mathlib documentation


The category of measurable spaces #

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

Main definitions #

Tags #

measurable space, giry monad, borel

def Meas  :
Type (u+1)

The category of measurable spaces and measurable functions.

def Meas.of (α : Type u) [measurable_space α] :

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

theorem Meas.coe_of (X : Type u) [measurable_space X] :

Measure X is the measurable space of measures over the measurable space X. It is the weakest measurable space, s.t. λμ, 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 Meas-morphisms X → Prob X are (sub-)Markov kernels (here Prob is the restriction of Measure to (sub-)probability space.)


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


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

def Borel  :

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