Category of measurable spaces

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In mathematics, the category of measurable spaces, often denoted Meas, is the category whose objects are measurable spaces and whose morphisms are measurable maps. [1] [2] [3] [4] This is a category because the composition of two measurable maps is again measurable, and the identity function is measurable.

Contents

N.B. Some authors reserve the name Meas for categories whose objects are measure spaces, and denote the category of measurable spaces as Mble, or other notations. Some authors also restrict the category only to particular well-behaved measurable spaces, such as standard Borel spaces.

As a concrete category

Like many categories, the category Meas is a concrete category, meaning its objects are sets with additional structure (i.e. sigma-algebras) and its morphisms are functions preserving this structure. There is a natural forgetful functor

U : Meas→Set

to the category of sets which assigns to each measurable space the underlying set and to each measurable map the underlying function.

The forgetful functor U has both a left adjoint

D : Set→Meas

which equips a given set with the discrete sigma-algebra, and a right adjoint

I : Set→Meas

which equips a given set with the indiscrete or trivial sigma-algebra. Both of these functors are, in fact, right inverses to U (meaning that UD and UI are equal to the identity functor on Set). Moreover, since any function between discrete or between indiscrete spaces is measurable, both of these functors give full embeddings of Set into Meas.

Limits and colimits

The category Meas is both complete and cocomplete, which means that all small limits and colimits exist in Meas. In fact, the forgetful functor U : Meas → Set uniquely lifts both limits and colimits and preserves them as well. Therefore, (co)limits in Meas are given by placing particular sigma-algebras on the corresponding (co)limits in Set.

Examples of limits and colimits in Meas include:

Other properties

See also

Citations

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