# Documentation

Mathlib.MeasureTheory.Category.MeasCat

# 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 #

def MeasCat :
Type (u + 1)

The category of measurable spaces and measurable functions.

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def MeasCat.of (α : Type u) [ms : ] :

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

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@[simp]
theorem MeasCat.coe_of (X : Type u) [] :
↑() = X

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.)

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

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

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@[reducible]

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

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