Category of measurable spaces
{{Short description|Category of measurable spaces, on the model of the page "Category of topological spaces".}}
In mathematics, the category of measurable spaces, often denoted Meas, is the category whose objects are measurable spaces and whose morphisms are measurable maps.{{harvp|Giry|1982|page=69}}{{harvp|Jacobs|2018|page=205}}{{harvp|Fritz|2020|page=20}}{{harvp|Moss|Perrone|2022|page=3}}
This is a category because the composition of two measurable maps is again measurable, and the identity function is measurable.
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:
- The empty set (considered as a measurable space) is the initial object of Meas; any singleton measurable space is a terminal object. There are thus no zero objects in Meas.
- The product in Meas is given by the product sigma-algebra on the Cartesian product. The coproduct is given by the disjoint union of measurable spaces.
- The equalizer of a pair of morphisms is given by placing the induced sigma-algebra on the subset given by the set-theoretic equalizer. Dually, the coequalizer is given by placing the quotient sigma-algebra on the set-theoretic coequalizer.
- Direct limits and inverse limits are the set-theoretic limits with the final and initial sigma-algebra respectively. Canonical examples of direct and inverse systems are the ones arising from filtrations in probability theory, and the limits and colimits of such systems are, respectively, the join and the intersection of sigma-algebras.
Other properties
- The monomorphisms in Meas are the injective measurable maps, the epimorphisms are the surjective measurable maps, and the isomorphisms are the isomorphisms of measurable spaces.
- The split monomorphisms are (essentially) the inclusions of measurable retracts into their ambient space.
- The split epimorphisms are (up to isomorphism) the measurable surjective maps of a measurable space onto one of its retracts.
- Meas is not cartesian closed (and therefore also not a topos) since it does not have exponential objects for all spaces.
See also
- {{annotated link|Category of topological spaces}}
- {{annotated link|Category of sets}}
- {{annotated link|Category of measure spaces}}
- {{annotated link|Category of Markov kernels}}
- {{annotated link|Measurable space}}
- {{annotated link|Measurable function}}
Citations
{{reflist}}
References
{{refbegin}}
- {{cite book
|last = Giry
|first = Michèle
|title = Categorical Aspects of Topology and Analysis
|chapter = A categorical approach to probability theory
|series = Lecture Notes in Mathematics
|date = 1982
|volume = 915
|pages = 68–85
|publisher = Springer
|doi = 10.1007/BFb0092872
|isbn = 978-3-540-11211-2
|chapter-url = https://link.springer.com/chapter/10.1007/BFb0092872
}}
- {{cite journal
| last = Jacobs | first = Bart
| title = From probability monads to commutative effectuses
| journal = Journal of Logical and Algebraic Methods in Programming
| volume = 94
| pages = 200–237
| date = 2018
| doi = 10.1016/j.jlamp.2016.11.006
| url = https://www.sciencedirect.com/science/article/pii/S2352220816301122
| hdl = 2066/182000
| hdl-access = free
}}
- {{cite journal
| last = Fritz | first = Tobias
| title = A synthetic approach to Markov kernels, conditional independence and theorems on sufficient statistics
| journal = Advances in Mathematics
| volume = 370
| date = 2020
| doi = 10.1016/j.aim.2020.107239
| arxiv = 1908.07021
| s2cid = 201103837
| url = https://www.sciencedirect.com/science/article/pii/S0001870820302656
}}
- {{cite conference
| last1 = Moss | first1 = Sean
| last2 = Perrone | first2 = Paolo
| title = Probability monads with submonads of deterministic states
| book-title = LICS '22: Proceedings of the 37th Annual ACM/IEEE Symposium on Logic in Computer Science
| date = 2022
| doi = 10.1145/3531130.3533355
| arxiv = 2204.07003
| url = https://dl.acm.org/doi/10.1145/3531130.3533355
}}
{{refend}}