## A Category of Probability Spaces

J. Math. Sci. Univ. Tokyo
Vol. 26 (2019), No. 2, Page 201-221.

We introduce a category $\textbf{Prob}$ of probability spaces whose objects are all probability spaces and whose arrows correspond to measurable functions satisfying an absolutely continuous requirement. We can consider a $\textbf{Prob}$-arrow as an evolving direction of information. We introduce a contravariant functor $\mathcal{E}$ from $\textbf{Prob}$ to $\textbf{Set}$, the category of sets. The functor $\mathcal{E}$ provides conditional expectations along arrows in $\textbf{Prob}$, which are generalizations of the classical conditional expectations. For a $\textbf{Prob}$-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 $\textbf{Prob}$.