A Category of Probability Spaces

Takanori Adachi, Yoshihiro Ryu

We introduce a category Prob of probability spaces whose objects are all
probability spaces and arrows are corresponding to measurable functions
satisfying an absolutely continuous requirement. We can consider a Prob-arrow
as an evolving direction of information with a way of its interpretation. We
introduce a contravariant functor E from Prob to Set, the category of sets. The
functor E provides conditional expectations along arrows in Prob, which are
generalizations of the classical conditional expectations. For a Prob arrow f,
we introduce two concepts f-measurability and f-independence and investigate
their interaction with conditional expectations along f. We also show that the
completion of probability spaces is naturally formulated as an endofunctor of
Prob.