Contravariant functors of probability spaces

A Category of Probability Spaces

We introduce a category of probability spaces whose objects are all probability spaces and arrows are corresponding to measurable functions satisfying an absolutely continuous requirement.We introduce a contravariant functor from the category of probability spaces to the category of sets, the category of sets.The functor provides conditional expectations along arrows in probability spaces, which are generalizations of the classical conditional expectations.We also show that the completion of probability spaces is naturally formulated as an endofunctor of probability spaces.