The category of measurable spaces

Measurable spaces and measurable functions form a (concrete) category MeasCat.

Main definitions

• Measure : MeasCat ⥤ MeasCat: the functor which sends a measurable space X to the space of measures on X; it is a monad (the "Giry monad").

• Borel : TopCat ⥤ MeasCat: sends a topological space X to X equipped with the σ-algebra of Borel sets (the σ-algebra generated by the open subsets of X).

Tags

measurable space, giry monad, borel

structure MeasCat : Type (u + 1)

The category of measurable spaces and measurable functions.

• carrier : Type u

  The underlying measurable space.

• str : MeasurableSpace self.carrier

instance MeasCat.instCoeSortType : CoeSort MeasCat (Type u_1)

Equations

• MeasCat.instCoeSortType = { coe := MeasCat.carrier }

@[reducible, inline]

abbrev MeasCat.of (α : Type u) [ms : MeasurableSpace α] : MeasCat

Construct a bundled MeasCat from the underlying type and the typeclass.

Equations

• MeasCat.of α = { carrier := α, str := ms }

theorem MeasCat.coe_of (X : Type u) [MeasurableSpace X] : (of X).carrier = X

instance MeasCat.instLargeCategory : CategoryTheory.LargeCategory MeasCat

Equations

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instance MeasCat.instFunLikeSubtypeForallCarrierMeasurable (X : MeasCat) (Y : MeasCat) : FunLike { f : X.carrier → Y.carrier // Measurable f } X.carrier Y.carrier

Equations

• X.instFunLikeSubtypeForallCarrierMeasurable Y = { coe := fun (f : { f : X.carrier → Y.carrier // Measurable f }) => ↑f, coe_injective' := ⋯ }

instance MeasCat.instConcreteCategorySubtypeForallCarrierMeasurable : CategoryTheory.ConcreteCategory MeasCat fun (x1 x2 : MeasCat) => { f : x1.carrier → x2.carrier // Measurable f }

Equations

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instance MeasCat.instInhabited : Inhabited MeasCat

Equations

• MeasCat.instInhabited = { default := MeasCat.of Empty }

def MeasCat.Measure : CategoryTheory.Functor MeasCat MeasCat

Measure X is the measurable space of measures over the measurable space X. It is the weakest measurable space, s.t. fun μ ↦ μ s is measurable for all measurable sets s in X. An important purpose is to assign a monadic structure on it, the Giry monad. In the Giry monad, the pure values are the Dirac measure, and the bind operation maps to the integral: (μ >>= ν) s = ∫ x. (ν x) s dμ.

In probability theory, the MeasCat-morphisms X → Prob X are (sub-)Markov kernels (here Prob is the restriction of Measure to (sub-)probability space.)

Equations

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def MeasCat.Giry : CategoryTheory.Monad MeasCat

The Giry monad, i.e. the monadic structure associated with Measure.

Equations

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def MeasCat.Integral : Giry.Algebra

An example for an algebra on Measure: the nonnegative Lebesgue integral is a hom, behaving nicely under the monad operations.

Equations

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instance TopCat.hasForgetToMeasCat : CategoryTheory.HasForget₂ TopCat MeasCat

Equations

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@[reducible, inline]

abbrev Borel : CategoryTheory.Functor TopCat MeasCat

The Borel functor, the canonical embedding of topological spaces into measurable spaces.

Equations

• Borel = CategoryTheory.forget₂ TopCat MeasCat