets are measurable;
Continuous.measurable : a continuous function is measurable;
Continuous.measurable2 : if f : α → β and g : α → γ are measurable and op : β × γ → δ is continuous, then fun x => op (f x, g y) is measurable;
Measurable.add etc : dot notation for arithmetic operations on Measurable predicates, and similarly for dist and edist;
AEMeasurable.add : similar dot notation for almost everywhere measurable functions;
Measurable.ennreal* : special cases for arithmetic operations on ℝ≥0∞.

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def borel (α : Type u) [TopologicalSpace α] :

MeasurableSpace α

MeasurableSpace structure generated by TopologicalSpace.

Equations

borel α = MeasurableSpace.generateFrom {s | IsOpen s}

Instances For

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theorem borel_anti {α : Type u_1} :

Antitone (@borel α)

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theorem borel_eq_top_of_discrete {α : Type u_1} [TopologicalSpace α] [DiscreteTopology α] :

borel α = ⊤

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theorem borel_eq_top_of_countable {α : Type u_1} [TopologicalSpace α] [T1Space α] [Countable α] :

borel α = ⊤

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theorem borel_eq_generateFrom_of_subbasis {α : Type u_1} {s : Set (Set α)} [t : TopologicalSpace α] [TopologicalSpace.SecondCountableTopology α] (hs : t = TopologicalSpace.generateFrom s) :

borel α = MeasurableSpace.generateFrom s

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theorem TopologicalSpace.IsTopologicalBasis.borel_eq_generateFrom {α : Type u_1} [TopologicalSpace α] [TopologicalSpace.SecondCountableTopology α] {s : Set (Set α)} (hs : TopologicalSpace.IsTopologicalBasis s) :

borel α = MeasurableSpace.generateFrom s

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theorem isPiSystem_isOpen {α : Type u_1} [TopologicalSpace α] :

IsPiSystem IsOpen

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theorem borel_eq_generateFrom_isClosed {α : Type u_1} [TopologicalSpace α] :

borel α = MeasurableSpace.generateFrom {s | IsClosed s}

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theorem borel_eq_generateFrom_Iio (α : Type u_1) [TopologicalSpace α] [TopologicalSpace.SecondCountableTopology α] [LinearOrder α] [OrderTopology α] :

borel α = MeasurableSpace.generateFrom (Set.range Set.Iio)

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theorem borel_eq_generateFrom_Ioi (α : Type u_1) [TopologicalSpace α] [TopologicalSpace.SecondCountableTopology α] [LinearOrder α] [OrderTopology α] :

borel α = MeasurableSpace.generateFrom (Set.range Set.Ioi)

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theorem borel_eq_generateFrom_Iic (α : Type u_1) [TopologicalSpace α] [TopologicalSpace.SecondCountableTopology α] [LinearOrder α] [OrderTopology α] :

borel α = MeasurableSpace.generateFrom (Set.range Set.Iic)

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theorem borel_eq_generateFrom_Ici (α : Type u_1) [TopologicalSpace α] [TopologicalSpace.SecondCountableTopology α] [LinearOrder α] [OrderTopology α] :

borel α = MeasurableSpace.generateFrom (Set.range Set.Ici)

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theorem borel_comap {α : Type u_1} {β : Type u_2} {f : α → β} {t : TopologicalSpace β} :

borel α = MeasurableSpace.comap f (borel β)

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theorem Continuous.borel_measurable {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} (hf : Continuous f) :

Measurable f

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class OpensMeasurableSpace (α : Type u_6) [TopologicalSpace α] [h : MeasurableSpace α] :

Prop

borel_le : borel α ≤ h
Borel-measurable sets are measurable.

A space with MeasurableSpace and TopologicalSpace structures such that all open sets are measurable.

Instances

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class BorelSpace (α : Type u_6) [TopologicalSpace α] [MeasurableSpace α] :

Prop

measurable_eq : inst✝ = borel α
The measurable sets are exactly the Borel-measurable sets.

A space with MeasurableSpace and TopologicalSpace structures such that the σ-algebra of measurable sets is exactly the σ-algebra generated by open sets.

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def Mathlib.Tactic.Borelize.tacticBorelize___ :

Lean.ParserDescr

The behaviour of borelize α depends on the existing assumptions on α.

if α is a topological space with instances [MeasurableSpace α] [BorelSpace α], then borelize α replaces the former instance by borel α;
otherwise, borelize α adds instances borel α : MeasurableSpace α and ⟨rfl⟩ : BorelSpace α.

Finally, borelize α β γ runs borelize α; borelize β; borelize γ.

Equations

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

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def Mathlib.Tactic.Borelize.addBorelInstance (e : Lean.Expr) :

Lean.Elab.Tactic.TacticM Unit

Add instances borel e : MeasurableSpace e and ⟨rfl⟩ : BorelSpace e.

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

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def Mathlib.Tactic.Borelize.borelToRefl (e : Lean.Expr) (i : Lean.FVarId) :

Lean.Elab.Tactic.TacticM Unit

Given a type e, an assumption i : MeasurableSpace e, and an instance [BorelSpace e], replace i with borel e.

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

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def Mathlib.Tactic.Borelize.borelize (t : Lean.Term) :

Lean.Elab.Tactic.TacticM Unit

Given a type $t, if there is an assumption [i : MeasurableSpace $t], then try to prove [BorelSpace $t] and replace i with borel $t. Otherwise, add instances borel $t : MeasurableSpace $t and ⟨rfl⟩ : BorelSpace $t.

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

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instance OrderDual.opensMeasurableSpace {α : Type u_6} [TopologicalSpace α] [MeasurableSpace α] [h : OpensMeasurableSpace α] :

OpensMeasurableSpace αᵒᵈ

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@OrderDual.opensMeasurableSpace = @OrderDual.opensMeasurableSpace.proof_1

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instance OrderDual.borelSpace {α : Type u_6} [TopologicalSpace α] [MeasurableSpace α] [h : BorelSpace α] :

BorelSpace αᵒᵈ

Equations

@OrderDual.borelSpace = @OrderDual.borelSpace.proof_1

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instance BorelSpace.opensMeasurable {α : Type u_6} [TopologicalSpace α] [MeasurableSpace α] [BorelSpace α] :

OpensMeasurableSpace α

In a BorelSpace all open sets are measurable.

Equations

@BorelSpace.opensMeasurable = @BorelSpace.opensMeasurable.proof_1

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instance Subtype.borelSpace {α : Type u_6} [TopologicalSpace α] [MeasurableSpace α] [hα : BorelSpace α] (s : Set α) :

BorelSpace ↑s

Equations

@Subtype.borelSpace = @Subtype.borelSpace.proof_1

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instance Subtype.opensMeasurableSpace {α : Type u_6} [TopologicalSpace α] [MeasurableSpace α] [h : OpensMeasurableSpace α] (s : Set α) :

OpensMeasurableSpace ↑s

Equations

@Subtype.opensMeasurableSpace = @Subtype.opensMeasurableSpace.proof_1

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theorem opensMeasurableSpace_iff_forall_measurableSet {α : Type u_1} [TopologicalSpace α] [MeasurableSpace α] :

OpensMeasurableSpace α ↔ ∀ (s : Set α), IsOpen s → MeasurableSet s

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instance BorelSpace.countablyGenerated {α : Type u_6} [TopologicalSpace α] [MeasurableSpace α] [BorelSpace α] [TopologicalSpace.SecondCountableTopology α] :

MeasurableSpace.CountablyGenerated α

Equations

@BorelSpace.countablyGenerated = @BorelSpace.countablyGenerated.proof_1

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theorem MeasurableSet.induction_on_open {α : Type u_1} [TopologicalSpace α] [MeasurableSpace α] [BorelSpace α] {C : Set α → Prop} (h_open : (U : Set α) → IsOpen U → C U) (h_compl : (t : Set α) → MeasurableSet t → C t → C tᶜ) (h_union : (f : ℕ → Set α) → Pairwise (Disjoint on f) → (∀ (i : ℕ), MeasurableSet (f i)) → ((i : ℕ) → C (f i)) → C (⋃ (i : ℕ), f i)) ⦃t : Set α⦄ :

MeasurableSet t → C t

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theorem IsOpen.measurableSet {α : Type u_1} {s : Set α} [TopologicalSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] (h : IsOpen s) :

MeasurableSet s

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instance instHasCountableSeparatingOnMeasurableSet {α : Type u_1} [TopologicalSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] {s : Set α} [HasCountableSeparatingOn α IsOpen s] :

HasCountableSeparatingOn α MeasurableSet s

Equations

@instHasCountableSeparatingOnMeasurableSet = @instHasCountableSeparatingOnMeasurableSet.proof_1

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theorem measurableSet_interior {α : Type u_1} {s : Set α} [TopologicalSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] :

MeasurableSet (interior s)

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theorem IsGδ.measurableSet {α : Type u_1} {s : Set α} [TopologicalSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] (h : IsGδ s) :

MeasurableSet s

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theorem measurableSet_of_continuousAt {α : Type u_1} [TopologicalSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] {β : Type u_6} [EMetricSpace β] (f : α → β) :

MeasurableSet {x | ContinuousAt f x}

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theorem IsClosed.measurableSet {α : Type u_1} {s : Set α} [TopologicalSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] (h : IsClosed s) :

MeasurableSet s

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theorem IsCompact.measurableSet {α : Type u_1} {s : Set α} [TopologicalSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] [T2Space α] (h : IsCompact s) :

MeasurableSet s

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theorem measurableSet_closure {α : Type u_1} {s : Set α} [TopologicalSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] :

MeasurableSet (closure s)

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theorem measurable_of_isOpen {γ : Type u_3} {δ : Type u_5} [TopologicalSpace γ] [MeasurableSpace γ] [BorelSpace γ] [MeasurableSpace δ] {f : δ → γ} (hf : ∀ (s : Set γ), IsOpen s → MeasurableSet (f ⁻¹' s)) :

Measurable f

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theorem measurable_of_isClosed {γ : Type u_3} {δ : Type u_5} [TopologicalSpace γ] [MeasurableSpace γ] [BorelSpace γ] [MeasurableSpace δ] {f : δ → γ} (hf : ∀ (s : Set γ), IsClosed s → MeasurableSet (f ⁻¹' s)) :

Measurable f

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theorem measurable_of_is_closed' {γ : Type u_3} {δ : Type u_5} [TopologicalSpace γ] [MeasurableSpace γ] [BorelSpace γ] [MeasurableSpace δ] {f : δ → γ} (hf : ∀ (s : Set γ), IsClosed s → Set.Nonempty s → s ≠ Set.univ → MeasurableSet (f ⁻¹' s)) :

Measurable f

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instance nhds_isMeasurablyGenerated {α : Type u_1} [TopologicalSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] (a : α) :

Filter.IsMeasurablyGenerated (nhds a)

Equations

@nhds_isMeasurablyGenerated = @nhds_isMeasurablyGenerated.proof_1

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theorem MeasurableSet.nhdsWithin_isMeasurablyGenerated {α : Type u_1} [TopologicalSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] {s : Set α} (hs : MeasurableSet s) (a : α) :

Filter.IsMeasurablyGenerated (nhdsWithin a s)

If s is a measurable set, then 𝓝[s] a is a measurably generated filter for each a. This cannot be an instance because it depends on a non-instance hs : MeasurableSet s.

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instance OpensMeasurableSpace.toMeasurableSingletonClass {α : Type u_1} [TopologicalSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] [T1Space α] :

MeasurableSingletonClass α

Equations

@OpensMeasurableSpace.toMeasurableSingletonClass = @OpensMeasurableSpace.toMeasurableSingletonClass.proof_1

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instance Pi.opensMeasurableSpace {ι : Type u_6} {π : ι → Type u_7} [Countable ι] [t' : (i : ι) → TopologicalSpace (π i)] [(i : ι) → MeasurableSpace (π i)] [∀ (i : ι), TopologicalSpace.SecondCountableTopology (π i)] [∀ (i : ι), OpensMeasurableSpace (π i)] :

OpensMeasurableSpace ((i : ι) → π i)

Equations

@Pi.opensMeasurableSpace = @Pi.opensMeasurableSpace.proof_1

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class SecondCountableTopologyEither (α : Type u_6) (β : Type u_7) [TopologicalSpace α] [TopologicalSpace β] :

Prop

out : TopologicalSpace.SecondCountableTopology α ∨ TopologicalSpace.SecondCountableTopology β
The projection out of SecondCountableTopologyEither

The typeclass SecondCountableTopologyEither α β registers the fact that at least one of the two spaces has second countable topology. This is the right assumption to ensure that continuous maps from α to β are strongly measurable.

Instances

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instance secondCountableTopologyEither_of_left (α : Type u_6) (β : Type u_7) [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace.SecondCountableTopology α] :

SecondCountableTopologyEither α β

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secondCountableTopologyEither_of_left = secondCountableTopologyEither_of_left.proof_1

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instance secondCountableTopologyEither_of_right (α : Type u_6) (β : Type u_7) [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace.SecondCountableTopology β] :

SecondCountableTopologyEither α β

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secondCountableTopologyEither_of_right = secondCountableTopologyEither_of_right.proof_1

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instance Prod.opensMeasurableSpace {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] [TopologicalSpace β] [MeasurableSpace β] [OpensMeasurableSpace β] [h : SecondCountableTopologyEither α β] :

OpensMeasurableSpace (α × β)

If either α or β has second-countable topology, then the open sets in α × β belong to the product sigma-algebra.

Equations

@Prod.opensMeasurableSpace = @Prod.opensMeasurableSpace.proof_1

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theorem interior_ae_eq_of_null_frontier {α' : Type u_6} [TopologicalSpace α'] [MeasurableSpace α'] {μ : MeasureTheory.Measure α'} {s : Set α'} (h : ↑↑μ (frontier s) = 0) :

interior s =ᶠ[MeasureTheory.Measure.ae μ] s

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theorem measure_interior_of_null_frontier {α' : Type u_6} [TopologicalSpace α'] [MeasurableSpace α'] {μ : MeasureTheory.Measure α'} {s : Set α'} (h : ↑↑μ (frontier s) = 0) :

↑↑μ (interior s) = ↑↑μ s

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theorem nullMeasurableSet_of_null_frontier {α : Type u_1} [TopologicalSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] {s : Set α} {μ : MeasureTheory.Measure α} (h : ↑↑μ (frontier s) = 0) :

MeasureTheory.NullMeasurableSet s

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theorem closure_ae_eq_of_null_frontier {α' : Type u_6} [TopologicalSpace α'] [MeasurableSpace α'] {μ : MeasureTheory.Measure α'} {s : Set α'} (h : ↑↑μ (frontier s) = 0) :

closure s =ᶠ[MeasureTheory.Measure.ae μ] s

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theorem measure_closure_of_null_frontier {α' : Type u_6} [TopologicalSpace α'] [MeasurableSpace α'] {μ : MeasureTheory.Measure α'} {s : Set α'} (h : ↑↑μ (frontier s) = 0) :

↑↑μ (closure s) = ↑↑μ s

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@[simp]

theorem measurableSet_Ici {α : Type u_1} [TopologicalSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] [Preorder α] [OrderClosedTopology α] {a : α} :

MeasurableSet (Set.Ici a)

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@[simp]

theorem measurableSet_Iic {α : Type u_1} [TopologicalSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] [Preorder α] [OrderClosedTopology α] {a : α} :

MeasurableSet (Set.Iic a)

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@[simp]

theorem measurableSet_Icc {α : Type u_1} [TopologicalSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] [Preorder α] [OrderClosedTopology α] {a : α} {b : α} :

MeasurableSet (Set.Icc a b)

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instance nhdsWithin_Ici_isMeasurablyGenerated {α : Type u_1} [TopologicalSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] [Preorder α] [OrderClosedTopology α] {a : α} {b : α} :

Filter.IsMeasurablyGenerated (nhdsWithin a (Set.Ici b))

Equations

@nhdsWithin_Ici_isMeasurablyGenerated = @nhdsWithin_Ici_isMeasurablyGenerated.proof_1

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instance nhdsWithin_Iic_isMeasurablyGenerated {α : Type u_1} [TopologicalSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] [Preorder α] [OrderClosedTopology α] {a : α} {b : α} :

Filter.IsMeasurablyGenerated (nhdsWithin a (Set.Iic b))

Equations

@nhdsWithin_Iic_isMeasurablyGenerated = @nhdsWithin_Iic_isMeasurablyGenerated.proof_1

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instance nhdsWithin_Icc_isMeasurablyGenerated {α : Type u_1} [TopologicalSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] [Preorder α] [OrderClosedTopology α] {a : α} {b : α} {x : α} :

Filter.IsMeasurablyGenerated (nhdsWithin x (Set.Icc a b))

Equations

@nhdsWithin_Icc_isMeasurablyGenerated = @nhdsWithin_Icc_isMeasurablyGenerated.proof_1

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instance atTop_isMeasurablyGenerated {α : Type u_1} [TopologicalSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] [Preorder α] [OrderClosedTopology α] :

Filter.IsMeasurablyGenerated Filter.atTop

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@atTop_isMeasurablyGenerated = @atTop_isMeasurablyGenerated.proof_1

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instance atBot_isMeasurablyGenerated {α : Type u_1} [TopologicalSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] [Preorder α] [OrderClosedTopology α] :

Filter.IsMeasurablyGenerated Filter.atBot

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@atBot_isMeasurablyGenerated = @atBot_isMeasurablyGenerated.proof_1

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theorem measurableSet_le' {α : Type u_1} [TopologicalSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] [PartialOrder α] [OrderClosedTopology α] [TopologicalSpace.SecondCountableTopology α] :

MeasurableSet {p | p.fst ≤ p.snd}

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theorem measurableSet_le {α : Type u_1} {δ : Type u_5} [TopologicalSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] [MeasurableSpace δ] [PartialOrder α] [OrderClosedTopology α] [TopologicalSpace.SecondCountableTopology α] {f : δ → α} {g : δ → α} (hf : Measurable f) (hg : Measurable g) :

MeasurableSet {a | f a ≤ g a}

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@[simp]

theorem measurableSet_Iio {α : Type u_1} [TopologicalSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] [LinearOrder α] [OrderClosedTopology α] {a : α} :

MeasurableSet (Set.Iio a)

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@[simp]

theorem measurableSet_Ioi {α : Type u_1} [TopologicalSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] [LinearOrder α] [OrderClosedTopology α] {a : α} :

MeasurableSet (Set.Ioi a)

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@[simp]

theorem measurableSet_Ioo {α : Type u_1} [TopologicalSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] [LinearOrder α] [OrderClosedTopology α] {a : α} {b : α} :

MeasurableSet (Set.Ioo a b)

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@[simp]

theorem measurableSet_Ioc {α : Type u_1} [TopologicalSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] [LinearOrder α] [OrderClosedTopology α] {a : α} {b : α} :

MeasurableSet (Set.Ioc a b)

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@[simp]

theorem measurableSet_Ico {α : Type u_1} [TopologicalSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] [LinearOrder α] [OrderClosedTopology α] {a : α} {b : α} :

MeasurableSet (Set.Ico a b)

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instance nhdsWithin_Ioi_isMeasurablyGenerated {α : Type u_1} [TopologicalSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] [LinearOrder α] [OrderClosedTopology α] {a : α} {b : α} :

Filter.IsMeasurablyGenerated (nhdsWithin a (Set.Ioi b))

Equations

@nhdsWithin_Ioi_isMeasurablyGenerated = @nhdsWithin_Ioi_isMeasurablyGenerated.proof_1

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instance nhdsWithin_Iio_isMeasurablyGenerated {α : Type u_1} [TopologicalSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] [LinearOrder α] [OrderClosedTopology α] {a : α} {b : α} :

Filter.IsMeasurablyGenerated (nhdsWithin a (Set.Iio b))

Equations

@nhdsWithin_Iio_isMeasurablyGenerated = @nhdsWithin_Iio_isMeasurablyGenerated.proof_1

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instance nhdsWithin_uIcc_isMeasurablyGenerated {α : Type u_1} [TopologicalSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] [LinearOrder α] [OrderClosedTopology α] {a : α} {b : α} {x : α} :

Filter.IsMeasurablyGenerated (nhdsWithin x (Set.uIcc a b))

Equations

@nhdsWithin_uIcc_isMeasurablyGenerated = @nhdsWithin_uIcc_isMeasurablyGenerated.proof_1

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theorem measurableSet_lt' {α : Type u_1} [TopologicalSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] [LinearOrder α] [OrderClosedTopology α] [TopologicalSpace.SecondCountableTopology α] :

MeasurableSet {p | p.fst < p.snd}

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theorem measurableSet_lt {α : Type u_1} {δ : Type u_5} [TopologicalSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] [MeasurableSpace δ] [LinearOrder α] [OrderClosedTopology α] [TopologicalSpace.SecondCountableTopology α] {f : δ → α} {g : δ → α} (hf : Measurable f) (hg : Measurable g) :

MeasurableSet {a | f a < g a}

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theorem nullMeasurableSet_lt {α : Type u_1} {δ : Type u_5} [TopologicalSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] [MeasurableSpace δ] [LinearOrder α] [OrderClosedTopology α] [TopologicalSpace.SecondCountableTopology α] {μ : MeasureTheory.Measure δ} {f : δ → α} {g : δ → α} (hf : AEMeasurable f) (hg : AEMeasurable g) :

MeasureTheory.NullMeasurableSet {a | f a < g a}

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theorem Set.OrdConnected.measurableSet {α : Type u_1} {s : Set α} [TopologicalSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] [LinearOrder α] [OrderClosedTopology α] (h : Set.OrdConnected s) :

MeasurableSet s

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theorem IsPreconnected.measurableSet {α : Type u_1} {s : Set α} [TopologicalSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] [LinearOrder α] [OrderClosedTopology α] (h : IsPreconnected s) :

MeasurableSet s

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theorem generateFrom_Ico_mem_le_borel {α : Type u_7} [TopologicalSpace α] [LinearOrder α] [OrderClosedTopology α] (s : Set α) (t : Set α) :

MeasurableSpace.generateFrom {S | ∃ l, l ∈ s ∧ ∃ u, u ∈ t ∧ l < u ∧ Set.Ico l u = S} ≤ borel α

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theorem Dense.borel_eq_generateFrom_Ico_mem_aux {α : Type u_7} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [TopologicalSpace.SecondCountableTopology α] {s : Set α} (hd : Dense s) (hbot : ∀ (x : α), IsBot x → x ∈ s) (hIoo : ∀ (x y : α), x < y → Set.Ioo x y = ∅ → y ∈ s) :

borel α = MeasurableSpace.generateFrom {S | ∃ l, l ∈ s ∧ ∃ u, u ∈ s ∧ l < u ∧ Set.Ico l u = S}

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theorem Dense.borel_eq_generateFrom_Ico_mem {α : Type u_7} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [TopologicalSpace.SecondCountableTopology α] [DenselyOrdered α] [NoMinOrder α] {s : Set α} (hd : Dense s) :

borel α = MeasurableSpace.generateFrom {S | ∃ l, l ∈ s ∧ ∃ u, u ∈ s ∧ l < u ∧ Set.Ico l u = S}

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theorem borel_eq_generateFrom_Ico (α : Type u_7) [TopologicalSpace α] [TopologicalSpace.SecondCountableTopology α] [LinearOrder α] [OrderTopology α] :

borel α = MeasurableSpace.generateFrom {S | ∃ l u, l < u ∧ Set.Ico l u = S}

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theorem Dense.borel_eq_generateFrom_Ioc_mem_aux {α : Type u_7} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [TopologicalSpace.SecondCountableTopology α] {s : Set α} (hd : Dense s) (hbot : ∀ (x : α), IsTop x → x ∈ s) (hIoo : ∀ (x y : α), x < y → Set.Ioo x y = ∅ → x ∈ s) :

borel α = MeasurableSpace.generateFrom {S | ∃ l, l ∈ s ∧ ∃ u, u ∈ s ∧ l < u ∧ Set.Ioc l u = S}

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theorem Dense.borel_eq_generateFrom_Ioc_mem {α : Type u_7} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [TopologicalSpace.SecondCountableTopology α] [DenselyOrdered α] [NoMaxOrder α] {s : Set α} (hd : Dense s) :

borel α = MeasurableSpace.generateFrom {S | ∃ l, l ∈ s ∧ ∃ u, u ∈ s ∧ l < u ∧ Set.Ioc l u = S}

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theorem borel_eq_generateFrom_Ioc (α : Type u_7) [TopologicalSpace α] [TopologicalSpace.SecondCountableTopology α] [LinearOrder α] [OrderTopology α] :

borel α = MeasurableSpace.generateFrom {S | ∃ l u, l < u ∧ Set.Ioc l u = S}

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theorem MeasureTheory.Measure.ext_of_Ico_finite {α : Type u_7} [TopologicalSpace α] {m : MeasurableSpace α} [TopologicalSpace.SecondCountableTopology α] [LinearOrder α] [OrderTopology α] [BorelSpace α] (μ : MeasureTheory.Measure α) (ν : MeasureTheory.Measure α) [MeasureTheory.IsFiniteMeasure μ] (hμν : ↑↑μ Set.univ = ↑↑ν Set.univ) (h : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Set.Ico a b) = ↑↑ν (Set.Ico a b)) :

μ = ν

Two finite measures on a Borel space are equal if they agree on all closed-open intervals. If α is a conditionally complete linear order with no top element, MeasureTheory.Measure.ext_of_Ico is an extensionality lemma with weaker assumptions on μ and ν.

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theorem MeasureTheory.Measure.ext_of_Ioc_finite {α : Type u_7} [TopologicalSpace α] {m : MeasurableSpace α} [TopologicalSpace.SecondCountableTopology α] [LinearOrder α] [OrderTopology α] [BorelSpace α] (μ : MeasureTheory.Measure α) (ν : MeasureTheory.Measure α) [MeasureTheory.IsFiniteMeasure μ] (hμν : ↑↑μ Set.univ = ↑↑ν Set.univ) (h : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Set.Ioc a b) = ↑↑ν (Set.Ioc a b)) :

μ = ν

Two finite measures on a Borel space are equal if they agree on all open-closed intervals. If α is a conditionally complete linear order with no top element, MeasureTheory.Measure.ext_of_Ioc is an extensionality lemma with weaker assumptions on μ and ν.

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theorem MeasureTheory.Measure.ext_of_Ico' {α : Type u_7} [TopologicalSpace α] {m : MeasurableSpace α} [TopologicalSpace.SecondCountableTopology α] [LinearOrder α] [OrderTopology α] [BorelSpace α] [NoMaxOrder α] (μ : MeasureTheory.Measure α) (ν : MeasureTheory.Measure α) (hμ : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Set.Ico a b) ≠ ⊤) (h : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Set.Ico a b) = ↑↑ν (Set.Ico a b)) :

μ = ν

Two measures which are finite on closed-open intervals are equal if the agree on all closed-open intervals.

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theorem MeasureTheory.Measure.ext_of_Ioc' {α : Type u_7} [TopologicalSpace α] {m : MeasurableSpace α} [TopologicalSpace.SecondCountableTopology α] [LinearOrder α] [OrderTopology α] [BorelSpace α] [NoMinOrder α] (μ : MeasureTheory.Measure α) (ν : MeasureTheory.Measure α) (hμ : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Set.Ioc a b) ≠ ⊤) (h : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Set.Ioc a b) = ↑↑ν (Set.Ioc a b)) :

μ = ν

Two measures which are finite on closed-open intervals are equal if the agree on all open-closed intervals.

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theorem MeasureTheory.Measure.ext_of_Ico {α : Type u_7} [TopologicalSpace α] {_m : MeasurableSpace α} [TopologicalSpace.SecondCountableTopology α] [ConditionallyCompleteLinearOrder α] [OrderTopology α] [BorelSpace α] [NoMaxOrder α] (μ : MeasureTheory.Measure α) (ν : MeasureTheory.Measure α) [MeasureTheory.IsLocallyFiniteMeasure μ] (h : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Set.Ico a b) = ↑↑ν (Set.Ico a b)) :

μ = ν

Two measures which are finite on closed-open intervals are equal if the agree on all closed-open intervals.

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theorem MeasureTheory.Measure.ext_of_Ioc {α : Type u_7} [TopologicalSpace α] {_m : MeasurableSpace α} [TopologicalSpace.SecondCountableTopology α] [ConditionallyCompleteLinearOrder α] [OrderTopology α] [BorelSpace α] [NoMinOrder α] (μ : MeasureTheory.Measure α) (ν : MeasureTheory.Measure α) [MeasureTheory.IsLocallyFiniteMeasure μ] (h : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Set.Ioc a b) = ↑↑ν (Set.Ioc a b)) :

μ = ν

Two measures which are finite on closed-open intervals are equal if the agree on all open-closed intervals.

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theorem MeasureTheory.Measure.ext_of_Iic {α : Type u_7} [TopologicalSpace α] {m : MeasurableSpace α} [TopologicalSpace.SecondCountableTopology α] [LinearOrder α] [OrderTopology α] [BorelSpace α] (μ : MeasureTheory.Measure α) (ν : MeasureTheory.Measure α) [MeasureTheory.IsFiniteMeasure μ] (h : ∀ (a : α), ↑↑μ (Set.Iic a) = ↑↑ν (Set.Iic a)) :

μ = ν

Two finite measures on a Borel space are equal if they agree on all left-infinite right-closed intervals.

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theorem MeasureTheory.Measure.ext_of_Ici {α : Type u_7} [TopologicalSpace α] {m : MeasurableSpace α} [TopologicalSpace.SecondCountableTopology α] [LinearOrder α] [OrderTopology α] [BorelSpace α] (μ : MeasureTheory.Measure α) (ν : MeasureTheory.Measure α) [MeasureTheory.IsFiniteMeasure μ] (h : ∀ (a : α), ↑↑μ (Set.Ici a) = ↑↑ν (Set.Ici a)) :

μ = ν

Two finite measures on a Borel space are equal if they agree on all left-closed right-infinite intervals.

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theorem measurableSet_uIcc {α : Type u_1} [TopologicalSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] [LinearOrder α] [OrderClosedTopology α] {a : α} {b : α} :

MeasurableSet (Set.uIcc a b)

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theorem measurableSet_uIoc {α : Type u_1} [TopologicalSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] [LinearOrder α] [OrderClosedTopology α] {a : α} {b : α} :

MeasurableSet (Ι a b)

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theorem Measurable.max {α : Type u_1} {δ : Type u_5} [TopologicalSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] [MeasurableSpace δ] [LinearOrder α] [OrderClosedTopology α] [TopologicalSpace.SecondCountableTopology α] {f : δ → α} {g : δ → α} (hf : Measurable f) (hg : Measurable g) :

Measurable fun a => max (f a) (g a)

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theorem AEMeasurable.max {α : Type u_1} {δ : Type u_5} [TopologicalSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] [MeasurableSpace δ] [LinearOrder α] [OrderClosedTopology α] [TopologicalSpace.SecondCountableTopology α] {f : δ → α} {g : δ → α} {μ : MeasureTheory.Measure δ} (hf : AEMeasurable f) (hg : AEMeasurable g) :

AEMeasurable fun a => max (f a) (g a)

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theorem Measurable.min {α : Type u_1} {δ : Type u_5} [TopologicalSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] [MeasurableSpace δ] [LinearOrder α] [OrderClosedTopology α] [TopologicalSpace.SecondCountableTopology α] {f : δ → α} {g : δ → α} (hf : Measurable f) (hg : Measurable g) :

Measurable fun a => min (f a) (g a)

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theorem AEMeasurable.min {α : Type u_1} {δ : Type u_5} [TopologicalSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] [MeasurableSpace δ] [LinearOrder α] [OrderClosedTopology α] [TopologicalSpace.SecondCountableTopology α] {f : δ → α} {g : δ → α} {μ : MeasureTheory.Measure δ} (hf : AEMeasurable f) (hg : AEMeasurable g) :

AEMeasurable fun a => min (f a) (g a)

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theorem Continuous.measurable {α : Type u_1} {γ : Type u_3} [TopologicalSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] [TopologicalSpace γ] [MeasurableSpace γ] [BorelSpace γ] {f : α → γ} (hf : Continuous f) :

Measurable f

A continuous function from an OpensMeasurableSpace to a BorelSpace is measurable.

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theorem Continuous.aemeasurable {α : Type u_1} {γ : Type u_3} [TopologicalSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] [TopologicalSpace γ] [MeasurableSpace γ] [BorelSpace γ] {f : α → γ} (h : Continuous f) {μ : MeasureTheory.Measure α} :

AEMeasurable f

A continuous function from an OpensMeasurableSpace to a BorelSpace is ae-measurable.

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theorem ClosedEmbedding.measurable {α : Type u_1} {γ : Type u_3} [TopologicalSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] [TopologicalSpace γ] [MeasurableSpace γ] [BorelSpace γ] {f : α → γ} (hf : ClosedEmbedding f) :

Measurable f

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theorem ContinuousOn.measurable_piecewise {α : Type u_1} {γ : Type u_3} [TopologicalSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] [TopologicalSpace γ] [MeasurableSpace γ] [BorelSpace γ] {f : α → γ} {g : α → γ} {s : Set α} [(j : α) → Decidable (j ∈ s)] (hf : ContinuousOn f s) (hg : ContinuousOn g sᶜ) (hs : MeasurableSet s) :

Measurable (Set.piecewise s f g)

If a function is defined piecewise in terms of functions which are continuous on their respective pieces, then it is measurable.

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instance ContinuousAdd.measurableAdd {γ : Type u_3} [TopologicalSpace γ] [MeasurableSpace γ] [BorelSpace γ] [Add γ] [ContinuousAdd γ] :

MeasurableAdd γ

Equations

@ContinuousAdd.measurableAdd = @ContinuousAdd.measurableAdd.proof_1

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theorem ContinuousAdd.measurableAdd.proof_1 {γ : Type u_1} [TopologicalSpace γ] [MeasurableSpace γ] [BorelSpace γ] [Add γ] [ContinuousAdd γ] :

MeasurableAdd γ

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instance ContinuousMul.measurableMul {γ : Type u_3} [TopologicalSpace γ] [MeasurableSpace γ] [BorelSpace γ] [Mul γ] [ContinuousMul γ] :

MeasurableMul γ

Equations

@ContinuousMul.measurableMul = @ContinuousMul.measurableMul.proof_1

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instance ContinuousSub.measurableSub {γ : Type u_3} [TopologicalSpace γ] [MeasurableSpace γ] [BorelSpace γ] [Sub γ] [ContinuousSub γ] :

MeasurableSub γ

Equations

@ContinuousSub.measurableSub = @ContinuousSub.measurableSub.proof_1

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instance TopologicalAddGroup.measurableNeg {γ : Type u_3} [TopologicalSpace γ] [MeasurableSpace γ] [BorelSpace γ] [AddGroup γ] [TopologicalAddGroup γ] :

MeasurableNeg γ

Equations

@TopologicalAddGroup.measurableNeg = @TopologicalAddGroup.measurableNeg.proof_1

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theorem TopologicalAddGroup.measurableNeg.proof_1 {γ : Type u_1} [TopologicalSpace γ] [MeasurableSpace γ] [BorelSpace γ] [AddGroup γ] [TopologicalAddGroup γ] :

MeasurableNeg γ

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instance TopologicalGroup.measurableInv {γ : Type u_3} [TopologicalSpace γ] [MeasurableSpace γ] [BorelSpace γ] [Group γ] [TopologicalGroup γ] :

MeasurableInv γ

Equations

@TopologicalGroup.measurableInv = @TopologicalGroup.measurableInv.proof_1

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instance ContinuousSMul.measurableSMul {M : Type u_7} {α : Type u_8} [TopologicalSpace M] [TopologicalSpace α] [MeasurableSpace M] [MeasurableSpace α] [OpensMeasurableSpace M] [BorelSpace α] [SMul M α] [ContinuousSMul M α] :

MeasurableSMul M α

Equations

@ContinuousSMul.measurableSMul = @ContinuousSMul.measurableSMul.proof_1

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instance ContinuousSup.measurableSup {γ : Type u_3} [TopologicalSpace γ] [MeasurableSpace γ] [BorelSpace γ] [Sup γ] [ContinuousSup γ] :

MeasurableSup γ

Equations

@ContinuousSup.measurableSup = @ContinuousSup.measurableSup.proof_1

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instance ContinuousSup.measurableSup₂ {γ : Type u_3} [TopologicalSpace γ] [MeasurableSpace γ] [BorelSpace γ] [TopologicalSpace.SecondCountableTopology γ] [Sup γ] [ContinuousSup γ] :

MeasurableSup₂ γ

Equations

@ContinuousSup.measurableSup₂ = @ContinuousSup.measurableSup₂.proof_1

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instance ContinuousInf.measurableInf {γ : Type u_3} [TopologicalSpace γ] [MeasurableSpace γ] [BorelSpace γ] [Inf γ] [ContinuousInf γ] :

MeasurableInf γ

Equations

@ContinuousInf.measurableInf = @ContinuousInf.measurableInf.proof_1

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instance ContinuousInf.measurableInf₂ {γ : Type u_3} [TopologicalSpace γ] [MeasurableSpace γ] [BorelSpace γ] [TopologicalSpace.SecondCountableTopology γ] [Inf γ] [ContinuousInf γ] :

MeasurableInf₂ γ

Equations

@ContinuousInf.measurableInf₂ = @ContinuousInf.measurableInf₂.proof_1

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theorem Homeomorph.measurable {α : Type u_1} {γ : Type u_3} [TopologicalSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] [TopologicalSpace γ] [MeasurableSpace γ] [BorelSpace γ] (h : α ≃ₜ γ) :

Measurable ↑h

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def Homeomorph.toMeasurableEquiv {γ : Type u_3} {γ₂ : Type u_4} [TopologicalSpace γ] [MeasurableSpace γ] [BorelSpace γ] [TopologicalSpace γ₂] [MeasurableSpace γ₂] [BorelSpace γ₂] (h : γ ≃ₜ γ₂) :

γ ≃ᵐ γ₂

A homeomorphism between two Borel spaces is a measurable equivalence.

Equations

Homeomorph.toMeasurableEquiv h = { toEquiv := h.toEquiv, measurable_toFun := (_ : Measurable ↑h), measurable_invFun := (_ : Measurable ↑(Homeomorph.symm h)) }

Instances For

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@[simp]

theorem Homeomorph.toMeasurableEquiv_coe {γ : Type u_3} {γ₂ : Type u_4} [TopologicalSpace γ] [MeasurableSpace γ] [BorelSpace γ] [TopologicalSpace γ₂] [MeasurableSpace γ₂] [BorelSpace γ₂] (h : γ ≃ₜ γ₂) :

↑(Homeomorph.toMeasurableEquiv h) = ↑h

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@[simp]

theorem Homeomorph.toMeasurableEquiv_symm_coe {γ : Type u_3} {γ₂ : Type u_4} [TopologicalSpace γ] [MeasurableSpace γ] [BorelSpace γ] [TopologicalSpace γ₂] [MeasurableSpace γ₂] [BorelSpace γ₂] (h : γ ≃ₜ γ₂) :

↑(MeasurableEquiv.symm (Homeomorph.toMeasurableEquiv h)) = ↑(Homeomorph.symm h)

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theorem ContinuousMap.measurable {α : Type u_1} {γ : Type u_3} [TopologicalSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] [TopologicalSpace γ] [MeasurableSpace γ] [BorelSpace γ] (f : C(α, γ)) :

Measurable ↑f

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theorem measurable_of_continuousOn_compl_singleton {α : Type u_1} {γ : Type u_3} [TopologicalSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] [TopologicalSpace γ] [MeasurableSpace γ] [BorelSpace γ] [T1Space α] {f : α → γ} (a : α) (hf : ContinuousOn f {a}ᶜ) :

Measurable f

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theorem Continuous.measurable2 {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_5} [TopologicalSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] [TopologicalSpace β] [MeasurableSpace β] [OpensMeasurableSpace β] [TopologicalSpace γ] [MeasurableSpace γ] [BorelSpace γ] [MeasurableSpace δ] [SecondCountableTopologyEither α β] {f : δ → α} {g : δ → β} {c : α → β → γ} (h : Continuous fun p => c p.fst p.snd) (hf : Measurable f) (hg : Measurable g) :

Measurable fun a => c (f a) (g a)

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theorem Continuous.aemeasurable2 {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_5} [TopologicalSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] [TopologicalSpace β] [MeasurableSpace β] [OpensMeasurableSpace β] [TopologicalSpace γ] [MeasurableSpace γ] [BorelSpace γ] [MeasurableSpace δ] [SecondCountableTopologyEither α β] {f : δ → α} {g : δ → β} {c : α → β → γ} {μ : MeasureTheory.Measure δ} (h : Continuous fun p => c p.fst p.snd) (hf : AEMeasurable f) (hg : AEMeasurable g) :

AEMeasurable fun a => c (f a) (g a)

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instance HasContinuousInv₀.measurableInv {γ : Type u_3} [TopologicalSpace γ] [MeasurableSpace γ] [BorelSpace γ] [GroupWithZero γ] [T1Space γ] [HasContinuousInv₀ γ] :

MeasurableInv γ

Equations

@HasContinuousInv₀.measurableInv = @HasContinuousInv₀.measurableInv.proof_1

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instance ContinuousAdd.measurableMul₂ {γ : Type u_3} [TopologicalSpace γ] [MeasurableSpace γ] [BorelSpace γ] [TopologicalSpace.SecondCountableTopology γ] [Add γ] [ContinuousAdd γ] :

MeasurableAdd₂ γ

Equations

@ContinuousAdd.measurableMul₂ = @ContinuousAdd.measurableMul₂.proof_1

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theorem ContinuousAdd.measurableMul₂.proof_1 {γ : Type u_1} [TopologicalSpace γ] [MeasurableSpace γ] [BorelSpace γ] [TopologicalSpace.SecondCountableTopology γ] [Add γ] [ContinuousAdd γ] :

MeasurableAdd₂ γ

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instance ContinuousMul.measurableMul₂ {γ : Type u_3} [TopologicalSpace γ] [MeasurableSpace γ] [BorelSpace γ] [TopologicalSpace.SecondCountableTopology γ] [Mul γ] [ContinuousMul γ] :

MeasurableMul₂ γ

Equations

@ContinuousMul.measurableMul₂ = @ContinuousMul.measurableMul₂.proof_1

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instance ContinuousSub.measurableSub₂ {γ : Type u_3} [TopologicalSpace γ] [MeasurableSpace γ] [BorelSpace γ] [TopologicalSpace.SecondCountableTopology γ] [Sub γ] [ContinuousSub γ] :

MeasurableSub₂ γ

Equations

@ContinuousSub.measurableSub₂ = @ContinuousSub.measurableSub₂.proof_1

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instance ContinuousSMul.measurableSMul₂ {M : Type u_7} {α : Type u_8} [TopologicalSpace M] [MeasurableSpace M] [OpensMeasurableSpace M] [TopologicalSpace α] [SecondCountableTopologyEither M α] [MeasurableSpace α] [BorelSpace α] [SMul M α] [ContinuousSMul M α] :

MeasurableSMul₂ M α

Equations

@ContinuousSMul.measurableSMul₂ = @ContinuousSMul.measurableSMul₂.proof_1

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theorem pi_le_borel_pi {ι : Type u_6} {π : ι → Type u_7} [(i : ι) → TopologicalSpace (π i)] [(i : ι) → MeasurableSpace (π i)] [∀ (i : ι), BorelSpace (π i)] :

MeasurableSpace.pi ≤ borel ((i : ι) → π i)

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theorem prod_le_borel_prod {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [MeasurableSpace α] [BorelSpace α] [TopologicalSpace β] [MeasurableSpace β] [BorelSpace β] :

Prod.instMeasurableSpace ≤ borel (α × β)

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instance Pi.borelSpace {ι : Type u_6} {π : ι → Type u_7} [Countable ι] [(i : ι) → TopologicalSpace (π i)] [(i : ι) → MeasurableSpace (π i)] [∀ (i : ι), TopologicalSpace.SecondCountableTopology (π i)] [∀ (i : ι), BorelSpace (π i)] :

BorelSpace ((i : ι) → π i)

Equations

@Pi.borelSpace = @Pi.borelSpace.proof_1

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instance Prod.borelSpace {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [MeasurableSpace α] [BorelSpace α] [TopologicalSpace β] [MeasurableSpace β] [BorelSpace β] [SecondCountableTopologyEither α β] :

BorelSpace (α × β)

Equations

@Prod.borelSpace = @Prod.borelSpace.proof_1

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theorem Embedding.measurableEmbedding {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [MeasurableSpace α] [BorelSpace α] [TopologicalSpace β] [MeasurableSpace β] [BorelSpace β] {f : α → β} (h₁ : Embedding f) (h₂ : MeasurableSet (Set.range f)) :

MeasurableEmbedding f

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theorem ClosedEmbedding.measurableEmbedding {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [MeasurableSpace α] [BorelSpace α] [TopologicalSpace β] [MeasurableSpace β] [BorelSpace β] {f : α → β} (h : ClosedEmbedding f) :

MeasurableEmbedding f

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theorem OpenEmbedding.measurableEmbedding {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [MeasurableSpace α] [BorelSpace α] [TopologicalSpace β] [MeasurableSpace β] [BorelSpace β] {f : α → β} (h : OpenEmbedding f) :

MeasurableEmbedding f

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theorem measurable_of_Iio {α : Type u_1} {δ : Type u_5} [TopologicalSpace α] [MeasurableSpace α] [BorelSpace α] [MeasurableSpace δ] [LinearOrder α] [OrderTopology α] [TopologicalSpace.SecondCountableTopology α] {f : δ → α} (hf : ∀ (x : α), MeasurableSet (f ⁻¹' Set.Iio x)) :

Measurable f

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theorem UpperSemicontinuous.measurable {α : Type u_1} {δ : Type u_5} [TopologicalSpace α] [MeasurableSpace α] [BorelSpace α] [MeasurableSpace δ] [LinearOrder α] [OrderTopology α] [TopologicalSpace.SecondCountableTopology α] [TopologicalSpace δ] [OpensMeasurableSpace δ] {f : δ → α} (hf : UpperSemicontinuous f) :

Measurable f

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theorem measurable_of_Ioi {α : Type u_1} {δ : Type u_5} [TopologicalSpace α] [MeasurableSpace α] [BorelSpace α] [MeasurableSpace δ] [LinearOrder α] [OrderTopology α] [TopologicalSpace.SecondCountableTopology α] {f : δ → α} (hf : ∀ (x : α), MeasurableSet (f ⁻¹' Set.Ioi x)) :

Measurable f

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theorem LowerSemicontinuous.measurable {α : Type u_1} {δ : Type u_5} [TopologicalSpace α] [MeasurableSpace α] [BorelSpace α] [MeasurableSpace δ] [LinearOrder α] [OrderTopology α] [TopologicalSpace.SecondCountableTopology α] [TopologicalSpace δ] [OpensMeasurableSpace δ] {f : δ → α} (hf : LowerSemicontinuous f) :

Measurable f

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theorem measurable_of_Iic {α : Type u_1} {δ : Type u_5} [TopologicalSpace α] [MeasurableSpace α] [BorelSpace α] [MeasurableSpace δ] [LinearOrder α] [OrderTopology α] [TopologicalSpace.SecondCountableTopology α] {f : δ → α} (hf : ∀ (x : α), MeasurableSet (f ⁻¹' Set.Iic x)) :

Measurable f

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theorem measurable_of_Ici {α : Type u_1} {δ : Type u_5} [TopologicalSpace α] [MeasurableSpace α] [BorelSpace α] [MeasurableSpace δ] [LinearOrder α] [OrderTopology α] [TopologicalSpace.SecondCountableTopology α] {f : δ → α} (hf : ∀ (x : α), MeasurableSet (f ⁻¹' Set.Ici x)) :

Measurable f

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theorem Measurable.isLUB {α : Type u_1} {δ : Type u_5} [TopologicalSpace α] [MeasurableSpace α] [BorelSpace α] [MeasurableSpace δ] [LinearOrder α] [OrderTopology α] [TopologicalSpace.SecondCountableTopology α] {ι : Sort u_6} [Countable ι] {f : ι → δ → α} {g : δ → α} (hf : ∀ (i : ι), Measurable (f i)) (hg : ∀ (b : δ), IsLUB {a | ∃ i, f i b = a} (g b)) :

Measurable g

If a function is the least upper bound of countably many measurable functions, then it is measurable.

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theorem Measurable.isLUB_of_mem {α : Type u_1} {δ : Type u_5} [TopologicalSpace α] [MeasurableSpace α] [BorelSpace α] [MeasurableSpace δ] [LinearOrder α] [OrderTopology α] [TopologicalSpace.SecondCountableTopology α] {ι : Sort u_6} [Countable ι] {f : ι → δ → α} {g : δ → α} {g' : δ → α} (hf : ∀ (i : ι), Measurable (f i)) {s : Set δ} (hs : MeasurableSet s) (hg : ∀ (b : δ), b ∈ s → IsLUB {a | ∃ i, f i b = a} (g b)) (hg' : Set.EqOn g g' sᶜ) (g'_meas : Measurable g') :

Measurable g

If a function is the least upper bound of countably many measurable functions on a measurable set s, and coincides with a measurable function outside of s, then it is measurable.

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theorem AEMeasurable.isLUB {α : Type u_1} {δ : Type u_5} [TopologicalSpace α] [MeasurableSpace α] [BorelSpace α] [MeasurableSpace δ] [LinearOrder α] [OrderTopology α] [TopologicalSpace.SecondCountableTopology α] {ι : Sort u_6} {μ : MeasureTheory.Measure δ} [Countable ι] {f : ι → δ → α} {g : δ → α} (hf : ∀ (i : ι), AEMeasurable (f i)) (hg : ∀ᵐ (b : δ) ∂μ, IsLUB {a | ∃ i, f i b = a} (g b)) :

AEMeasurable g

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theorem Measurable.isGLB {α : Type u_1} {δ : Type u_5} [TopologicalSpace α] [MeasurableSpace α] [BorelSpace α] [MeasurableSpace δ] [LinearOrder α] [OrderTopology α] [TopologicalSpace.SecondCountableTopology α] {ι : Sort u_6} [Countable ι] {f : ι → δ → α} {g : δ → α} (hf : ∀ (i : ι), Measurable (f i)) (hg : ∀ (b : δ), IsGLB {a | ∃ i, f i b = a} (g b)) :

Measurable g

If a function is the greatest lower bound of countably many measurable functions, then it is measurable.

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theorem Measurable.isGLB_of_mem {α : Type u_1} {δ : Type u_5} [TopologicalSpace α] [MeasurableSpace α] [BorelSpace α] [MeasurableSpace δ] [LinearOrder α] [OrderTopology α] [TopologicalSpace.SecondCountableTopology α] {ι : Sort u_6} [Countable ι] {f : ι → δ → α} {g : δ → α} {g' : δ → α} (hf : ∀ (i : ι), Measurable (f i)) {s : Set δ} (hs : MeasurableSet s) (hg : ∀ (b : δ), b ∈ s → IsGLB {a | ∃ i, f i b = a} (g b)) (hg' : Set.EqOn g g' sᶜ) (g'_meas : Measurable g') :

Measurable g

If a function is the greatest lower bound of countably many measurable functions on a measurable set s, and coincides with a measurable function outside of s, then it is measurable.

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theorem AEMeasurable.isGLB {α : Type u_1} {δ : Type u_5} [TopologicalSpace α] [MeasurableSpace α] [BorelSpace α] [MeasurableSpace δ] [LinearOrder α] [OrderTopology α] [TopologicalSpace.SecondCountableTopology α] {ι : Sort u_6} {μ : MeasureTheory.Measure δ} [Countable ι] {f : ι → δ → α} {g : δ → α} (hf : ∀ (i : ι), AEMeasurable (f i)) (hg : ∀ᵐ (b : δ) ∂μ, IsGLB {a | ∃ i, f i b = a} (g b)) :

AEMeasurable g

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theorem Monotone.measurable {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [MeasurableSpace α] [BorelSpace α] [TopologicalSpace β] [MeasurableSpace β] [BorelSpace β] [LinearOrder α] [OrderTopology α] [TopologicalSpace.SecondCountableTopology α] [LinearOrder β] [OrderClosedTopology β] {f : β → α} (hf : Monotone f) :

Measurable f

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theorem aemeasurable_restrict_of_monotoneOn {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [MeasurableSpace α] [BorelSpace α] [TopologicalSpace β] [MeasurableSpace β] [BorelSpace β] [LinearOrder α] [OrderTopology α] [TopologicalSpace.SecondCountableTopology α] [LinearOrder β] [OrderClosedTopology β] {μ : MeasureTheory.Measure β} {s : Set β} (hs : MeasurableSet s) {f : β → α} (hf : MonotoneOn f s) :

AEMeasurable f

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theorem Antitone.measurable {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [MeasurableSpace α] [BorelSpace α] [TopologicalSpace β] [MeasurableSpace β] [BorelSpace β] [LinearOrder α] [OrderTopology α] [TopologicalSpace.SecondCountableTopology α] [LinearOrder β] [OrderClosedTopology β] {f : β → α} (hf : Antitone f) :

Measurable f

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theorem aemeasurable_restrict_of_antitoneOn {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [MeasurableSpace α] [BorelSpace α] [TopologicalSpace β] [MeasurableSpace β] [BorelSpace β] [LinearOrder α] [OrderTopology α] [TopologicalSpace.SecondCountableTopology α] [LinearOrder β] [OrderClosedTopology β] {μ : MeasureTheory.Measure β} {s : Set β} (hs : MeasurableSet s) {f : β → α} (hf : AntitoneOn f s) :

AEMeasurable f

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theorem measurableSet_of_mem_nhdsWithin_Ioi_aux {α : Type u_1} [TopologicalSpace α] [MeasurableSpace α] [BorelSpace α] [LinearOrder α] [OrderTopology α] [TopologicalSpace.SecondCountableTopology α] {s : Set α} (h : ∀ (x : α), x ∈ s → s ∈ nhdsWithin x (Set.Ioi x)) (h' : ∀ (x : α), x ∈ s → ∃ y, x < y) :

MeasurableSet s

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theorem measurableSet_of_mem_nhdsWithin_Ioi {α : Type u_1} [TopologicalSpace α] [MeasurableSpace α] [BorelSpace α] [LinearOrder α] [OrderTopology α] [TopologicalSpace.SecondCountableTopology α] {s : Set α} (h : ∀ (x : α), x ∈ s → s ∈ nhdsWithin x (Set.Ioi x)) :

MeasurableSet s

If a set is a right-neighborhood of all of its points, then it is measurable.

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theorem measurableSet_bddAbove_range {α : Type u_1} {δ : Type u_5} [TopologicalSpace α] [MeasurableSpace α] [BorelSpace α] [MeasurableSpace δ] [LinearOrder α] [OrderTopology α] [TopologicalSpace.SecondCountableTopology α] {ι : Sort u_6} [Countable ι] {f : ι → δ → α} (hf : ∀ (i : ι), Measurable (f i)) :

MeasurableSet {b | BddAbove (Set.range fun i => f i b)}

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theorem measurableSet_bddBelow_range {α : Type u_1} {δ : Type u_5} [TopologicalSpace α] [MeasurableSpace α] [BorelSpace α] [MeasurableSpace δ] [LinearOrder α] [OrderTopology α] [TopologicalSpace.SecondCountableTopology α] {ι : Sort u_6} [Countable ι] {f : ι → δ → α} (hf : ∀ (i : ι), Measurable (f i)) :

MeasurableSet {b | BddBelow (Set.range fun i => f i b)}

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theorem Measurable.iSup_Prop {δ : Type u_5} [MeasurableSpace δ] {α : Type u_6} [MeasurableSpace α] [ConditionallyCompleteLattice α] (p : Prop) {f : δ → α} (hf : Measurable f) :

Measurable fun b => ⨆ (x : p), f b

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theorem Measurable.iInf_Prop {δ : Type u_5} [MeasurableSpace δ] {α : Type u_6} [MeasurableSpace α] [ConditionallyCompleteLattice α] (p : Prop) {f : δ → α} (hf : Measurable f) :

Measurable fun b => ⨅ (x : p), f b

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theorem measurable_iSup {α : Type u_1} {δ : Type u_5} [TopologicalSpace α] [MeasurableSpace α] [BorelSpace α] [MeasurableSpace δ] [ConditionallyCompleteLinearOrder α] [OrderTopology α] [TopologicalSpace.SecondCountableTopology α] {ι : Sort u_6} [Countable ι] {f : ι → δ → α} (hf : ∀ (i : ι), Measurable (f i)) :

Measurable fun b => ⨆ (i : ι), f i b

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theorem aemeasurable_iSup {α : Type u_1} {δ : Type u_5} [TopologicalSpace α] [MeasurableSpace α] [BorelSpace α] [MeasurableSpace δ] [ConditionallyCompleteLinearOrder α] [OrderTopology α] [TopologicalSpace.SecondCountableTopology α] {ι : Sort u_6} {μ : MeasureTheory.Measure δ} [Countable ι] {f : ι → δ → α} (hf : ∀ (i : ι), AEMeasurable (f i)) :

AEMeasurable fun b => ⨆ (i : ι), f i b

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theorem measurable_iInf {α : Type u_1} {δ : Type u_5} [TopologicalSpace α] [MeasurableSpace α] [BorelSpace α] [MeasurableSpace δ] [ConditionallyCompleteLinearOrder α] [OrderTopology α] [TopologicalSpace.SecondCountableTopology α] {ι : Sort u_6} [Countable ι] {f : ι → δ → α} (hf : ∀ (i : ι), Measurable (f i)) :

Measurable fun b => ⨅ (i : ι), f i b

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theorem aemeasurable_iInf {α : Type u_1} {δ : Type u_5} [TopologicalSpace α] [MeasurableSpace α] [BorelSpace α] [MeasurableSpace δ] [ConditionallyCompleteLinearOrder α] [OrderTopology α] [TopologicalSpace.SecondCountableTopology α] {ι : Sort u_6} {μ : MeasureTheory.Measure δ} [Countable ι] {f : ι → δ → α} (hf : ∀ (i : ι), AEMeasurable (f i)) :

AEMeasurable fun b => ⨅ (i : ι), f i b

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theorem measurable_sSup {α : Type u_1} {δ : Type u_5} [TopologicalSpace α] [MeasurableSpace α] [BorelSpace α] [MeasurableSpace δ] [ConditionallyCompleteLinearOrder α] [OrderTopology α] [TopologicalSpace.SecondCountableTopology α] {ι : Type u_6} {f : ι → δ → α} {s : Set ι} (hs : Set.Countable s) (hf : ∀ (i : ι), i ∈ s → Measurable (f i)) :

Measurable fun x => sSup ((fun i => f i x) '' s)

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theorem measurable_sInf {α : Type u_1} {δ : Type u_5} [TopologicalSpace α] [MeasurableSpace α] [BorelSpace α] [MeasurableSpace δ] [ConditionallyCompleteLinearOrder α] [OrderTopology α] [TopologicalSpace.SecondCountableTopology α] {ι : Type u_6} {f : ι → δ → α} {s : Set ι} (hs : Set.Countable s) (hf : ∀ (i : ι), i ∈ s → Measurable (f i)) :

Measurable fun x => sInf ((fun i => f i x) '' s)

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theorem measurable_biSup {α : Type u_1} {δ : Type u_5} [TopologicalSpace α] [MeasurableSpace α] [BorelSpace α] [MeasurableSpace δ] [ConditionallyCompleteLinearOrder α] [OrderTopology α] [TopologicalSpace.SecondCountableTopology α] {ι : Type u_6} (s : Set ι) {f : ι → δ → α} (hs : Set.Countable s) (hf : ∀ (i : ι), i ∈ s → Measurable (f i)) :

Measurable fun b => ⨆ (i : ι) (_ : i ∈ s), f i b

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theorem aemeasurable_biSup {α : Type u_1} {δ : Type u_5} [TopologicalSpace α] [MeasurableSpace α] [BorelSpace α] [MeasurableSpace δ] [ConditionallyCompleteLinearOrder α] [OrderTopology α] [TopologicalSpace.SecondCountableTopology α] {ι : Type u_6} {μ : MeasureTheory.Measure δ} (s : Set ι) {f : ι → δ → α} (hs : Set.Countable s) (hf : ∀ (i : ι), i ∈ s → AEMeasurable (f i)) :

AEMeasurable fun b => ⨆ (i : ι) (_ : i ∈ s), f i b

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theorem measurable_biInf {α : Type u_1} {δ : Type u_5} [TopologicalSpace α] [MeasurableSpace α] [BorelSpace α] [MeasurableSpace δ] [ConditionallyCompleteLinearOrder α] [OrderTopology α] [TopologicalSpace.SecondCountableTopology α] {ι : Type u_6} (s : Set ι) {f : ι → δ → α} (hs : Set.Countable s) (hf : ∀ (i : ι), i ∈ s → Measurable (f i)) :

Measurable fun b => ⨅ (i : ι) (_ : i ∈ s), f i b

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theorem aemeasurable_biInf {α : Type u_1} {δ : Type u_5} [TopologicalSpace α] [MeasurableSpace α] [BorelSpace α] [MeasurableSpace δ] [ConditionallyCompleteLinearOrder α] [OrderTopology α] [TopologicalSpace.SecondCountableTopology α] {ι : Type u_6} {μ : MeasureTheory.Measure δ} (s : Set ι) {f : ι → δ → α} (hs : Set.Countable s) (hf : ∀ (i : ι), i ∈ s → AEMeasurable (f i)) :

AEMeasurable fun b => ⨅ (i : ι) (_ : i ∈ s), f i b

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theorem measurable_liminf' {α : Type u_1} {δ : Type u_5} [TopologicalSpace α] [MeasurableSpace α] [BorelSpace α] [MeasurableSpace δ] [ConditionallyCompleteLinearOrder α] [OrderTopology α] [TopologicalSpace.SecondCountableTopology α] {ι : Type u_6} {ι' : Type u_7} {f : ι → δ → α} {v : Filter ι} (hf : ∀ (i : ι), Measurable (f i)) {p : ι' → Prop} {s : ι' → Set ι} (hv : Filter.HasCountableBasis v p s) (hs : ∀ (j : ι'), Set.Countable (s j)) :

Measurable fun x => Filter.liminf (fun i => f i x) v

liminf over a general filter is measurable. See measurable_liminf for the version over ℕ.

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theorem measurable_limsup' {α : Type u_1} {δ : Type u_5} [TopologicalSpace α] [MeasurableSpace α] [BorelSpace α] [MeasurableSpace δ] [ConditionallyCompleteLinearOrder α] [OrderTopology α] [TopologicalSpace.SecondCountableTopology α] {ι : Type u_6} {ι' : Type u_7} {f : ι → δ → α} {u : Filter ι} (hf : ∀ (i : ι), Measurable (f i)) {p : ι' → Prop} {s : ι' → Set ι} (hu : Filter.HasCountableBasis u p s) (hs : ∀ (i : ι'), Set.Countable (s i)) :

Measurable fun x => Filter.limsup (fun i => f i x) u

limsup over a general filter is measurable. See measurable_limsup for the version over ℕ.

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theorem measurable_liminf {α : Type u_1} {δ : Type u_5} [TopologicalSpace α] [MeasurableSpace α] [BorelSpace α] [MeasurableSpace δ] [ConditionallyCompleteLinearOrder α] [OrderTopology α] [TopologicalSpace.SecondCountableTopology α] {f : ℕ → δ → α} (hf : ∀ (i : ℕ), Measurable (f i)) :

Measurable fun x => Filter.liminf (fun i => f i x) Filter.atTop

liminf over ℕ is measurable. See measurable_liminf' for a version with a general filter.

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theorem measurable_limsup {α : Type u_1} {δ : Type u_5} [TopologicalSpace α] [MeasurableSpace α] [BorelSpace α] [MeasurableSpace δ] [ConditionallyCompleteLinearOrder α] [OrderTopology α] [TopologicalSpace.SecondCountableTopology α] {f : ℕ → δ → α} (hf : ∀ (i : ℕ), Measurable (f i)) :

Measurable fun x => Filter.limsup (fun i => f i x) Filter.atTop

limsup over ℕ is measurable. See measurable_limsup' for a version with a general filter.

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def Homemorph.toMeasurableEquiv {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [MeasurableSpace α] [BorelSpace α] [TopologicalSpace β] [MeasurableSpace β] [BorelSpace β] (h : α ≃ₜ β) :

α ≃ᵐ β

Convert a Homeomorph to a MeasurableEquiv.

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Homemorph.toMeasurableEquiv h = { toEquiv := h.toEquiv, measurable_toFun := (_ : Measurable h.toFun), measurable_invFun := (_ : Measurable h.invFun) }

Instances For

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theorem IsFiniteMeasureOnCompacts.map {α : Type u_6} {m0 : MeasurableSpace α} [TopologicalSpace α] [OpensMeasurableSpace α] {β : Type u_7} [MeasurableSpace β] [TopologicalSpace β] [BorelSpace β] [T2Space β] (μ : MeasureTheory.Measure α) [MeasureTheory.IsFiniteMeasureOnCompacts μ] (f : α ≃ₜ β) :

MeasureTheory.IsFiniteMeasureOnCompacts (MeasureTheory.Measure.map (↑f) μ)

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instance Empty.borelSpace :

BorelSpace Empty

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Empty.borelSpace = Empty.borelSpace.proof_1

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instance Unit.borelSpace :

BorelSpace Unit

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Unit.borelSpace = Unit.borelSpace.proof_1

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instance Bool.borelSpace :

BorelSpace Bool

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Bool.borelSpace = Bool.borelSpace.proof_1

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instance Nat.borelSpace :

BorelSpace ℕ

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Nat.borelSpace = Nat.borelSpace.proof_1

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instance Int.borelSpace :

BorelSpace ℤ

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Int.borelSpace = Int.borelSpace.proof_1

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instance Rat.borelSpace :

BorelSpace ℚ

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Rat.borelSpace = Rat.borelSpace.proof_1

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instance Real.measurableSpace :

MeasurableSpace ℝ

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Real.measurableSpace = borel ℝ

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instance Real.borelSpace :

BorelSpace ℝ

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Real.borelSpace = Real.borelSpace.proof_1

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instance NNReal.measurableSpace :

MeasurableSpace NNReal

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NNReal.measurableSpace = Subtype.instMeasurableSpace

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instance NNReal.borelSpace :

BorelSpace NNReal

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NNReal.borelSpace = NNReal.borelSpace.proof_1

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instance ENNReal.measurableSpace :

MeasurableSpace ENNReal

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ENNReal.measurableSpace = borel ENNReal

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instance ENNReal.borelSpace :

BorelSpace ENNReal

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ENNReal.borelSpace = ENNReal.borelSpace.proof_1

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instance EReal.measurableSpace :

MeasurableSpace EReal

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EReal.measurableSpace = borel EReal

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instance EReal.borelSpace :

BorelSpace EReal

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EReal.borelSpace = EReal.borelSpace.proof_1

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theorem measure_eq_measure_preimage_add_measure_tsum_Ico_zpow {α : Type u_1} [MeasurableSpace α] (μ : MeasureTheory.Measure α) {f : α → ENNReal} (hf : Measurable f) {s : Set α} (hs : MeasurableSet s) {t : NNReal} (ht : 1 < t) :

↑↑μ s = ↑↑μ (s ∩ f ⁻¹' {0}) + ↑↑μ (s ∩ f ⁻¹' {⊤}) + ∑' (n : ℤ), ↑↑μ (s ∩ f ⁻¹' Set.Ico (↑t ^ n) (↑t ^ (n + 1)))

One can cut out ℝ≥0∞ into the sets {0}, Ico (t^n) (t^(n+1)) for n : ℤ and {∞}. This gives a way to compute the measure of a set in terms of sets on which a given function f does not fluctuate by more than t.

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theorem measurableSet_ball {α : Type u_1} [PseudoMetricSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] {x : α} {ε : ℝ} :

MeasurableSet (Metric.ball x ε)

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theorem measurableSet_closedBall {α : Type u_1} [PseudoMetricSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] {x : α} {ε : ℝ} :

MeasurableSet (Metric.closedBall x ε)

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theorem measurable_infDist {α : Type u_1} [PseudoMetricSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] {s : Set α} :

Measurable fun x => Metric.infDist x s

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theorem Measurable.infDist {α : Type u_1} {β : Type u_2} [PseudoMetricSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] [MeasurableSpace β] {f : β → α} (hf : Measurable f) {s : Set α} :

Measurable fun x => Metric.infDist (f x) s

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theorem measurable_infNndist {α : Type u_1} [PseudoMetricSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] {s : Set α} :

Measurable fun x => Metric.infNndist x s

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theorem Measurable.infNndist {α : Type u_1} {β : Type u_2} [PseudoMetricSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] [MeasurableSpace β] {f : β → α} (hf : Measurable f) {s : Set α} :

Measurable fun x => Metric.infNndist (f x) s

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theorem measurable_dist {α : Type u_1} [PseudoMetricSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] [TopologicalSpace.SecondCountableTopology α] :

Measurable fun p => dist p.fst p.snd

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theorem Measurable.dist {α : Type u_1} {β : Type u_2} [PseudoMetricSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] [MeasurableSpace β] [TopologicalSpace.SecondCountableTopology α] {f : β → α} {g : β → α} (hf : Measurable f) (hg : Measurable g) :

Measurable fun b => dist (f b) (g b)

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theorem measurable_nndist {α : Type u_1} [PseudoMetricSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] [TopologicalSpace.SecondCountableTopology α] :

Measurable fun p => nndist p.fst p.snd

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theorem Measurable.nndist {α : Type u_1} {β : Type u_2} [PseudoMetricSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] [MeasurableSpace β] [TopologicalSpace.SecondCountableTopology α] {f : β → α} {g : β → α} (hf : Measurable f) (hg : Measurable g) :

Measurable fun b => nndist (f b) (g b)

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theorem measurableSet_eball {α : Type u_1} [PseudoEMetricSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] {x : α} {ε : ENNReal} :

MeasurableSet (EMetric.ball x ε)

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theorem measurable_edist_right {α : Type u_1} [PseudoEMetricSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] {x : α} :

Measurable (edist x)

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theorem measurable_edist_left {α : Type u_1} [PseudoEMetricSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] {x : α} :

Measurable fun y => edist y x

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theorem measurable_infEdist {α : Type u_1} [PseudoEMetricSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] {s : Set α} :

Measurable fun x => EMetric.infEdist x s

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theorem Measurable.infEdist {α : Type u_1} {β : Type u_2} [PseudoEMetricSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] [MeasurableSpace β] {f : β → α} (hf : Measurable f) {s : Set α} :

Measurable fun x => EMetric.infEdist (f x) s

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theorem tendsto_measure_cthickening {α : Type u_1} [PseudoEMetricSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] {μ : MeasureTheory.Measure α} {s : Set α} (hs : ∃ R, R > 0 ∧ ↑↑μ (Metric.cthickening R s) ≠ ⊤) :

Filter.Tendsto (fun r => ↑↑μ (Metric.cthickening r s)) (nhds 0) (nhds (↑↑μ (closure s)))

If a set has a closed thickening with finite measure, then the measure of its r-closed thickenings converges to the measure of its closure as r tends to 0.

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theorem tendsto_measure_cthickening_of_isClosed {α : Type u_1} [PseudoEMetricSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] {μ : MeasureTheory.Measure α} {s : Set α} (hs : ∃ R, R > 0 ∧ ↑↑μ (Metric.cthickening R s) ≠ ⊤) (h's : IsClosed s) :

Filter.Tendsto (fun r => ↑↑μ (Metric.cthickening r s)) (nhds 0) (nhds (↑↑μ s))

If a closed set has a closed thickening with finite measure, then the measure of its closed r-thickenings converge to its measure as r tends to 0.

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theorem tendsto_measure_thickening {α : Type u_1} [PseudoEMetricSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] {μ : MeasureTheory.Measure α} {s : Set α} (hs : ∃ R, R > 0 ∧ ↑↑μ (Metric.thickening R s) ≠ ⊤) :

Filter.Tendsto (fun r => ↑↑μ (Metric.thickening r s)) (nhdsWithin 0 (Set.Ioi 0)) (nhds (↑↑μ (closure s)))

If a set has a thickening with finite measure, then the measures of its r-thickenings converge to the measure of its closure as r > 0 tends to 0.

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theorem tendsto_measure_thickening_of_isClosed {α : Type u_1} [PseudoEMetricSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] {μ : MeasureTheory.Measure α} {s : Set α} (hs : ∃ R, R > 0 ∧ ↑↑μ (Metric.thickening R s) ≠ ⊤) (h's : IsClosed s) :

Filter.Tendsto (fun r => ↑↑μ (Metric.thickening r s)) (nhdsWithin 0 (Set.Ioi 0)) (nhds (↑↑μ s))

If a closed set has a thickening with finite measure, then the measure of its r-thickenings converge to its measure as r > 0 tends to 0.

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theorem measurable_edist {α : Type u_1} [PseudoEMetricSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] [TopologicalSpace.SecondCountableTopology α] :

Measurable fun p => edist p.fst p.snd

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theorem Measurable.edist {α : Type u_1} {β : Type u_2} [PseudoEMetricSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] [MeasurableSpace β] [TopologicalSpace.SecondCountableTopology α] {f : β → α} {g : β → α} (hf : Measurable f) (hg : Measurable g) :

Measurable fun b => edist (f b) (g b)

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theorem AEMeasurable.edist {α : Type u_1} {β : Type u_2} [PseudoEMetricSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] [MeasurableSpace β] [TopologicalSpace.SecondCountableTopology α] {f : β → α} {g : β → α} {μ : MeasureTheory.Measure β} (hf : AEMeasurable f) (hg : AEMeasurable g) :

AEMeasurable fun a => edist (f a) (g a)

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theorem tendsto_measure_cthickening_of_isCompact {α : Type u_1} [MetricSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] [ProperSpace α] {μ : MeasureTheory.Measure α} [MeasureTheory.IsFiniteMeasureOnCompacts μ] {s : Set α} (hs : IsCompact s) :

Filter.Tendsto (fun r => ↑↑μ (Metric.cthickening r s)) (nhds 0) (nhds (↑↑μ s))

Given a compact set in a proper space, the measure of its r-closed thickenings converges to its measure as r tends to 0.

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theorem Real.borel_eq_generateFrom_Ioo_rat :

borel ℝ = MeasurableSpace.generateFrom (⋃ (a : ℚ) (b : ℚ) (_ : a < b), {Set.Ioo ↑a ↑b})

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theorem Real.borel_eq_generateFrom_Iio_rat :

borel ℝ = MeasurableSpace.generateFrom (⋃ (a : ℚ), {Set.Iio ↑a})

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theorem Real.borel_eq_generateFrom_Ioi_rat :

borel ℝ = MeasurableSpace.generateFrom (⋃ (a : ℚ), {Set.Ioi ↑a})

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theorem Real.borel_eq_generateFrom_Iic_rat :

borel ℝ = MeasurableSpace.generateFrom (⋃ (a : ℚ), {Set.Iic ↑a})

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theorem Real.borel_eq_generateFrom_Ici_rat :

borel ℝ = MeasurableSpace.generateFrom (⋃ (a : ℚ), {Set.Ici ↑a})

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theorem Real.isPiSystem_Ioo_rat :

IsPiSystem (⋃ (a : ℚ) (b : ℚ) (_ : a < b), {Set.Ioo ↑a ↑b})

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theorem Real.isPiSystem_Iio_rat :

IsPiSystem (⋃ (a : ℚ), {Set.Iio ↑a})

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theorem Real.isPiSystem_Ioi_rat :

IsPiSystem (⋃ (a : ℚ), {Set.Ioi ↑a})

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theorem Real.isPiSystem_Iic_rat :

IsPiSystem (⋃ (a : ℚ), {Set.Iic ↑a})

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theorem Real.isPiSystem_Ici_rat :

IsPiSystem (⋃ (a : ℚ), {Set.Ici ↑a})

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def Real.finiteSpanningSetsInIooRat (μ : MeasureTheory.Measure ℝ) [MeasureTheory.IsLocallyFiniteMeasure μ] :

MeasureTheory.Measure.FiniteSpanningSetsIn μ (⋃ (a : ℚ) (b : ℚ) (_ : a < b), {Set.Ioo ↑a ↑b})

The intervals (-(n + 1), (n + 1)) form a finite spanning sets in the set of open intervals with rational endpoints for a locally finite measure μ on ℝ.

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

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theorem Real.measure_ext_Ioo_rat {μ : MeasureTheory.Measure ℝ} {ν : MeasureTheory.Measure ℝ} [MeasureTheory.IsLocallyFiniteMeasure μ] (h : ∀ (a b : ℚ), ↑↑μ (Set.Ioo ↑a ↑b) = ↑↑ν (Set.Ioo ↑a ↑b)) :

μ = ν

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theorem measurable_real_toNNReal :

Measurable Real.toNNReal

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theorem Measurable.real_toNNReal {α : Type u_1} [MeasurableSpace α] {f : α → ℝ} (hf : Measurable f) :

Measurable fun x => Real.toNNReal (f x)

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theorem AEMeasurable.real_toNNReal {α : Type u_1} [MeasurableSpace α] {f : α → ℝ} {μ : MeasureTheory.Measure α} (hf : AEMeasurable f) :

AEMeasurable fun x => Real.toNNReal (f x)

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theorem measurable_coe_nnreal_real :

Measurable NNReal.toReal

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theorem Measurable.coe_nnreal_real {α : Type u_1} [MeasurableSpace α] {f : α → NNReal} (hf : Measurable f) :

Measurable fun x => ↑(f x)

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theorem AEMeasurable.coe_nnreal_real {α : Type u_1} [MeasurableSpace α] {f : α → NNReal} {μ : MeasureTheory.Measure α} (hf : AEMeasurable f) :

AEMeasurable fun x => ↑(f x)

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theorem measurable_coe_nnreal_ennreal :

Measurable ENNReal.some

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theorem Measurable.coe_nnreal_ennreal {α : Type u_1} [MeasurableSpace α] {f : α → NNReal} (hf : Measurable f) :

Measurable fun x => ↑(f x)

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theorem AEMeasurable.coe_nnreal_ennreal {α : Type u_1} [MeasurableSpace α] {f : α → NNReal} {μ : MeasureTheory.Measure α} (hf : AEMeasurable f) :

AEMeasurable fun x => ↑(f x)

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theorem Measurable.ennreal_ofReal {α : Type u_1} [MeasurableSpace α] {f : α → ℝ} (hf : Measurable f) :

Measurable fun x => ENNReal.ofReal (f x)

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@[simp]

theorem measurable_coe_nnreal_real_iff {α : Type u_1} [MeasurableSpace α] {f : α → NNReal} :

(Measurable fun x => ↑(f x)) ↔ Measurable f

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@[simp]

theorem aEMeasurable_coe_nnreal_real_iff {α : Type u_1} [MeasurableSpace α] {f : α → NNReal} {μ : MeasureTheory.Measure α} :

(AEMeasurable fun x => ↑(f x)) ↔ AEMeasurable f

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def MeasurableEquiv.ennrealEquivNNReal :

↑{r | r ≠ ⊤} ≃ᵐ NNReal

The set of finite ℝ≥0∞ numbers is MeasurableEquiv to ℝ≥0.

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MeasurableEquiv.ennrealEquivNNReal = Homeomorph.toMeasurableEquiv ENNReal.neTopHomeomorphNNReal

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theorem ENNReal.measurable_of_measurable_nnreal {α : Type u_1} [MeasurableSpace α] {f : ENNReal → α} (h : Measurable fun p => f ↑p) :

Measurable f

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def ENNReal.ennrealEquivSum :

ENNReal ≃ᵐ NNReal ⊕ Unit

ℝ≥0∞ is MeasurableEquiv to ℝ≥0 ⊕ Unit.

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

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theorem ENNReal.measurable_of_measurable_nnreal_prod {β : Type u_2} {γ : Type u_3} [MeasurableSpace β] [MeasurableSpace γ] {f : ENNReal × β → γ} (H₁ : Measurable fun p => f (↑p.fst, p.snd)) (H₂ : Measurable fun x => f (⊤, x)) :

Measurable f

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theorem ENNReal.measurable_of_measurable_nnreal_nnreal {β : Type u_2} [MeasurableSpace β] {f : ENNReal × ENNReal → β} (h₁ : Measurable fun p => f (↑p.fst, ↑p.snd)) (h₂ : Measurable fun r => f (⊤, ↑r)) (h₃ : Measurable fun r => f (↑r, ⊤)) :

Measurable f

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theorem ENNReal.measurable_ofReal :

Measurable ENNReal.ofReal

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theorem ENNReal.measurable_toReal :

Measurable ENNReal.toReal

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theorem ENNReal.measurable_toNNReal :

Measurable ENNReal.toNNReal

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instance ENNReal.instMeasurableMul₂ :

MeasurableMul₂ ENNReal

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ENNReal.instMeasurableMul₂ = ENNReal.instMeasurableMul₂.proof_1

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instance ENNReal.instMeasurableSub₂ :

MeasurableSub₂ ENNReal

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ENNReal.instMeasurableSub₂ = ENNReal.instMeasurableSub₂.proof_1

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instance ENNReal.instMeasurableInv :

MeasurableInv ENNReal

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ENNReal.instMeasurableInv = ENNReal.instMeasurableInv.proof_1

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theorem Measurable.ennreal_toNNReal {α : Type u_1} [MeasurableSpace α] {f : α → ENNReal} (hf : Measurable f) :

Measurable fun x => ENNReal.toNNReal (f x)

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theorem AEMeasurable.ennreal_toNNReal {α : Type u_1} [MeasurableSpace α] {f : α → ENNReal} {μ : MeasureTheory.Measure α} (hf : AEMeasurable f) :

AEMeasurable fun x => ENNReal.toNNReal (f x)

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@[simp]

theorem measurable_coe_nnreal_ennreal_iff {α : Type u_1} [MeasurableSpace α] {f : α → NNReal} :

(Measurable fun x => ↑(f x)) ↔ Measurable f

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@[simp]

theorem aemeasurable_coe_nnreal_ennreal_iff {α : Type u_1} [MeasurableSpace α] {f : α → NNReal} {μ : MeasureTheory.Measure α} :

(AEMeasurable fun x => ↑(f x)) ↔ AEMeasurable f

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theorem Measurable.ennreal_toReal {α : Type u_1} [MeasurableSpace α] {f : α → ENNReal} (hf : Measurable f) :

Measurable fun x => ENNReal.toReal (f x)

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theorem AEMeasurable.ennreal_toReal {α : Type u_1} [MeasurableSpace α] {f : α → ENNReal} {μ : MeasureTheory.Measure α} (hf : AEMeasurable f) :

AEMeasurable fun x => ENNReal.toReal (f x)

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theorem Measurable.ennreal_tsum {α : Type u_1} [MeasurableSpace α] {ι : Type u_6} [Countable ι] {f : ι → α → ENNReal} (h : ∀ (i : ι), Measurable (f i)) :

Measurable fun x => ∑' (i : ι), f i x

note: ℝ≥0∞ can probably be generalized in a future version of this lemma.

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theorem Measurable.ennreal_tsum' {α : Type u_1} [MeasurableSpace α] {ι : Type u_6} [Countable ι] {f : ι → α → ENNReal} (h : ∀ (i : ι), Measurable (f i)) :

Measurable (∑' (i : ι), f i)

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theorem Measurable.nnreal_tsum {α : Type u_1} [MeasurableSpace α] {ι : Type u_6} [Countable ι] {f : ι → α → NNReal} (h : ∀ (i : ι), Measurable (f i)) :

Measurable fun x => ∑' (i : ι), f i x

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theorem AEMeasurable.ennreal_tsum {α : Type u_1} [MeasurableSpace α] {ι : Type u_6} [Countable ι] {f : ι → α → ENNReal} {μ : MeasureTheory.Measure α} (h : ∀ (i : ι), AEMeasurable (f i)) :

AEMeasurable fun x => ∑' (i : ι), f i x

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theorem AEMeasurable.nnreal_tsum {α : Type u_6} [MeasurableSpace α] {ι : Type u_7} [Countable ι] {f : ι → α → NNReal} {μ : MeasureTheory.Measure α} (h : ∀ (i : ι), AEMeasurable (f i)) :

AEMeasurable fun x => ∑' (i : ι), f i x

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theorem measurable_coe_real_ereal :

Measurable Real.toEReal

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theorem Measurable.coe_real_ereal {α : Type u_1} [MeasurableSpace α] {f : α → ℝ} (hf : Measurable f) :

Measurable fun x => ↑(f x)

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theorem AEMeasurable.coe_real_ereal {α : Type u_1} [MeasurableSpace α] {f : α → ℝ} {μ : MeasureTheory.Measure α} (hf : AEMeasurable f) :

AEMeasurable fun x => ↑(f x)

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def MeasurableEquiv.erealEquivReal :

↑{⊥, ⊤}ᶜ ≃ᵐ ℝ

The set of finite EReal numbers is MeasurableEquiv to ℝ.

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MeasurableEquiv.erealEquivReal = Homeomorph.toMeasurableEquiv EReal.neBotTopHomeomorphReal

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theorem EReal.measurable_of_measurable_real {α : Type u_1} [MeasurableSpace α] {f : EReal → α} (h : Measurable fun p => f ↑p) :

Measurable f

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theorem measurable_ereal_toReal :

Measurable EReal.toReal

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theorem Measurable.ereal_toReal {α : Type u_1} [MeasurableSpace α] {f : α → EReal} (hf : Measurable f) :

Measurable fun x => EReal.toReal (f x)

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theorem AEMeasurable.ereal_toReal {α : Type u_1} [MeasurableSpace α] {f : α → EReal} {μ : MeasureTheory.Measure α} (hf : AEMeasurable f) :

AEMeasurable fun x => EReal.toReal (f x)

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theorem measurable_coe_ennreal_ereal :

Measurable ENNReal.toEReal

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theorem Measurable.coe_ereal_ennreal {α : Type u_1} [MeasurableSpace α] {f : α → ENNReal} (hf : Measurable f) :

Measurable fun x => ↑(f x)

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theorem AEMeasurable.coe_ereal_ennreal {α : Type u_1} [MeasurableSpace α] {f : α → ENNReal} {μ : MeasureTheory.Measure α} (hf : AEMeasurable f) :

AEMeasurable fun x => ↑(f x)

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theorem measurable_norm {α : Type u_1} [MeasurableSpace α] [NormedAddCommGroup α] [OpensMeasurableSpace α] :

Measurable norm

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theorem Measurable.norm {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [NormedAddCommGroup α] [OpensMeasurableSpace α] [MeasurableSpace β] {f : β → α} (hf : Measurable f) :

Measurable fun a => ‖f a‖

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theorem AEMeasurable.norm {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [NormedAddCommGroup α] [OpensMeasurableSpace α] [MeasurableSpace β] {f : β → α} {μ : MeasureTheory.Measure β} (hf : AEMeasurable f) :

AEMeasurable fun a => ‖f a‖

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theorem measurable_nnnorm {α : Type u_1} [MeasurableSpace α] [NormedAddCommGroup α] [OpensMeasurableSpace α] :

Measurable nnnorm

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theorem Measurable.nnnorm {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [NormedAddCommGroup α] [OpensMeasurableSpace α] [MeasurableSpace β] {f : β → α} (hf : Measurable f) :

Measurable fun a => ‖f a‖₊

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theorem AEMeasurable.nnnorm {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [NormedAddCommGroup α] [OpensMeasurableSpace α] [MeasurableSpace β] {f : β → α} {μ : MeasureTheory.Measure β} (hf : AEMeasurable f) :

AEMeasurable fun a => ‖f a‖₊

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theorem measurable_ennnorm {α : Type u_1} [MeasurableSpace α] [NormedAddCommGroup α] [OpensMeasurableSpace α] :

Measurable fun x => ↑‖x‖₊

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theorem Measurable.ennnorm {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [NormedAddCommGroup α] [OpensMeasurableSpace α] [MeasurableSpace β] {f : β → α} (hf : Measurable f) :

Measurable fun a => ↑‖f a‖₊

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theorem AEMeasurable.ennnorm {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [NormedAddCommGroup α] [OpensMeasurableSpace α] [MeasurableSpace β] {f : β → α} {μ : MeasureTheory.Measure β} (hf : AEMeasurable f) :

AEMeasurable fun a => ↑‖f a‖₊